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PHYSICS
1. Physics and Measurement ........................................................................................................................................................ 1
2. Kinematics .................................................................................................................................................................................... 3
3. Laws of Motion ............................................................................................................................................................................ 9
4. Work, Energy and Power ........................................................................................................................................................ 14
5. Rotational Motion ...................................................................................................................................................................... 20
6. Gravitation .................................................................................................................................................................................. 26
7. Properties of Solids and Liquids ............................................................................................................................................ 30
8. Thermodynamics ....................................................................................................................................................................... 37
9. Kinetic Theory of Gases .......................................................................................................................................................... 42
10. Oscillations and Waves ........................................................................................................................................................... 44
11. Electrostatics .............................................................................................................................................................................. 54
12. Current Electricity...................................................................................................................................................................... 64
13. Magnetic Effects of Current and Magnetism ........................................................................................................................ 72
14. Electromagnetic Induction and Alternating Currents ........................................................................................................... 79
15. Electromagnetic W aves ........................................................................................................................................................... 86
16. Optics .......................................................................................................................................................................................... 88
17. Dual Nature of Matter and Radiation .................................................................................................................................... 94
18. Atoms and Nuclei ..................................................................................................................................................................... 99
19. Electronic Devices .................................................................................................................................................................. 106
20. Experimental Skills................................................................................................................................................................... 111
CHEMISTRY
1. Some Basic Concepts in Chemistry ........................................................................................................................................ 1
2. States of Matter ........................................................................................................................................................................... 3
3. Atomic Structure .......................................................................................................................................................................... 7
4. Chemical Bonding and Molecular Structure ..........................................................................................................................11
5. Chemical Thermodynamics ..................................................................................................................................................... 16
6. Solutions ..................................................................................................................................................................................... 22
7. Equilibrium .................................................................................................................................................................................. 28
8. Redox Reactions and Electrochemistry ................................................................................................................................ 35
9. Chemical Kinetics ..................................................................................................................................................................... 41
10. Surface Chemistry .................................................................................................................................................................... 45
11. Nuclear Chemistry .................................................................................................................................................................... 47
12. Classification of Elements and Periodicity in Properties ................................................................................................... 49
iv
13. General Principles and Processes of Isolation of Metals .................................................................................................. 53
14. Hydrogen .................................................................................................................................................................................... 55
15. sBlock Elements ...................................................................................................................................................................... 57
16. pBlock Elements ...................................................................................................................................................................... 59
17. d and fBlock Elements .......................................................................................................................................................... 65
18. Coordination Compounds ........................................................................................................................................................ 70
19. Environmental Chemistry ......................................................................................................................................................... 74
20. Purification and Characterisation of Organic Compounds ................................................................................................. 76
21. Some Basic Principles of Organic Chemistry ...................................................................................................................... 78
22. Hydrocarbons ............................................................................................................................................................................. 82
23. Organic Compounds Containing Halogens ........................................................................................................................... 87
24. Alcohols, Phenols and Ethers ................................................................................................................................................ 91
25. Aldehydes, Ketones and Carboxylic Acids ........................................................................................................................... 95
26. Organic Compounds Containing Nitrogen ............................................................................................................................ 99
27. Polymers ................................................................................................................................................................................... 101
28. Biomolecules ............................................................................................................................................................................ 103
29. Chemistry in Everyday Life ................................................................................................................................................... 107
30. Principles Related to Practical Chemistry .......................................................................................................................... 109
MATHEMATICS
1. Sets, Relations and Functions .................................................................................................................................................. 1
2. Complex Numbers ...................................................................................................................................................................... 7
3. Matrices and Determinants ..................................................................................................................................................... 12
4. Quadratic Equations ................................................................................................................................................................. 19
5. Permutations and Combinations ............................................................................................................................................. 24
6. Mathematical Induction and its Application .......................................................................................................................... 28
7. Binomial Theorem ..................................................................................................................................................................... 30
8. Sequences and Series ............................................................................................................................................................. 34
9. Differential Calculus .................................................................................................................................................................. 40
10. Integral Calculus ....................................................................................................................................................................... 52
11. Differential Equations ............................................................................................................................................................... 63
12. Two Dimensional Geometry .................................................................................................................................................... 67
13. Three Dimensional Geometry ................................................................................................................................................. 82
14. Vector Algebra ........................................................................................................................................................................... 90
15. Statistics ..................................................................................................................................................................................... 96
16. Probability ................................................................................................................................................................................. 100
17. Trigonometry ............................................................................................................................................................................ 105
18. Mathematical Logic .................................................................................................................................................................. 111
v
S Y L L A B U S«
PHYSICS
SECTION A
Unit 1: Physics and Measurement
Physics, technology and society, S. I. units, fundamental and derived units, least count, accuracy and precision of measuring
instruments, errors in measurement.
Dimensions of physical quantities, dimensional analysis and its applications.
Unit 2: Kinematics
Frame of reference, motion in a straight line: positiontime graph, speed and velocity, uniform and nonuniform motion,
average speed and instantaneous velocity.
Uniformly accelerated motion, velocitytime, positiontime graphs, relations for uniformly accelerated motion.
Scalars and vectors, vector addition and subtraction, zero vector, scalar and vector products, unit vector, resolution of
a vector, relative velocity, motion in a plane, projectile motion, uniform circular motion.
Unit 3: Laws of Motion
Force and inertia, Newton’s f irst law of motion, m omentum , Newton’s second law of m otion, im pulse, Newton’s third law
of motion, law of conservation of linear momentum and its applications, equilibrium of concurrent forces.
Static and kinetic friction, laws of friction, rolling friction.
Dynamics of uniform circular motion: centripetal force and its applications.
Unit 4: Work, Energy and Power
Work done by a constant force and a variable force, kinetic and potential energies, workenergy theorem, power.
Potential energy of a spring, conservation of mechanical energy, conservative and nonconservative forces, elastic and
inelastic collisions in one and two dimensions.
Unit 5: Rotational Motion
Centre of m ass of a twoparticle system , centre of m ass of a rigid body, basic concepts of rotational m otion, mom ent of
a force, torque, angular momentum, conservation of angular momentum and its applications, moment of inertia, radius
of gyration, values of moments of inertia for simple geometrical objects, parallel and perpendicular axes theorems and
their applications.
Rigid body rotation, equations of rotational motion.
Unit 6: Gravitation
The universal law of gravitation.
Acceleration due to gravity and its variation with altitude and depth.
Kepler’s laws of planetary motion.
Gravitational potential energy, gravitational potential.
Escape velocity, orbital velocity of a satellite, geostationary satellites.
Unit 7: Properties of Solids and Liquids
Elastic behaviour, stressstrain relationship, Hooke’s law, Young’s modulus, bulk modulus, modulus of rigidity.
Pressure due to a fluid column, Pascal’s law and its applications.
Viscosity, Stokes’s law, terminal velocity, streamline and turbulent flow, Reynolds number, Bernoulli’s principle and its
applications. « For latest information refer prospectus 2014
vi
Surface energy and surface tension, angle of contact, application of surface tension drops, bubbles and capillary rise.
Heat, temperature, thermal expansion, specific heat capacity, calorimetry, change of state, latent heat.
Heat transferconduction, convection and radiation, Newton’s law of cooling.
Unit 8: Thermodynamics
Thermal equilibrium, zeroth law of thermodynamics, concept of temperature, heat, work and internal energy, first law of
thermodynamics.
Second law of thermodynamics, reversible and irreversible processes, Carnot engine and its efficiency.
Unit 9: Kinetic Theory of Gases
Equation of state of a perfect gas, work done on compressing a gas.
Kinetic theory of gases assumptions, concept of pressure, kinetic energy and temperature, rms speed of gas molecules,
degrees of freedom, law of equipartition of energy, applications to specific heat capacities of gases, mean free path,
Avogadro’s number.
Unit 10: Oscillations and Waves
Periodic motion period, frequency, displacement as a function of time, periodic functions, simple harmonic motion (S.H.M.)
and its equation, phase, oscillations of a spring restoring force and force constant, energy in S.H.M. kinetic and potential
energies, simple pendulum derivation of expression for its time period, free, forced and damped oscillations, resonance.
Wave motion, longitudinal and transverse waves, speed of a wave, displacement relation for a progressive wave, principle
of superposition of waves, reflection of waves, standing waves in strings and organ pipes, fundamental mode and harmonics,
beats, Doppler effect in sound.
Unit 11: Electrostatics
Electric charges, conservation of charge, Coulomb’s lawforces between two point charges, forces between multiple charges,
superposition principle and continuous charge distribution.
Electric field, electric field due to a point charge, electric field lines, electric dipole, electric field due to a dipole, torque
on a dipole in a uniform electric field.
Electric f lux, Gauss’s law and its applications to f ind f ield due to inf initely long, unif orm ly charged straight wire, unif orm ly
charged infinite plane sheet and uniformly charged thin spherical shell.
Electric potential and its calculation for a point charge, electric dipole and system of charges, equipotential surfaces,
electrical potential energy of a system of two point charges in an electrostatic field.
Conductors and insulators, dielectrics and electric polarization, capacitor, combination of capacitors in series and in parallel,
capacitance of a parallel plate capacitor with and without dielectric medium between the plates, energy stored in a capacitor.
Unit 12: Current Electricity
Electric current, drift velocity, Ohm’s law, electrical resistance, resistances of different materials,
VI characteristics of ohmic and nonohmic conductors, electrical energy and power, electrical resistivity, colour code for
resistors, series and parallel combinations of resistors, temperature dependence of resistance.
Electric cell and its internal resistance, potential difference and emf of a cell, combination of cells in series and in parallel.
Kirchhoff’s laws and their applications, Wheatstone bridge, metre bridge.
Potentiometer principle and its applications.
Unit 13: Magnetic Effects of Current and Magnetism
Biot Savart law and its application to current carrying circular loop.
Ampere’s law and its applications to infinitely long current carrying straight wire and solenoid.
Force on a moving charge in uniform magnetic and electric fields, cyclotron.
Force on a currentcarrying conductor in a uniform magnetic field, force between two parallel currentcarrying
conductorsdefinition of ampere, torque experienced by a current loop in uniform magnetic field, moving coil galvanometer,
its current sensitivity and conversion to ammeter and voltmeter.
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Current loop as a magnetic dipole and its magnetic dipole moment, bar magnet as an equivalent solenoid, magnetic field
lines, earth’s magnetic field and magnetic elements, para, dia and ferro magnetic substances .
Magnetic susceptibility and permeability, hysteresis, electromagnets and permanent magnets.
Unit 14: Electromagnetic Induction and Alternating Currents
Electromagnetic induction, Faraday’s law, induced emf and current, Lenz’s law, Eddy currents, self and mutual inductance.
Alternating currents, peak and rms value of alternating current/ voltage, reactance and impedance, LCR series circuit,
resonance, quality factor, power in AC circuits, wattless current.
AC generator and transformer.
Unit 15: Electromagnetic Waves
Electromagnetic waves and their characteristics, transverse nature of electrom agnetic waves, electromagnetic spectrum
(radio waves, microwaves, infrared, visible, ultraviolet, Xrays, gamma rays). Applications of electromagnetic waves.
Unit 16: Optics
Reflection and refraction of light at plane and spherical surfaces, mirror formula, total internal reflection and its
applications, deviation and dispersion of light by a prism, lens formula, magnification, power of a lens, combination of
thin lenses in contact, microscope and astronomical telescope (reflecting and refracting) and their magnifying powers.
Wave optics wavefront and Huygens principle, laws of reflection and refraction using Huygens principle, interference,
Young’s double slit experiment and expression for fringe width, coherent sources and sustained interference of light,
diffraction due to a single slit, width of central maximum, resolving power of microscopes and astronomical telescopes,
polarisation, plane polarized light, Brewster’s law, uses of plane polarized light and polaroids.
Unit 17: Dual Nature of Matter and Radiation
Dual nature of radiation, photoelectric effect, Hertz and Lenard’s observations, Einstein’s photoelectric equation, particle
nature of light.
Matter waveswave nature of particle, de Broglie relation, DavissonGermer experiment.
Unit 18: Atoms and Nuclei
Alphaparticle scattering experiment, Rutherford’s model of atom, Bohr model, energy levels, hydrogen spectrum.
Composition and size of nucleus, atomic masses, isotopes, isobars, isotones, radioactivityalpha, beta and gamma
particles/rays and their properties, radioactive decay law, massenergy relation, mass defect, binding energy per nucleon
and its variation with mass number, nuclear fission and fusion.
Unit 19: Electronic Devices
Semiconductors, semiconductor diode IV characteristics in forward and reverse bias, diode as a rectifier, IV characteristics
of LED, photodiode, solar cell. Zener diode, Zener diode as a voltage regulator, junction transistor, transistor action,
characteristics of a transistor, transistor as an amplifier (common emitter configuration) and oscillator, logic gates (OR,
AND, NOT, NAND and NOR), transistor as a switch.
Unit 20: Communication Systems
Propagation of electromagnetic waves in the atmosphere, sky and space wave propagation, need for modulation,
amplitude and frequency modulation, bandwidth of signals, bandwidth of transmission medium, basic elements of a
communication system (Block Diagram only).
SECTION B
Unit 21 : Experimental Skills
Familiarity with the basic approach and observations of the experiments and activities:
l Vernier callipersits use to measure internal and external diameter and depth of a vessel.
l Screw gaugeits use to determine thickness/diameter of thin sheet/wire.
l Simple Pendulumdissipation of energy by plotting a graph between square of amplitude and time.
viii
l Metre Scale mass of a given object by principle of moments.
l Young’s modulus of elasticity of the material of a metallic wire.
l Surface tension of water by capillary rise and effect of detergents.
l Coefficient of Viscosity of a given viscous liquid by measuring terminal velocity of a given spherical body.
l Plotting a cooling curve for the relationship between the temperature of a hot body and time.
l Speed of sound in air at room temperature using a resonance tube.
l Specific heat capacity of a given (i) solid and (ii) liquid by method of mixtures.
l Resistivity of the material of a given wire using metre bridge.
l Resistance of a given wire using Ohm’s law.
l Potentiometer :
(i) Comparison of emf of two primary cells.
(ii) Determination of internal resistance of a cell.
l Resistance and figure of merit of a galvanometer by half deflection method.
l Focal length of:
(i) Convex mirror (ii) Concave mirror and (iii) Convex lens using parallax method.
l Plot of angle of deviation vs angle of incidence for a triangular prism.
l Refractive index of a glass slab using a travelling microscope.
l Characteristic curves of a pn junction diode in forward and reverse bias.
l Characteristic curves of a Zener diode and finding reverse break down voltage.
l Characteristic curves of a transistor and finding current gain and voltage gain.
l Identification of Diode, LED, Transistor, IC, Resistor, Capacitor from mixed collection of such items.
l Using multimeter to:
(i) Identify base of a transistor
(ii) Distinguish between npn and pnp type transistor
(iii) See the unidirectional flow of current in case of a diode and an LED.
(iv) Check the correctness or otherwise of a given electronic component (diode, transistor or IC).
CHEMISTRY
SECTION A (Physical Chemistry)
UNIT 1: SOME BASIC CONCEPTS IN CHEMISTRY
Matter and its nature, Dalton’s atomic theory, concept of atom, molecule, element and compound, physical quantities
and their measurements in chemistry, precision and accuracy, significant figures, S.I. units, dimensional analysis, Laws
of chemical combination, atomic and molecular masses, mole concept, molar mass, percentage composition, empirical
and molecular formulae, chemical equations and stoichiometry.
UNIT 2: STATES OF MATTER
Classification of matter into solid, liquid and gaseous states.
Gaseous State Measurable properties of gases, Gas laws Boyle’s law, Charle’s law, Graham’s law of diffusion, Avogadro’s
law, Dalton’s law of partial pressure, concept of absolute scale of temperature, Ideal gas equation, kinetic theory of gases
(only postulates), concept of average, root mean square and most probable velocities, real gases, deviation from Ideal
behaviour, compressibility factor, van der Waals equation.
Liquid State Properties of liquids vapour pressure, viscosity and surface tension and effect of temperature on them
(qualitative treatment only).
Solid State Classification of solids molecular, ionic, covalent and metallic solids, amorphous and crystalline solids
(elementary idea), Bragg’s Law and its applications, unit cell and lattices, packing in solids (fcc, bcc and hcp lattices),
voids, calculations involving unit cell parameters, imperfection in solids, electrical, magnetic and dielectric properties.
ix
UNIT 3: ATOMIC STRUCTURE
Thomson and Rutherford atomic models and their limitations, nature of electromagnetic radiation, photoelectric effect,
spectrum of hydrogen atom, Bohr model of hydrogen atom its postulates, derivation of the relations for energy of the
electron and radii of the different orbits, limitations of Bohr’s model, dual nature of matter, deBroglie’s relationship, Heisenberg
uncertainty principle, elementary ideas of quantum mechanics, quantum mechanical model of atom, its important features,
concept of atomic orbitals as one electron wave functions, variation of Y and Y2 with r for 1s and 2s orbitals, various
quantum numbers (principal, angular momentum and magnetic quantum numbers) and their significance, shapes of s, p
and d orbitals, electron spin and spin quantum number, rules for filling electrons in orbitals – Aufbau principle, Pauli’s
exclusion principle and Hund’s rule, electronic configuration of elements, extra stability of half filled and completely filled
orbitals.
UNIT 4: CHEMICAL BONDING AND MOLECULAR STRUCTURE
Kossel Lewis approach to chemical bond formation, concept of ionic and covalent bonds.
Ionic Bonding Formation of ionic bonds, factors affecting the formation of ionic bonds, calculation of lattice enthalpy.
Covalent Bonding concept of electronegativity, Fajan’s rule, dipole moment, Valence Shell Electron Pair Repulsion
(VSEPR) theory and shapes of simple molecules.
Quantum mechanical approach to covalent bonding valence bond theory its important features, concept of hybridization
involving s, p and d orbitals, Resonance.
Molecular Orbital Theory its im portant features, LCAOs, types of m olecular orbitals (bonding, antibonding), sigm a and
pibonds, molecular orbital electronic configurations of homonuclear diatomic molecules, concept of bond order, bond
length and bond energy.
Elementary idea of metallic bonding, hydrogen bonding and its applications.
UNIT 5: CHEMICAL THERMODYNAMICS
Fundamentals of thermodynamics: system and surroundings, extensive and intensive properties, state functions, types
of processes.
First law of thermodynamics Concept of work, heat internal energy and enthalpy, heat capacity, molar heat capacity,
Hess’s law of constant heat summation, enthalpies of bond dissociation, combustion, formation, atomization, sublimation,
phase transition, hydration, ionization and solution.
Second law of thermodynamics Spontaneity of processes, DS of the universe and DG of the system as criteria for
spontaneity, DG° (standard Gibbs energy change) and equilibrium constant.
UNIT 6: SOLUTIONS
Different methods for expressing concentration of solution molality, molarity, mole fraction, percentage (by volume and
m ass both), vapour pressure of solutions and Raoult’s law Ideal and nonideal solutions, vapour pressure composition
plots for ideal and nonideal solutions, colligative properties of dilute solutions relative lowering of vapour pressure,
depression of freezing point, elevation of boiling point and osmotic pressure, determination of molecular mass using
colligative properties, abnormal value of molar mass, van’t Hoff factor and its significance.
UNIT 7: EQUILIBRIUM
Meaning of equilibrium, concept of dynamic equilibrium.
Equilibria involving physical processes Solid liquid, liquid gas and solid gas equilibria, Henry’s law, general
characteristics of equilibrium involving physical processes.
Equilibria involving chemical processes Law of chemical equilibrium, equilibrium constants
(Kp and K c) and their significance, significance of DG and DG° in chemical equilibria, factors affecting equilibrium concentration,
pressure, temperature, effect of catalyst, Le Chatelier’s principle.
Ionic equilibrium Weak and strong electrolytes, ionization of electrolytes, various concepts of acids and bases (Arrhenius,
Bronsted Lowry and Lewis) and their ionization, acid base equilibria (including multistage ionization) and ionization
x
constants, ionization of water, pH scale, common ion effect, hydrolysis of salts and pH of their solutions, solubility of
sparingly soluble salts and solubility products, buffer solutions. .
UNIT 8 : REDOX REACTIONS AND ELECTROCHEMISTRY
Electronic concepts of oxidation and reduction, redox reactions, oxidation number, rules for assigning oxidation number,
balancing of redox reactions.
Electrolytic and metallic conduction, conductance in electrolytic solutions, molar conductivities and their variation with
concentration: Kohlrausch’s law and its applications.
Electrochemical cells electrolytic and galvanic cells, different types of electrodes, electrode potentials including standard
electrode potential, half cell and cell reactions, emf of a galvanic cell and its measurement, Nernst equation and its
applications, relationship between cell potential and Gibbs’ energy change, dry cell and lead accumulator, fuel cells.
UNIT 9 : CHEMICAL KINETICS
Rate of a chemical reaction, factors affecting the rate of reactions concentration, temperature, pressure and catalyst,
elementary and complex reactions, order and molecularity of reactions, rate law, rate constant and its units, differential
and integral f orms of zero and first order reactions, their characteristics and half lives, eff ect of tem perature on rate of
reactions Arrhenius theory, activation energy and its calculation, collision theory of bimolecular gaseous reactions (no
derivation).
UNIT 10 : SURFACE CHEMISTRY
Adsorption Physisorption and chemisorption and their characteristics, factors affecting adsorption of gases on solids,
Freundlich and Langmuir adsorption isotherms, adsorption from solutions.
Catalysis Homogeneous and heterogeneous, activity and selectivity of solid catalysts, enzyme catalysis and its mechanism.
Colloidal state distinction among true solutions, colloids and suspensions, classification of colloids lyophilic, lyophobic,
multi molecular, macromolecular and associated colloids (micelles), preparation and properties of colloids Tyndall effect,
Brownian movement, electrophoresis, dialysis, coagulation and flocculation, emulsions and their characteristics.
Section B (Inorganic Chemistry)
UNIT 11: CLASSIFICATION OF ELEMENTS AND PERIODICITY IN PROPERTIES
Modern periodic law and present form of the periodic table, s, p, d and f block elements, periodic trends in properties of
elementsatomic and ionic radii, ionization enthalpy, electron gain enthalpy, valence, oxidation states and chemical reactivity.
UNIT 12: GENERAL PRINCIPLES AND PROCESSES OF ISOLATION OF METALS
Modes of occurrence of elements in nature, minerals, ores, steps involved in the extraction of metals concentration,
reduction (chemical and electrolytic methods) and refining with special reference to the extraction of Al, Cu, Zn and Fe,
thermodynamic and electrochemical principles involved in the extraction of metals.
UNIT 13: HYDROGEN
Position of hydrogen in periodic table, isotopes, preparation, properties and uses of hydrogen, physical and chemical
properties of water and heavy water, structure, preparation, reactions and uses of hydrogen peroxide, classification of
hydrides ionic, covalent and interstitial, hydrogen as a fuel.
UNIT 14: s BLOCK ELEMENTS (ALKALI AND ALKALINE EARTH METALS)
Group 1 and 2 Elements
General introduction, electronic configuration and general trends in physical and chemical properties of elements, anomalous
properties of the first element of each group, diagonal relationships.
Preparation and properties of some important compounds sodium carbonate, sodium hydroxide and sodium hydrogen
carbonate, Industrial uses of lime, limestone, Plaster of Paris and cement, Biological significance of Na, K, Mg and Ca.
UNIT 15: p BLOCK ELEMENTS
Group 13 to Group 18 Elements
General Introduction Electronic configuration and general trends in physical and chemical properties of elements across
the periods and down the groups, unique behaviour of the first element in each group.
xi
Group 13
Preparation, properties and uses of boron and aluminium, structure, properties and uses of borax, boric acid, diborane,
boron trifluoride, aluminium chloride and alums.
Group 14
Tendency f or catenation, structure, properties and uses of allotropes and oxides of carbon, silicon tetrachloride, silicates,
zeolites and silicones.
Group 15
Properties and uses of nitrogen and phosphorus, allotropic forms of phosphorus, preparation, properties, structure and
uses of ammonia, nitric acid, phosphine and phosphorus halides, (PCl 3 , PCl 5 ), structures of oxides and oxoacids of nitrogen
and phosphorus.
Group 16
Preparation, properties, structures and uses of ozone, allotropic f orm s of sulphur, preparation, properties, structures and
uses of sulphuric acid (including its industrial preparation), Structures of oxoacids of sulphur.
Group 17
Preparation, properties and uses of hydrochloric acid, trends in the acidic nature of hydrogen halides, structures of interhalogen
compounds and oxides and oxoacids of halogens.
Group 18
Occurrence and uses of noble gases, structures of fluorides and oxides of xenon.
UNIT 16: d and f BLOCK ELEMENTS
Transition Elements
General introduction, electronic conf iguration, occurrence and characteristics, general trends in properties of the first row
transition elements physical properties, ionization enthalpy, oxidation states, atomic radii, colour, catalytic behaviour,
magnetic properties, complex formation, interstitial compounds, alloy formation, preparation, properties and uses of K 2 Cr 2 O 7
and KMnO 4 .
Inner Transition Elements
Lanthanoids Electronic configuration, oxidation states and lanthanoid contraction.
Actinoids Electronic configuration and oxidation states.
UNIT 17: COORDINATION COMPOUNDS
Introduction to coordination compounds, Werner’s theory, ligands, coordination number, denticity, chelation, IUPAC nomenclature
of mononuclear coordination compounds, isomerism, bonding valence bond approach and basic ideas of crystal field
theory, colour and magnetic properties, importance of coordination compounds (in qualitative analysis, extraction of metals
and in biological systems).
UNIT 18: ENVIRONMENTAL CHEMISTRY
Environmental Pollution Atmospheric, water and soil. Atmospheric pollution tropospheric and stratospheric.
Tropospheric pollutants Gaseous pollutants: oxides of carbon, nitrogen and sulphur, hydrocarbons, their sources,
harmful effects and prevention, green house effect and global warming, acid rain.
Particulate pollutants Smoke, dust, smog, fumes, mist, their sources, harmful effects and prevention.
Stratospheric pollution Formation and breakdown of ozone, depletion of ozone layer its mechanism and effects.
Water Pollution Major pollutants such as, pathogens, organic wastes and chemical pollutants, their harmful effects and
prevention.
Soil Pollution Major pollutants like pesticides (insecticides, herbicides and fungicides), their harmful effects and prevention.
Strategies to control environmental pollution.
xii
SECTION C (Organic Chemistry)
UNIT 19: PURIFICATION AND CHARACTERISATION OF ORGANIC COMPOUNDS
Purification Crystallization, sublimation, distillation, differential extraction and chromatography principles and their
applications.
Qualitative analysis Detection of nitrogen, sulphur, phosphorus and halogens.
Quantitative analysis (Basic principles only) Estimation of carbon, hydrogen, nitrogen, halogens, sulphur, phosphorus.
Calculations of empirical formulae and molecular formulae, numerical problems in organic quantitative analysis.
UNIT 20: SOME BASIC PRINCIPLES OF ORGANIC CHEMISTRY
Tetravalency of carbon, shapes of simple molecules hybridization (s and p), classification of organic compounds based
on functional groups: and those containing halogens, oxygen, nitrogen and sulphur, homologous series, Isomerism
structural and stereoisomerism.
Nomenclature (trivial and IUPAC)
Covalent bond fission Homolytic and heterolytic: free radicals, carbocations and carbanions, stability of carbocations
and free radicals, electrophiles and nucleophiles.
Electronic displacement in a covalent bond Inductive effect, electromeric effect, resonance and hyperconjugation.
Common types of organic reactions Substitution, addition, elimination and rearrangement.
UNIT 21: HYDROCARBONS
Classification, isomerism, IUPAC nomenclature, general methods of preparation, properties and reactions.
Alkanes Conformations: Sawhorse and Newman projections (of ethane), mechanism of halogenation of alkanes.
Alkenes Geometrical isomerism, mechanism of electrophilic addition: addition of hydrogen, halogens, water, hydrogen
halides (Markownikoff’s and peroxide effect), ozonolysis, oxidation, and polymerization.
Alkynes Acidic character, addition of hydrogen, halogens, water and hydrogen halides, polymerization.
Aromatic hydrocarbons Nomenclature, benzene structure and aromaticity, mechanism of electrophilic substitution:
halogenation, nitration, Friedel – Craft’s alkylation and acylation, directive influence of functional group in monosubstituted
benzene.
UNIT 22: ORGANIC COMPOUNDS CONTAINING HALOGENS
General methods of preparation, properties and reactions, nature of CX bond, mechanisms of substitution reactions.
Uses/environmental effects of chloroform, iodoform, freons and DDT.
UNIT 23: ORGANIC COMPOUNDS CONTAINING OXYGEN
General methods of preparation, properties, reactions and uses.
Alcohols Identification of primary, secondary and tertiary alcohols, mechanism of dehydration.
Phenols Acidic nature, electrophilic substitution reactions: halogenation, nitration and sulphonation, Reimer Tiemann
reaction.
Ethers Structure.
Aldehydes and Ketones Nature of carbonyl group, nucleophilic addition to >C O group, relative reactivities of aldehydes
and ketones, important reactions such as nucleophilic addition reactions (addition of HCN, NH 3 and its derivatives),
Grignard reagent, oxidation, reduction (Wolff Kishner and Clemmensen), acidity of ahydrogen, aldol condensation, Cannizzaro
reaction, haloform reaction, chemical tests to distinguish between aldehydes and ketones.
Carboxylic acids Acidic strength and factors affecting it.
UNIT 24: ORGANIC COMPOUNDS CONTAINING NITROGEN
General methods of preparation, properties, reactions and uses.
Amines Nomenclature, classification, structure basic character and identification of primary, secondary and tertiary
amines and their basic character.
Diazonium Salts Importance in synthetic organic chemistry.
xiii
UNIT 25: POLYMERS
General introduction and classification of polymers, general methods of polymerization addition and condensation,
copolymerization, natural and synthetic rubber and vulcanization, some important polymers with emphasis on their monomers
and uses polythene, nylon, polyester and bakelite.
UNIT 26: BIOMOLECULES
General introduction and importance of biomolecules.
Carbohydrates Classification: aldoses and ketoses, monosaccharides (glucose and fructose), constituent monosaccharides
of oligosaccharides (sucrose, lactose, maltose).
Proteins Elementary Idea of a amino acids, peptide bond, polypeptides, proteins primary, secondary, tertiary and
quaternary structure (qualitative idea only), denaturation of proteins, enzymes.
Vitamins Classification and functions.
Nucleic acids Chemical constitution of DNA and RNA, biological functions of nucleic acids.
UNIT 27: CHEMISTRY IN EVERYDAY LIFE
Chemicals in medicines Analgesics, tranquilizers, antiseptics, disinfectants, antimicrobials, antifertility drugs, antibiotics,
antacids, antihistamins their meaning and common examples.
Chemicals in food Preservatives, artificial sweetening agents common examples.
Cleansing agents Soaps and detergents, cleansing action.
UNIT 28: PRINCIPLES RELATED TO PRACTICAL CHEMISTRY
Detection of extra elements (N, S, halogens) in organic compounds, detection of the following functional groups: hydroxyl
(alcoholic and phenolic), carbonyl (aldehyde and ketone), carboxyl and amino groups in organic compounds.
Chemistry involved in the preparation of the following:
Inorganic compounds Mohr’s salt, potash alum.
Organic compounds Acetanilide, pnitroacetanilide, aniline yellow, iodoform.
Chem istry involved in the titrim etric exercises Acids, bases and the use of indicators, oxalic acid vs KMnO 4 , Mohr’s salt
vs KMnO 4 .
Chemical principles involved in the qualitative salt analysis:
Cations Pb 2+ , Cu2 + , AI3 + , Fe 3+ , Zn2 + , Ni 2+ , Ca 2+ , Ba 2+ , Mg 2+ , NH 4 + .
Anions – CO 3 2– , S 2– , SO 4 2– , NO 2 – , NO 3 – , CI– , Br – , I– (insoluble salts excluded).
Chemical principles involved in the following experiments:
1. Enthalpy of solution of CuSO 4
2. Enthalpy of neutralization of strong acid and strong base.
3. Preparation of lyophilic and lyophobic sols.
4. Kinetic study of reaction of iodide ion with hydrogen peroxide at room temperature.
MATHEMATICS
UNIT 1 : SETS, RELATIONS AND FUNCTIONS
Sets and their representation, union, intersection and complement of sets and their algebraic properties, power set, relations,
types of relations, equivalence relations, functions, oneone, into and onto functions, composition of functions.
UNIT 2 : COMPLEX NUMBERS AND QUADRATIC EQUATIONS
Complex numbers as ordered pairs of reals, representation of complex numbers in the form a + ib and their representation
in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number,
square root of a complex number, triangle inequality, quadratic equations in real and complex number system and their
solutions, relation between roots and coefficients, nature of roots, formation of quadratic equations with given roots.
xiv
UNIT 3 : MATRICES AND DETERMINANTS
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants,
evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix
using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in
two or three variables using determinants and matrices.
UNIT 4 : PERMUTATIONS AND COMBINATIONS
Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and
C (n,r), simple applications.
UNIT 5 : MATHEMATICAL INDUCTION
Principle of Mathematical Induction and its simple applications.
UNIT 6 : BINOMIAL THEOREM AND ITS SIMPLE APPLICATIONS
Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and
simple applications.
UNIT 7 : SEQUENCES AND SERIES
Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers. Relation
between A.M. and G.M. Sum upto n terms of special series: å n, å ån2, n 3 ,. Arithmetic Geometric progression.
UNIT 8 : LIMITS, CONTINUITY AND DIFFERENTIABILITY
Real valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic and exponential functions,
inverse f unctions. Graphs of sim ple f unctions. Limits, continuity and diff erentiability. Diff erentiation of the sum, diff erence,
product and quotient of two functions. Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential,
composite and implicit functions, derivatives of order upto two. Rolle’s and Lagrange’s Mean value theorems. Applications
of derivatives: Rate of change of quantities, monotonic increasing and decreasing functions, Maxima and minima of
functions of one variable, tangents and normals.
UNIT 9 : INTEGRAL CALCULUS
Integral as an anti derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions.
Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities.
Evaluation of simple integrals of the type
dx , dx , dx , dx , dx , dx , a(xp2x ++bqx)d+xc , (px + q)d x
2 ± a2 x2 ± a2 a2 - x a2 - x2 + bx c ax2 + bx + c ax2 + dx + c
ò ò ò ò ò ò ò òx
2 ax2 +
ò òa2 ± x2 dx and x2 - a2 dx
Integral as limit of a sum. Fundamental theorem of calculus. Properties of definite integrals. Evaluation of definite integrals,
determining areas of the regions bounded by simple curves in standard form.
UNIT 10: DIFFERENTIAL EQUATIONS
Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations
by the method of separation of variables, solution of homogeneous and linear differential equations of the type:
dy + p( x )y = q( x )
dx
UNIT 11: COORDINATE GEOMETRY
Cartesian system of rectangular coordinates in a plane, distance formula, section formula, locus and its equation, translation
of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence
of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines,
coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point
of intersection of two lines.
xv
Circles, conic sections Standard form of equation of a circle, general form of the equation of a circle, its radius and
centre, equation of a circle when the end points of a diam eter are given, points of intersection of a line and a circle with
the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones,
equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent
and point(s) of tangency.
UNIT 12: THREE DIMENSIONAL GEOMETRY
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle
between two intersecting lines. Skew lines, the shortest distance between them and its equation. Equations of a line and
a plane in different forms, intersection of a line and a plane, coplanar lines.
UNIT 13: VECTOR ALGEBRA
Vectors and scalars, addition of vectors, components of a vector in two dim ensions and three dimensional space, scalar
and vector products, scalar and vector triple product.
UNIT 14: STATISTICS AND PROBABILITY
Measures of Dispersion Calculation of mean, m edian, m ode of grouped and ungrouped data. Calculation of standard
deviation, variance and mean deviation for grouped and ungrouped data.
Probability Probability of an event, addition and multiplication theorems of probability, Baye’s theorem, probability distribution
of a random variate, Bernoulli trials and Binomial distribution.
UNIT 15: TRIGONOMETRY
Trigonometrical identities and equations. Trigonom etrical f unctions. Inverse trigonom etrical f unctions and their properties.
Heights and Distances.
UNIT 16: MATHEMATICAL REASONING
Statements, logical operations and, or, implies, implied by, if and only if. Understanding of tautology, contradiction, converse
and contrapositive.
vvv
xvi
PHYSICS
Physics and Measurement 1
CHAPTER PHYSICS AND
MEASUREMENT
1
1. Let [Î 0] denote the dimensional formula of the permittivity 6. Out of the following pairs, which one does not have identical
of vacuum. If M = mass, L = length, T = time and A = electric
dimensions?
current, then
(a) [Î 0] = [M –1 L2 T– 1 A] (b) [Î 0] = [M– 1 L– 3 T2 A] (a) moment of inertia and moment of a force
(c) [Î0 ] = [M –1 L– 3 T4 A 2] (d) [Î0 ] = [M– 1 L2 T– 1 A –2]
(b) work and torque
(2013)
(c) angular momentum and Planck’s constant
(d) impulse and momentum (2005)
2. Resistance of a given wire is obtained by measuring the 7. Which one of the following represents the correct dimensions
current flowing in it and the voltage difference applied across of the coefficient of viscosity?
it. If the percentage errors in the measurement of the current (a) ML– 1 T– 2 (b) MLT– 1
and the voltage difference are 3% each, then error in the (c) ML– 1 T– 1 (d) ML– 2 T– 2 . (2004)
value of resistance of the wire is 8. The physical quantities not having same dimensions are
(a) zero (b) 1% (c) 3% (d) 6% (a) torque and work
(2012) (b) momentum and Planck's constant
3. The respective number of significant figures for the numbers (c) stress and Young's modulus
23.023, 0.0003 and 2.1 × 10– 3 are (d) speed and (m 0e 0) –1/2. (2003)
(a) 4, 4, 2 (b) 5, 1, 2
(c) 5, 1, 5 (d) 5, 5, 2. (2010) 9. Dimensions of 1 , where symbols have their usual
m0e 0
4. The dimension of magnetic field in M, L, T and C (coulomb)
meaning, are
is given as (a) [L –1 T] (b) [L –2 T2 ] (c) [L 2T –2 ] (d) [LT– 1 ].
(a) MT– 2C – 1 (b) MLT– 1 C– 1 (2003)
(c) MT2 C – 2 (d) MT– 1 C –1. (2008) 10. Identify the pair whose dimensions are equal.
5. Which of the following units denotes the dimensions ML2 / Q 2, (a) torque and work (b) stress and energy
where Q denotes the electric charge? (c) force and stress (d) force and work.
(a) weber (Wb) (b) Wb/m2 (2002)
(c) henry (H) (d) H/m2 . (2006)
Answer Key
1. (c) 2. (d) 3. (b) 4. (d) 5. (c) 6. (a)
7. (c) 8. (b) 9. (c) 10. (a)
2 JEE MAIN CHAPTERWISE EXPLORER
1. (c) : According to Coulomb’s law [henry] = [ML 2 T– 2 A –2 ]
F = 1 q1q2 \ Î0 = 1 q1q2 é H ùúû = [MT -2 A-2 ]
4p Î0 r 2 4p Fr 2 ëê m2
[Î0 ] = [AT][AT] = [M -1L-3T4A2 ] Obviously henry (H) has dimensions ML2 .
[MLT-2 ][L]2 Q 2
2. (d) : R = VI \ 6. (a) : Moment of inertia (I) = mr 2
DR = DV + DI \ [I] = [ML 2 ]
R V I
Moment of force (C) = r F
The percentage error in R is
\ [C] = [r][F] = [L][MLT –2] or [C] = [ML2 T –2 ]
DR ´ 100 = DV ´ 100 + DI ´ 100 = 3% + 3% = 6% Moment of inertia and moment of a force do not have identical
R V I
dimensions.
3. (b) : (i) All the nonzero digits are significant. 7. (c) : Viscous force F = 6phrv
(ii) All the zeros between two nonzero digits are significant, \ h = F or [h] = [r[ F ]
6 p rv ][v]
no matter where the decimal point is, if at all.
(iii)If the number is less than 1, the zero(s) on the right of or [h] = [MLT-2 ] or [h] = [ML –1 T– 1] .
decimal point but to the left of the first nonzero digit are [L][LT-1 ]
not significant.
8. (b) : [Momentum] = [MLT –1 ]
(iv)The power of 10 is irrelevant to the determination of [Planck's constant] = [ML 2 T– 1 ]
significant figures. Momentum and Planck's constant do not have same
dimensions.
According to the above rules,
23.023 has 5 significant figures.
0.0003 has 1 significant figures. 9. (c) : Velocity of light in vacuum = 1
m0e 0
2.1 × 10 –3 has 2 significant figures.
r r
4. (d) : Lorentz force =| F |=| qvr ´ B | or [LT -1 ] = é 1 ù
ê ú
êë m0e0 úû
\ [B ] = [q[F][ v] ] = MLT-2 = MLT -2 = [MT –1 C –1 ]
C ´ LT-1 C LT -1
or [L2 T -2 ] = é 1 ù
5. (c) : [ML2 Q –2 ] = [ML 2 A –2 T –2 ] êë m 0 e0 úû
[Wb] = [ML2 T– 2 A –1 ]
\ Dimensions of 1 = [L2T -2 ] .
m0e 0
é Wm2b úûù = –2 –1 ]
ëê [M T A
10. (a) : Torque and work have the same dimensions.
Kinematics 3
CHAPTER KINEMATICS
2
1. A projectile is given an initial velocity of ($i + 2$j )m s, where 7. For a particle in uniform circular motion, the acceleration ar
$i is along the ground and $j is along the vertical. If at a point P(R, q) on the circle of radius R is (Here q is
measured from the xaxis)
g = 10 m/s2 , the equation of its trajectory is
(a) 4y = 2x – 25x2 (b) y = x – 5x 2 (a) v2 ^ + v 2 ^ (b) - v2 ^ + v 2 ^
R R R R
(c) y = 2x – 5x2 (d) 4y = 2x – 5x2 (2013) i j cosq i sin q j
2. A boy can throw a stone up to a maximum height of 10 m. (c) - v2 ^ + v 2 ^ (d) - v2 ^ - v 2 ^
The maximum horizontal distance that the boy can throw the R R R R
sin q i cos q j cosq i sin q j
same stone up to will be (2010)
(a) 10 m (b) 10 2 m (c) 20 m (d) 20 2 m 8. A point P moves in counterclockwise direction on a circular
(2012) path as shown in the figure. The movement of P is such that
it sweeps out a length s = t 3 + 5, where s is in metres and t
3. An object moving with a speed of 6.25 m s– 1 , is decelerated is in seconds. The radius of the path is 20 m. The acceleration
at a rate given by ddvt = -2 .5 v , where v is the instantaneous of P when t = 2 s is nearly
speed. The time taken by the object, to come to rest, would be
(a) 14 m s– 2 y
(a) 1 s (b) 2 s (c) 4 s (d) 8 s (b) 13 m s– 2 B
(2011) (c) 12 m s –2 20 m P(x, y )
4. A water fountain on the ground sprinkles water all around it. (d) 7.2 m s –2 x (2010)
If the speed of water coming out of the fountain is v, the total O
area around the fountain that gets wet is 9. A particle has an initial velocity 3$i + 4 $j and an acceleration
(a) p vg 2 (b) p gv 42 p v 4 (d) p gv 22 of 0.4$i + 0.3 $j . Its speed after 10 s is
(c) 2 g2
(a) 10 units (b) 7 2 units
(2011) (c) 7 units (d) 8.5 units (2009)
5. A small particle of mass m is projected at an angle q with the 10. A body is at rest at x = 0. At t = 0, it starts moving in the
xaxis with an initial velocity v0 in the xy plane as shown in positive xdirection with a constant acceleration. At the same
the figure. At a time t < v0 sin q , the angular momentum of instant another body passes through x = 0 moving in the
g
y positive xdirection with a constant speed. The position of
the particle is the first body is given by x1 (t) after time t and that of the
(a) 1 mgv0 t 2 cos q ^ v0 second body by x2 (t) after the same time interval. Which of
2
i the following graphs correctly describes (x1 – x 2) as a function
(b) 2 ^ of time t?
(x1 – x2 ) (x1 – x2 )
- mgv0 t cos q j q x
(c) ^ (d) - 12 mg v0 t2 cos q k^ (a) (b)
mgv0 t cos q k
where ^ ^ ^ are unit vectors along x, y and z‑axis O t O t
i, j and k
respectively. (x1 – x2 )
(2010) (x1 – x 2 )
6. A particle is moving with velocity vr = K ( y i^ + x j^) , where K
is a constant. The general equation for its path is (c) (d) t (2008)
(a) y2 = x2 + constant (b) y = x 2 + constant O t O
(c) y 2 = x + constant (d) xy = constant (2010)
4 JEE MAIN CHAPTERWISE EXPLORER
11. The velocity of a particle is v = v 0 + gt + ft 2. If its position is (a) h/9 metre from the ground
x = 0 at t = 0, then its displacement after unit time (t = 1) is
(b) 7h/9 metre from the ground
(a) v0 + g/2 + f (b) v 0 + 2g + 3f (2007) (c) 8h/9 metre from the ground
(c) v 0 + g/2 + f/3 (d) v0 + g + f
(d) 17h/18 metre from the ground. (2004)
12. A particle located at x = 0 at time t = 0, starts moving along r r r r
20. If A ´ B = B ´ A , then the angle between A and B is
the positive xdirection with a velocity v that varies as v = a x.
The displacement of the particle varies with time as (a) p (b) p/3 (c) p/2 (d) p/4.
(a) t3 (b) t2 (c) t (d) t 1/2. (2004)
(2006) 21. A projectile can have the same range R for two angles of
13. A parachutist after bailing out falls 50 m without friction. projection. If T1 and T 2 be the time of flights in the two cases,
When parachute opens, it decelerates at 2 m/s 2. He reaches the then the product of the two time of flights is directly
ground with a speed of 3 m/s. At what height, did he bail out?
proportional to
(a) 293 m (b) 111 m (c) 91 m (d) 182 m (a) 1/R 2 (b) 1/R (c) R (d) R2 .
(2005)
(2004)
14. A car, starting from rest, accelerates at the rate f through a 22. Which of the following statements is false for a particle
moving in a circle with a constant angular speed?
distance s, then continues at constant speed for time t and (a) The velocity vector is tangent to the circle.
(b) The acceleration vector is tangent to the circle
then decelerates at the rate f/2 to come to rest. If the total (c) the acceleration vector points to the centre of the circle
(d) the velocity and acceleration vectors are perpendicular
distance traversed is 15 s, then to each other.
(2004)
(a) s = 1 f t2 (b) s = 1 f t2
2 4
(c) s = ft (d) s = 1 f t 2 (2005)
6
15. The relation between time t and distance x is t = ax2 + bx 23. A ball is thrown from a point with a speed v 0 at an angle of
projection q. From the same point and at the same instant a
where a and b are constants. The acceleration is
(a) –2av3 (b) 2av2 (c) –2av2 (d) 2bv3 person starts running with a constant speed v 0 /2 to catch the
ball. Will the person be able to catch the ball? If yes, what
(2005)
16. A particle is moving eastwards with a velocity of should be the angle of projection?
5 m/s. In 10 s the velocity changes to 5 m/s northwards. The
average acceleration in this time is (a) yes, 60° (b) yes, 30°
(a) zero
(c) no (d) yes, 45°. (2004)
1 24. A car moving with a speed of 50 km/hr, can be stopped by
(b) 2 ms –2 towards northwest
brakes after at least 6 m. If the same car is moving at a speed
1 of 100 km/hr, the minimum stopping distance is
(c) ms– 2 towards northeast
(a) 12 m (b) 18 m (c) 24 m (d) 6 m.
2
(2003)
1 (2005) 25. A boy playing on the roof of a 10 m high building throws a
(d) ms –2 towards north ball with a speed of 10 m/s at an angle of 30° with the
horizontal. How far from the throwing point will the ball be
2 at the height of 10 m from the ground ? [g = 10 m/s 2 ,
17. A projectile can have the same range R for two angles of
projection. If t1 and t2 be the time of flights in the two cases, sin30° = 1/2, cos30° = 3 / 2 ]
then the product of the two time of flights is proportional to
(a) 1/R (b) R (c) R 2 (d) 1/R 2 .
(2005) (a) 5.20 m (b) 4.33 m
(c) 2.60 m (d) 8.66 m.
18. An automobile travelling with a speed of 60 km/h, can brake (2003)
to stop within a distance of 20 m. If the car is going twice 26. The coordinates of a moving particle at any time t are given
by x = at 3 and y = bt 3. The speed of the particle at time t
as fast, i.e. 120 km/h, the stopping distance will be is given by
(a) 20 m (b) 40 m (c) 60 m (d) 80 m.
(2004)
19. A ball is released from the top of a tower of height h metre. (a) 3t a2 + b 2 (b) 3t 2 a2 + b 2
It takes T second to reach the ground. What is the position
of the ball in T/3 second? (c) t2 a2 + b 2 (d) a2 + b 2 . (2003)
Kinematics 5
27. From a building two balls A and B are thrown such that A is (a) 1 : 1 (b) 1 : 4 (c) 1 : 8 (d) 1 : 16.
(2002)
thrown upwards and B downwards (both vertically). If v A and
v B are their respective velocities on reaching the ground, then 29. If a body looses half of its velocity on penetrating 3 cm in
(a) v B > v A a wooden block, then how much will it penetrate more before
(b) v A = v B coming to rest?
(c) v A > v B (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm.
(2002)
(d) their velocities depend on their masses. (2002) 30. Two forces are such that the sum of their magnitudes is 18 N
28. Speeds of two identical cars are u and 4u at a specific instant. and their resultant is 12 N which is perpendicular to the
If the same deceleration is applied on both the cars, the ratio
of the respective distances in which the two cars are stopped smaller force. Then the magnitudes of the forces are
from that instant is
(a) 12 N, 6 N (b) 13 N, 5 N
(c) 10 N, 8 N (d) 16 N, 2 N. (2002)
Answer Key
1. (c) 2. (c) 3. (b) 4. (b) 5. (d) 6. (a)
7. (d) 8. (a) 9. (b) 10. (c) 11. (c) 12. (b)
13. (a) 14. (*) 15. (a) 16. (b) 17. (b) 18. (d)
19. (c) 20. (a) 21. (c) 22. (b) 23. (a) 24. (c)
25. (d) 26. (b) 27. (b) 28. (d) 29. (a) 30. (b)
6 JEE MAIN CHAPTERWISE EXPLORER
1. (c) : Given : u = $i + 2 $j = m éëv02 sin q cos qt k^ - v0 gt 2 cos q k^
r - v02 sin q cos qt k^ + 12 v0 gt2 cos q k^ ûùú
As u = ux $i + u y $j
\ ux = 1 and uy = 2 = :m Hëéêe-re12, vvr0 g=t 2 cos q k^ ûúù =- 1 mgv0 t 2 cos q k^
Also x = ux t (a) K ( y i^ + x ^j ) 2
and y = u y t - 12 gt 2 6. vr = Ky i^ + Kx ^j
\ x = t
r ...(i)
and y = 2t - 12 ´10 ´ t2 = 2t - 5t 2 Q v = dx i^ + dy ^j ...(ii)
Equation of trajectory is dt dt
y = 2x – 5x 2 Equating equations (i) and (ii), we get
dx = Ky; ddyt = Kx
dt
2. (c) : Let u be the velocity of projection of the stone.
The maximum height a boy can throw a stone is \ dy = dy ´ ddxt
dx dt
u 2
H max = 2 g = 10 m ...(i) dy Kx x
dx Ky y
The maximum horizontal distance the boy can throw the same = = ...(iii)
stone is Integrating both sides of the above equation, we get
Rm ax = u 2 = 20 m (Using (i)) ò ydy = ò xdx
g y 2 = x2 + constant
3. (b) : dv = - 2.5 v or 1 dv = - 2.5 dt y
dt v
On integrating, within limit (v 1 = 6.25 m s –1 to v2 = 0)
\ v2 = 0 2 dv = t acoqs q P(R, q)
ò v -1/ -2.5 ò dt
v1 = 6.25 m s -1 0 a
q
[ ] v1/ 2 0 -2 ´ (6.25)1 /2 7. (d) : asinq x
6.25 - 2.5
2 ´ = - (2.5) t Þ t = = 2 s O
4. (b) : Rm ax = v2 sin 90° = v 2
g g
For a particle in uniform circular motion,
p(Rm ax ) 2 pv 4
Area = = g 2 v 2
R
Acceleration, a = towards the centre
5. (d) : The position vector of the particle from the origin at From figure,
any time t is ar = v2 v 2
R R
r v0 cosqt i^ + (v0 sin qt - 12 rgt 2 ) ^j - a cos qi^ - a sin q ^j = - cosq i^ - sin q ^j
r= Velocity vector, vr = dr
\ dt 8. (a) : s = t3 + 3
vr = \ v = ds = d (t3 + 3) = 3t 2
dt dt
d i^ 12 gt 2 ) ^j )
dt (v0 cos qt + (v0 sin qt - Tangential acceleration, at = dv = d (3t 2 ) = 6 t
dt dt
= v0 cos q i^ + (v0 sin q - gt) ^j
At t = 2 s,
TLLrrh==e mrran´(grrmu´lvravr r ) momentum of the particle about the origin is
v = 3(2)2 = 12 m/s, a t = 6(2) = 12 m/s 2
Centripetal acceleration,
= m êëé(v0 cos qt i^ + (v0 sin qt - 21 gt2 ) ^j ) ac = v 2 = (12) 2 = 144 = 7.2 m/s 2
´ (v0 cos q i^ + (v0 sin q - gt) ^j ) ùû R 20 20
= m éë(v02 cosq sin qt - v0 gt2 cosq) k^ Net acceleration,
+ (v02 sin q cos qt - 12 gt 2v 0 cos q)(- k^ ) ûùú
a = (ac )2 + (at )2 = (7.2)2 + (12)2 » 14 m/s 2
9. (b) : v = u + at
r
v = (3$i + 4$j) + (0.4$i + 0.3$j ) ´ 10
Kinematics 7
r = (3 + 4)$i + (4 + 3) $j s3 = 1 æ 2f öø÷ ( 2t1 ) 2 or s3 = 1 ´ 4 f t1 2
v 2 çè 2 2
Þ | vr | = 49 + 49 = 98 = 7 2 units or s 3 = 2s 1 = 2s............ (iv)
s 1 + s2 + s3 = 15s
(This value is about 9.9 units close to 10 units. If (a) is given
that is also not wrong). or s + f t 1t + 2s = 15s
or f t 1 t = 12s
10. (c) : As u = 0, v 1 = at, v2 = constant for the other particle. ........... (v)
Initially both are zero. Relative velocity of particle 1 w.r.t.
2 is velocity of 1 – velocity of 2. At first the velocity of first From (i) and (v),
particle is less than that of 2. Then the distance travelled by
particle 1 increases as s = ft1 2
12 s 2 ´ ft1t
x1 = (1/2) at1 2. For the second it is proportional to t. Therefore
it is a parabola after crossing xaxis again. Curve (c) satisfies or t1 = t or s = 1 ft12 = 1 f çèæ t öø÷ 2 = ft 2
this. 6 2 2 6 72
or s = ft 2
72
11. (c) : Given : velocity v = v 0 + gt + ft 2 None of the given options provide this answer.
\ v = dx x t 15. (a) : t = ax 2 + bx
dt Differentiate the equation with respect to t
or ò dx = ò vdt
dx + b ddxt
0 0 dt
t \ 1 = 2 ax
or x = ò (v0 + gt + ft2 ) dt dx
dt
0 or 1 = 2axv + bv as = v
x = v0 t + gt 2 + ft 3 + C v = 1
2 3 2 ax + b
or
where C is the constant of integration
Given : x = 0, t = 0. \ C = 0 or dv = -2a(dx / dt ) = -2 av ´ v 2
dt (2ax + b) 2
gt 2 ft 3
or x = v0 t + 2 + 3 or Acceleration = –2av 3 .
At t = 1 sec 16. (b) : Velocity in eastward direction = 5iˆ
\ x = v0 + g + 3f . velocity in northward direction = 5 ˆj
2
N
12. (b) : v = a x 5 ˆj
a
or dx = a x or dx = a dt
dt x 45°
or ò dx = aò dt or 2 x1/ 2 = a t W E
x -5 iˆ
+ 5iˆ
or x = çæè a2 ÷öø 2 t2 \ Acceleration ar = 5 ˆj - 5i ˆ
or displacement is proportional to t2 . 10
or ar = 1 ˆj - 12 iˆ or | ar |= çèæ 1 2 + çèæ - 21 ÷øö2
2 2
13. (a) : Initially, the parachutist falls under gravity ÷øö
\ u2 = 2ah = 2 × 9.8 × 50 = 980 m 2 s –2 or | ar |= 1 ms -2 towards northwest.
2
He reaches the ground with speed
17. (b) : Range is same for angles of projection q and (90 – q)
= 3 m/s, a = –2 ms –2
\ (3) 2 = u 2 – 2 × 2 × h 1 \ t1 = 2u sin q and t 2 = 2u sin (90 - q)
or 9 = 980 – 4 h 1 g g
or h1 = 971 or h 1 = 242.75 m 4u2 sin qcos q 2 æ u2 sin 2q ö 2 R
4 g 2 g ç g
\ t1t 2 = = ´ ÷ =
\ Total height = 50 + 242.75 g ø
è
= 292.75 = 293 m.
\ t1 t 2 is proportional to R.
14. (*) : For first part of journey, s = s1 , 18. (d) : Let a be the retardation for both the vehicles
s1 = 1 f t12 = s ............ (i) For automobile, v2 = u 2 – 2 as
2 ........... (ii)
\ u1 2 – 2as1 = 0 Þ u1 2 = 2as 1
v = f t 1 ........... (iii) Similarly for car, u 22 = 2as 2
For second part of journey,
æ u2 ö2 ss12 Þ èæç 16200 ÷öø2 s 2
s 2 = vt \ çè u1 ÷ø = = 20
or s2 = f t1 t
For the third part of journey, or s 2 = 80 m.
8 JEE MAIN CHAPTERWISE EXPLORER
19. (c) : Equation of motion : s = ut + 1 gt2 26. (b) : Q x = at 3
2
\ dx = 3at2 Þ v x = 3a t 2
h = 0 + 1 gT 2 dt
\ 2
Again y = bt 3
or 2h = gT 2 ..... (i) \ dy Þ v = 3b t 2 \ v 2 = v x 2 + v y 2
....... (ii) dt
After T/3 sec, y
s = 0 + 12 ´ g æèç T ÷øö 2 = gT 2 or v 2 = (3at2 )2 + (3bt 2 )2 = (3t2 ) 2 (a2 + b2 )
3 18
or v = 3t2 a2 + b 2 .
or 18 s = gT 2 27. (b) : Ball A projected upwards with velocity u falls back
From (i) and (ii), to building top with velocity u downwards. It completes its
18 s = 2h journey to ground under gravity.
or s = h m from top. \ v A2 = u2 + 2gh ..............(i)
9
h - h = 8 h m. Ball B starts with downwards velocity u and reaches ground
9 9
\ Height from ground = after travelling a vertical distance h
(a) : r r r r
20. A´B = B´ A \ v B2 = u2 + 2gh ............(ii)
From (i) and (ii)
or AB sin q nˆ = AB sin(-q ) nˆ
v A = v B .
or sinq = – sinq 28. (d) : Both are given the same deceleration simultaneously
or 2 sinq = 0 and both finally stop.
or q = 0, p, 2p..... Formula relevant to motion : u 2 = 2 as
\ q = p.
21. (c) : Range is same for angles of projection q and (90º – q) \ For first car, s1 = u 2
2 a
2u sin q 2u sin (90º -q)
\ T1 = g and T2 = g (4u ) 2 16u 2
For second car, s 2 = 2a = 2 a
\ T1T2 = 4u 2 sin qcos q = 2 ´ æ u2 sin 2q ö = 2 R \ s1 = 116 .
g 2 g ç ÷ g s2
è g ø
\ T 1 T 2 is proportional to R. 29. (a) : For first part of penetration, by equation of motion,
22. (b) : The acceleration vector acts along the radius of the èçæ u2 ø÷ö 2 = u2 – 2a(3)
circle. The given statement is false. or 3u2 = 24a Þ u 2 = 8a
23. (a) : The person will catch the ball if his speed and horizontal For latter part of penetration, ........... (i)
speed of the ball are same
= v0 cos q = v0 Þ cos q = 1 = cos 60° \ q = 60°. 0 = æèç u2 ÷öø 2 – 2 ax
2 2 or u2 = 8ax
24. (c) : For first case, u1 = 50 km = 50 ´ 1000 = 125 m From (i) and (ii) ........... (ii)
hour 60´ 60 9 sec 8ax = 8a Þ x = 1 cm.
\ Acceleration a = - u1 2 = - çèæ 125 2 ´ 2 1´ 6 = -16 m/sec 2 30. (b) : Resultant R is perpendicular to smaller force Q and
2s1 9 (P + Q) = 18 N
÷öø \ P2 = Q 2 + R2 by right angled triangle
For second case, u2 = 100 km = 100 ´1000 = 250 m
hour 60 ´ 60 9 sec
-u2 2 -1 èçæ 250 ö÷ø 2 èæç 116 ø÷ö =
2a 2 9
\ s 2 = = ´ - 24 m
or s 2 = 24 m. Q R
90°
25. (d) : Height of building = 10 m
The ball projected from the roof of building will be back P
to roof height of 10 m after covering the maximum horizontal or (P2 – Q 2) = R 2
range.
or (P + Q)(P – Q) = R 2
Maximum horizontal range ( R ) = u 2 sin 2 q or (18)(P – Q) = (12) 2 [Q P + Q = 18]
g
or (P – Q) = 8
or R = (10)2 ´ sin 60° = 10 ´ 0.866 or R = 8.66 m. Hence P = 13 N and Q = 5 N.
10
Laws of Motion 9
CHAPTER LAWS OF MOTION
3
1. Two cars of masses m 1 and m2 are moving in circles of radii (a) 0.2 N s (b) 0.4 N s (c) 0.8 N s (d) 1.6 N s
r1 and r 2 , respectively. Their speeds are such that they make (2010)
complete circles in the same time t. The ratio of their
5. A body of mass m = 3.513 kg is moving along the xaxis
centripetal acceleration is
with a speed of 5.00 ms –1 . The magnitude of its momentum
(a) m 1 : m 2 (b) r1 : r2
(c) 1 : 1 (d) m 1 r1 : m 2r 2 (2012) is recorded as
2. A particle of mass m is at rest at the origin at time t = 0. It (a) 17.57 kg ms –1 (b) 17.6 kg ms –1
is subjected to a force F(t) = Fo e – bt in the x direction. Its (c) 17.565 kg ms –1 (d) 17.56 kg ms –1 . (2008)
speed v(t) is depicted by which of the following curves?
6. A block of mass m is connected to another block of mass M
Fo b Fo by a spring (massless) of spring constant k. The blocks are
m mb
kept on a smooth horizontal plane. Initially the blocks are at
(a) v(t) (b) v(t )
rest and the spring is unstretched. Then a constant force F
t t starts acting on the block of mass M to pull it. Find the force
of the block of mass m.
Fo Fo MF (b) mF
mb mb (a) (m + M ) M
(c) (d) ( M + m) F mF (2007)
(c) m (d) (m + M )
v(t ) v(t )
t t
(2012) 7. A ball of mass 0.2 kg is thrown vertically upwards by applying
3. Two fixed frictionless inclined planes making an angle 30° a force by hand. If the hand moves 0.2 m which applying the
and 60° with the vertical are shown in the figure. Two blocks
A and B are placed on the two planes. What is the relative force and the ball goes upto 2 m height further, find the
vertical acceleration of A with respect to B?
magnitude of the force. Consider g = 10 m/s 2
A
(a) 22 N (b) 4 N (c) 16 N (d) 20 N.
(2006)
8. A player caught a cricket ball of mass 150 g moving at a rate
B of 20 m/s. If the catching process is completed in 0.1 s, the
force of the blow exerted by the ball on the hand of the player
is equal to
60° 30° (a) 300 N (b) 150 N (c) 3 N (d) 30 N.
(2006)
(a) 4.9 ms– 2 in vertical direction 9. Consider a car moving on a straight road with a speed of
(b) 4.9 ms– 2 in horizontal direction
(c) 9.8 ms– 2 in vertical direction 100 m/s. The distance at which car can be stopped is
(d) zero (2010) [m k = 0.5] (d) 1000 m
(a) 100 m (b) 400 m (c) 800 m
4. The figure shows the position time (xt) graph of one ma cosa (2005)
dimensional motion of a body of mass 0.4 kg. The magnitude
of each impulse is 10. A block is kept on a frictionless
inclined surface with angle of ma
2 inclination a. The incline is given mg sina
x (m)
an acceleration a to keep the block
stationary. Then a is equal to a
0 2 4 6 8 10 12 14 16 (a) g (b) g tana
t(s)
(c) g/tana (d) g coseca (2005)
10 JEE MAIN CHAPTERWISE EXPLORER
11. A particle of mass 0.3 kg is subjected to a force F = – kx with 18. A machine gun fires a bullet of mass 40 g with a velocity
k = 15 N/m. What will be its initial acceleration if it is
released from a point 20 cm away from the origin? 1200 ms –1. The man holding it can exert a maximum force
(a) 5 m/s2 (b) 10 m/s 2 (c) 3 m/s 2 (d) 15 m/s2
(2005) of 144 N on the gun. How many bullets can he fire per
second at the most?
(a) one (b) four (c) two (d) three.
12. A bullet fired into a fixed target loses half its velocity after (2004)
penetrating 3 cm. How much further it will penetrate before
coming to rest assuming that it faces constant resistance to 19. A rocket with a liftoff mass 3.5 × 10 4 kg is blasted upwards
motion?
(a) 1.5 cm (b) 1.0 cm (c) 3.0 cm (d) 2.0 cm with an initial acceleration of 10 m/s 2 . Then the initial thrust
(2005)
of the blast is
(a) 3.5 × 10 5 N (b) 7.0 × 105 N
(c) 14.0 × 105 N (d) 1.75 × 105 N. (2003)
13. The upper half of an inclined plane with inclination f is 20. A light spring balance hangs from the hook of the other light
perfectly smooth while the lower half is rough. A body starting
from rest at the top will again come to rest at the bottom if spring balance and a block of mass M kg hangs from the
the coefficient of friction for the lower half is given by
(a) 2tanf (b) tanf (c) 2sinf (d) 2cosf former one. Then the true statement about the scale reading is
(2005)
(a) both the scales read M kg each
(b) the scale of the lower one reads M kg and of the upper
one zero
14. A smooth block is released at rest on a 45o incline and then (c) the reading of the two scales can be anything but the sum
slides a distance d. The time taken to slide is n times as much of the reading will be M kg
to slide on rough incline than on a smooth incline. The (d) both the scales read M/2 kg. (2003)
coefficient of friction is 21. A block of mass M is pulled along a horizontal frictionless
(a) ms = 1 - 1 (b) ms = 1 - 1 surface by a rope of mass m. If a force P is applied at the free
n2 n2
end of the rope, the force exerted by the rope on the block is
(c) m k = 1 - 1 (d) mk = 1 - n12 (2005) (a) Pm (b) Pm
n2 M + m M - m
15. An annular ring with inner and outer radii R1 and R2 is rolling (c) P (d) MPM+ m . (2003)
without slipping with a uniform angular speed. The ratio of
the forces experienced by the two particles situated on the 22. A marble block of mass 2 kg lying on ice when given a
inner and outer parts of the ring, F1 / F2 is ö2 velocity of 6 m/s is stopped by friction in 10 s. Then the
÷
(a) 1 (b) R1 (c) R2 (d) æ R1 ø coefficient of friction is
R2 R1 ç R2
è (a) 0.02 (b) 0.03 (c) 0.06 (d) 0.01.
(2005) (2003)
16. A block rests on a rough inclined plane making an angle of 23. A horizontal force of 10 N is necessary to
30° with the horizontal. The coefficient of static friction just hold a block stationary against a
between the block and the plane is 0.8. If the frictional force wall. The coefficient of friction between 10 N
the block and the wall is 0.2. The weight
on the block is 10 N, the mass of the block (in kg) is
(take g = 10 m/s 2) of the block is
(a) 2.0 (b) 4.0 (c) 1.6 (d) 2.5 (a) 20 N (b) 50 N
(2004) (c) 100 N (d) 2 N.
17. Two masses m 1 = 5 kg and m2 = 4.8 kg tied (2003)
to a string are hanging over a light
24. A spring balance is attached to the ceiling of a lift. A man
frictionless pulley. What is the acceleration
of the masses when lift free to move? hangs his bag on the spring and the spring reads 49 N,
(g = 9.8 m/s2 )
(a) 0.2 m/s2 when the lift is stationary. If the lift moves downward with
(b) 9.8 m/s2
(c) 5 m/s2 an acceleration of 5 m/s 2, the reading of the spring balance
(d) 4.8 m/s2 .
m 1 will be
m 2
(a) 24 N (b) 74 N
(2004)
(c) 15 N (d) 49 N.
(2003)
Laws of Motion 11
With what value of maximum safe acceleration (in ms –2 ) can
25. Twhitrhe ev feolrocceitsy s tvra r.t Tachteisneg fsoimrcueslt aanree oruepslrye soenn ate pda irnti cmlea mgnoivtuindge a man of 60 kg climb on the rope?
and direction by the three sides of a triangle ABC (as shown). P
The plg| eavrrsr e s|ta i ctitenlhre a ttnhwh eai vrlndl inrvreo cwti omn oovfe thwei thla rvgeelsotc ity C C
(a)
(b) AB (a) 16 (b) 6 (c) 4 (d) 8.
(c) (2002)
(2003)
(d) fvro r,c ere mBCai ning unchanged.
26. Three identical blocks of masses m = 2 kg are drawn by a 29. When forces F1 , F2 , F3 are acting on a particle of mass m
force F = 10.2 N with an acceleration of 0.6 ms– 2 on a such that F2 and F3 are mutually perpendicular, then the
frictionless surface, then what is the tension (in N) in the particle remains stationary. If the force F1 is now removed
string between the blocks B and C? then the acceleration of the particle is
C B A F (a) F1 /m (b) F2 F3 /mF1 (2002)
(c) (F2 – F3 )/m (d) F2 /m.
(a) 9.2 (b) 7.8 (c) 4 (d) 9.8 30. A lift is moving down with acceleration a. A man in the lift
(2002)
drops a ball inside the lift. The acceleration of the ball as
observed by the man in the lift and a man standing stationary
27. A light string passing over a smooth light pulley connects on the ground are respectively
two blocks of masses m 1 and m 2 (vertically). If the acceleration (a) g, g (b) g – a, g – a
of the system is g/8, then the ratio of the masses is
(c) g – a, g (d) a, g. (2002)
(a) 8 : 1 (b) 9 : 7 (c) 4 : 3 (d) 5 : 3. 31. The minimum velocity (in ms –1 ) with which a car driver must
(2002) traverse a flat curve of radius 150 m and coefficient of friction
28. One end of a massless rope, which passes over a massless 0.6 to avoid skidding is
and frictionless pulley P is tied to a hook C while the other
end is free. Maximum tension that the rope can bear is 960 N. (a) 60 (b) 30 (c) 15 (d) 25.
(2002)
Answer Key
1. (b) 2. (b) 3. (a) 4. (c) 5. (a) 6. (d)
7. (d) 8. (d) 9. (d) 10. (b) 11. (b) 12. (b)
13. (a) 14. (c) 15. (b) 16. (a) 17. (a) 18. (d)
19. (a) 20. (a) 21. (d) 22. (c) 23. (d) 24. (a)
25. (d) 26. (b) 27. (b) 28. (b) 29. (a) 30. (c)
31. (b)
12 JEE MAIN CHAPTERWISE EXPLORER
1. (b): Centripetal acceleration, a c = w 2r 5. (a) : Momentum is mv.
m = 3.513 kg ; v = 5.00 m/s
Q w = 2 p \ mv = 17.57 m s –1
T Because the values will be accurate up to second decimel
place only, 17.565 = 17.57.
As T 1 = T 2 Þ w1 = w2
Q ac 1 = r1 F
ac2 r2 + M
6. (d) : Acceleration of the system a =
m
2. (b) : F(t) = Fo e– bt (Given)
ma = Fo e –bt Force on block of mass m = ma = mF .
m + M
Fo e -bt
a = m 7. (d) : Work done by hand = Potential energy of the ball
dv = Fo e -bt or dv = Fo e-bt dt \ FS = mgh Þ F = mgh = 0.2 ´10 ´ 2 = 20 N .
dt m m s 0.2
Integrating both sides, we get 8. (d) : Force × time = Impulse = Change of momentum
v dv = t Fo e - bt dt Þ v = Fo é e -bt ù t = Fo [1 - e - bt ] \ Force = Impulse = 3 = 30 N .
m m ëê - b úû 0 mb time 0.1
ò ò
0 0
3. (a) : The acceleration of the body down the smooth inclined 9. (d) : Retardation due to friction = mg
plane is a = gsinq Q v2 = u 2 + 2as
\ 0 = (100) 2 – 2(mg)s or 2 mgs = 100 × 100
It is along the inclined plane.
where q is the angle of inclination or s = 100 ´100 = 1000 m .
2 ´ 0.5 ´ 10
\ The vertical component of acceleration a is
a (along vertical) = (gsinq)sinq = gsin2 q 10. (b) : The incline is given an acceleration a. Acceleration
of the block is to the right. Pseudo acceleration a acts on
For block A block to the left. Equate resolved parts of a and g along
a A(along vertical) = gsin2 6 0° incline.
\ macos a = mgsina or a = gtana.
For block B
a B(along vertical) = gsin2 30° 11. (b) : F = – kx
The relative vertical acceleration of A with respect to B is or F = -15 ´ æèç 12000 öø÷ = -3 N
Initial acceleration is over come by retarding force.
a AB(along vertical) = a A(along vertical) – aB (along vertical)
= gsin 26 0° – gsin2 30°
( ) ( ) æ 3 2 1 2 ö
2 2 ÷ø
= g çè -
g m s-2 in or m × (acceleration a) = 3
2
= = 4.9 vertical direction. or a = 3 = 3 = 10 ms-2 .
m 0.3
4. (c) : Here, mass of the body, m = 0.4 kg
Since positiontime (xt) graph is a straight line, so motion is 12. (b) : For first part of penetration, by equation of motion,
uniform. Because of impulse direction of velocity changes as
can be seen from the slopes of the graph. çèæ u2 ÷øö 2 = (u)2 - 2 f (3)
or 3u2 = 24 f
From graph, For latter part of penetration, .... (i)
Initial velocity, u = (2 - 0) = 1 m s -1
(2 - 0)
0 = èçæ u2 ÷øö 2 - 2 fx
Final velocity, v = (0 - 2) = -1m s -1 or u 2 = 8fx
(4 - 2)
From (i) and (ii) ......... (ii)
Initial momentum, pi = mu = 0.4 × 1 = 0.4 N s 3 × (8 fx) = 24 f
or x = 1 cm.
Final momentum, pf = mv = 0.4 × (–1) = – 0.4 N s
Impulse = Change in momentum = pf – pi 13. (a) : For upper half smooth incline, component of g down
= – 0.4 – (0.4) N s = – 0.8 N s the incline = gsinf
|Impulse| = 0.8 N s
Laws of Motion 13
\ v 2 = 2(gsinf) 2l R mk g cos f 19. (a) : Initial thrust = (Lift off mass) × acceleration
For lower half rough g sin f f gcosf = (3.5 × 10 4 ) × (10) = 3.5 × 10 5 N.
g
incline, f r i ct i o n a l 20. (a) : Both the scales read M kg each.
retardation = m k gcosf f
\ Resultant acceleration = gsinf – mk g cosf 21. (d) : Acceleration of block ( a ) = Force applied
Total mass
\ 0 = v 2 + 2 (gsinf – mk gcos f) l or a = (M P
2 + m)
or 0 = 2(gsinf) l + 2g(sinf – m k cos f) l \ Force on block
2 2
MP
or 0 = sinf + sinf – m k cosf = Mass of block × a = ( M + m) .
or m k cosf = 2sinf
or mk = 2tanf. 22. (c) : Frictional force provides the retarding force
\ m mg = ma
14. (c) : Component of g down the plane = gsinq
\ For smooth plane, or m = a = u/ t = 61/100 = 0.06 .
g g
d = 21 ( g sin q ) t2
For rough plane, ........ (i) 23. (d) : Weight of the block is balanced by force of friction
Frictional retardation up the plane = mk (gcosq) \ Weight of the block = mR = 0.2 × 10 = 2 N.
R mk gcosq 24. (a) : When lift is standing, W1 = mg acceleration a ,
When the lift descends with
gsinq W2 = m (g – a)
q
q \ W2 = m(g - a ) = 9.8 - 5 = 4.8
g gcosq W1 mg 9.8 9.8
\ d = 12 ( g sin q - mk g cos q)(nt ) 2 or W2 = W1 ´ 4.8 = 499´.84 .8 = 24 N .
9.8
1 (g sin q)t2 12 (g sin q - mk g cosq ) n2t2 25. (d) : By triangle of forces, the particle will be in equilibrium
2
\ = under the three forces. Obviously the resultant force on the
or sinq = n 2 (sinq – m k cosq) particle will be zero. Consequently the acceleration will be
Putting q = 45°
zero. the particle velocity remains unchanged at vr .
Hence
or sin45° = n 2 (sin45° – m kc os45°) 26. (b) : Q Force = mass × acceleration
\ F – TA B = ma
or 1 = n2 (1 - m k ) and TA B – TB C = ma
2 2 \ TB C = F – 2 ma
or TB C = 10.2 – (2 × 2 × 0.6)
or mk = 1 - n12 .
or TB C = 7.8 N.
15. (b) : Centripetal force on particle = mRw 2
\ F1 = mR1w2 = R1 27. (b) : a = (m1 - m2 )
F2 mR2 w2 R2 . g (m1 + m2 )
16. (a) : For equilibrium of block, R f \ 1 = (m1 - m2 ) or m1 = 9 .
8 (m1 + m2 ) m2 7
f = mgsinq
\ 10 = m × 10 × sin30° mgsinq mg q sq 28. (b) : T – 60g = 60a
or m = 2 kg. q mgco or 960 – (60 × 10) = 60a
or 60a = 360
17. (a) : a = (m1 - m2 ) = (5 - 4.8) = 0.2 or a = 6 ms –2 .
g (m1 + m2 ) (5 + 4.8) 9.8
29. (a) : F2 and F3 have a resultant equivalent to F1
or a = g ´ 0.2 = 9.8 ´ 0.2 = 0.2 ms-2 .
9.8 9.8 \ Acceleration = F1 .
m
18. (d) : Suppose he can fire n bullets per second
\ Force = Change in momentum per second 30. (c) : For observer in the lift, acceleration = (g – a)
For observer standing outside, acceleration = g.
144 = n ´ çæè 104000 ÷öø ´ (1200)
31. (b) : For no skidding along curved track,
144 ´1000 v = mRg
40 ´ 1200
or n = v = 0.6 ´ 150 ´ 10 = 30 m .
s
or n = 3. \
14 JEE MAIN CHAPTERWISE EXPLORER
CHAPTER WORK, ENERGY
AND POWER
4
1. This question has StatementI and StatementII. Of the four collision is negligible and the collision with the plate is totally
choices given after the Statements, choose the one that best elastic. Then the velocity as a function of time and the height
describes the two Statements. as function of time will be
Statement1 : A point particle of mass m moving with speed v y
v collides with stationary point particle of mass M. If the v1 h
t
maximum energy loss possible is given as (a)
y
h æ 12 mv2 øö then h = æ m ö O t
è è + m ø v
M
Statement2 : Maximum energy loss occurs when the particles
get stuck together as a result of the collision. +v1 h t
(a) StatementI is false, StatementII is true. t
(b) StatementI is true, StatementII is true, StatementII is a (b) O
y
correct explanation of StatementI. –v1
(c) StatementI is true, StatementII is true, StatementII is
v
not a correct explanation of statementI. +v1 h
(d) StatementI is true, StatementII is false. t
(2013) (c) O t1 2t 1 3t 1 4t1
–v 1
2. This question has Statement 1 and Statement 2. Of the four t
choices given after the statements, choose the one that best y
describes the two statements. (d) t 1 2t1 3t 1 4t 1 h (2009)
If two springs S1 and S2 of force constants k1 and k2 , respectively, t
are stretched by the same force, it is found that more work is t
done on spring S 1 than on spring S2 .
Statement 1 : If stretched by the same amount, work done on 5. A block of mass 0.50 kg is moving with a speed of 2.00 ms– 1 on
S1 , will be more than that on S2 .
a smooth surface. It strikes another mass of 1.00 kg and then they
Statement 2 : k 1 < k 2.
(a) Statement 1 is true, Statement 2 is false. move together as a single body. The energy loss during the collision
(b) Statement 1 is true, Statement 2 is true, Statement 2 is the is
(a) 0.34 J (b) 0.16 J (c) 1.00 J (d) 0.67 J.
correct explanation of Statement 1.
(c) Statement 1 is true, Statement 2 is true, Statement 2 is not (2008)
the correct explanation of Statement 1. 6. An athlete in the olympic games covers a distance of 100 m
(d) Statement 1 is false, Statement 2 is true.
(2012) in 10 s. His kinetic energy can be estimated to be in the range
3. Statement1 : Two particles moving in the same direction do (a) 2,000 J 5,000 J (b) 200 J 500 J
not lose all their energy in a completely inelastic collision.
(c) 2 × 105 J 3 × 105 J (d) 20,000 J 50,000 J. (2008)
Statement2 : Principle of conservation of momentum holds 7. A particle is projected at 60° to the horizontal with a kinetic
true for all kinds of collisions. energy K. The kinetic energy at the highest point is
(a) Statement1 is true, Statement2 is false. (a) K/2 (b) K (c) zero (d) K/4
(b) Statement1 is true, Statement2 is true; Statement2 is the (2007)
correct explanation of Statement1.
(c) Statement1 is true, Statement2 is true; Statement2 is 8. A 2 kg block slides on a horizontal floor with a speed of 4 m/s.
not the correct explanation of Statement1. It strikes a uncompressed spring, and compresses it till the
(d) Statement1 is false, Statement2 is true. (2010) block is motionless. The kinetic friction force is 15 N and
spring constant is 10,000 N/m. The spring compresses by
4. Consider a rubber ball freely falling from a height h = 4.9 m (a) 8.5 cm (b) 5.5 cm (c) 2.5 cm (d) 11.0 cm
onto a horizontal elastic plate. Assume that the duration of (2007)
Work, Energy and Power 15
9. The potential energy of a 1 kg particle free to move along the ML2 KL2
(b) (c) MK L (d)
xaxis is given by (a) zero
K 2 M
æ x4 x 2 ö (2005)
V (x) = ç 4 - 2 ÷ J .
è ø
16. A spherical ball of mass 20 kg is stationary at the top of a hill
The total mechanical energy of the particle 2 J. Then, the of height 100 m. It rolls down a smooth surface to the ground,
then climbs up another hill of height 30 m and finally rolls
maximum speed (in m/s) is down to a horizontal base at a height of 20 m above the
ground. The velocity attained by the ball is
(a) 2 (b) 3/ 2 (c) 2 (d) 1/ 2 . (a) 10 m/s (b) 34 m/s (c) 40 m/s (d) 20 m/s
(2006) (2005)
10. A particle of mass 100 g is thrown vertically upwards with 17. A body of mass m, accelerates uniformly from rest to v1 in
a speed of 5 m/s. The work done by the force of gravity time t1 . The instantaneous power delivered to the body as a
during the time the particle goes up is function of time t is
(a) 0.5 J (b) –0.5 J (c) –1.25 J (d) 1.25 J.
(2006)
11. A bomb of mass 16 kg at rest explodes into two pieces of (a) mv1t (b) mv12 t
masses of 4 kg and 12 kg. The velocity of the 12 kg mass is t1 t1 2
4 ms –1. The kinetic energy of the other mass is
(a) 96 J (b) 144 J (c) 288 J (d) 192 J. (c) mv1t 2 (d) mv12 t . (2004)
(2006) t1 r t1
A force F 2kˆ ) N is applied over a particle which
12. A mass of M kg is suspended by a weightless string. The 18. = (5iˆ + 3 ˆj +
horizontal force that is required to displace it until the string displaces it from its origin to the point rr = (2iˆ - ˆj ) m . The
making an angle of 45º with the initial vertical direction is
work done on the particle in joule is
(a) –7 (b) +7 (c) +10 (d) +13.
(a) Mg( 2 - 1) (b) Mg( 2 + 1) (2004)
(c) Mg 2 Mg (2006) 19. A uniform chain of length 2 m is kept on a table such that a
(d) 2 . length of 60 cm hangs freely from the edge of the table. The
total mass of the chain is 4 kg. What is the work done in
13. A body of mass m is accelerated uniformly from rest to a pulling the entire chain on the table?
(a) 7.2 J (b) 3.6 J (c) 120 J (d) 1200 J.
speed v in a time T. The instantaneous power delivered to the (2004)
body as a function of time is given by
1 mv 2 (b) 12 mTv 2 2 t 2 20. A particle is acted upon by a force of constant magnitude
(a) 2 T 2 t
(d) mv 2 × t 2 (2005) which is always perpendicular to the velocity of the particle,
mv 2 T 2
(c) T 2 × t the motion of the particle takes place in a plane. It follows
14. A mass m moves with a that
velocity v and collides (a) its velocity is constant
inelastically with another
identical mass. After v / 3 (b) its acceleration is constant
after collision
(c) its kinetic energy is constant
collision the first mass (d) it moves in a straight line. (2004)
moves with velocity in a m m 21. A particle moves in a straight line with retardation proportional
direction perpendicular to before collision
the initial direction of to its displacement. Its loss of kinetic energy for any
displacement x is proportional to
motion. Find the speed of the 2n d mass after collision (a) x2 (b) ex (c) x (d) loge x.
2 v v (2004)
3 (b) 3
(a) (c) v (d) 3v 22. A spring of spring constant 5 × 10 3 N/m is stretched initially
(2005)
by 5 cm from the unstretched position. Then the work required
15. The block of mass M to stretch it further by another 5 cm is
moving on the frictionless (a) 12.50 Nm (b) 18.75 Nm
horizontal surface collides (c) 25.00 Nm (d) 6.25 Nm. (2003)
with the spring of spring
M 23. A body is moved along a straight line by a machine delivering
constant K and compresses a constant power. The distance moved by the body in time
it by length L. The t is proportional to
maximum momentum of the block after collision is (a) t 3/4 (b) t3 /2 (c) t 1/4 (d) t 1/2.
(2003)
16 JEE MAIN CHAPTERWISE EXPLORER
24. Consider the following two statements. 26. If massenergy equivalence is taken into account, when water
A. Linear momentum of a system of particles is zero. is cooled to form ice, the mass of water should
B. Kinetic energy of a system of particles is zero. (a) increase
Then (b) remain unchanged
(a) A does not imply B and B does not imply A (c) decrease
(b) A implies B but B does not imply A (d) first increase then decrease. (2002)
(c) A does not imply B but B implies A 27. A ball whose kinetic energy is E, is projected at an angle of
(d) A implies B and B implies A. (2003) 45° to the horizontal. The kinetic energy of the ball at the
25. A spring of force constant 800 N/m has an extension of highest point of its flight will be
5 cm. The work done in extending it from 5 cm to 15 cm (a) E (b) E / 2
is (c) E/2 (d) zero.
(a) 16 J (b) 8 J (c) 32 J (d) 24 J. (2002)
(2002)
Answer Key
1. (a) 2. (d) 3. (b) 4. (c) 5. (d) 6. (a)
7. (d) 8. (b) 9. (b) 10. (c) 11. (c) 12. (a)
13. (c) 14. (a) 15. (c) 16. (b) 17. (b) 18. (b)
19. (b) 20. (c) 21. (a) 22. (b) 23. (b) 24. (c)
25. (b) 26. (c) 27. (c)
Work, Energy and Power 17
1. (a) : Loss of energy is maximum when collision is inelastic. Assuming an athelete has about 50 to 100 kg, his kinetic
energy would have been 12 mv a2v .
Maximum energy loss = 1 mM u 2
2 (M + m) v v
\ f = mM
(M + m)
Hence, Statement1 is false, Statement2 is true.
2. (d) : For the same force, F = k1 x 1 = k2 x2 ...(i) v/2
(Using (i))
Work done on spring S1 is (Using (i)) t time
W1 = 12 k1x1 2 = (k1x1 ) 2 = F 2 ...(ii) (1/2)mv a2 v = (1/2) × 50 × 100 = 2500 J.
2k1 2k 1 (Using (ii)) For 100 kg, (1/2) × 100 × 100 = 5000 J.
(Using (ii) It could be in the range 2000 to 5000 J.
Work done on spring S 2 is
W2 = 12 k2x2 2 = (k2 x2 ) 2 = F 2 7. (d) : The kinetic energy of a particle is K
2k2 2 k2 At highest point velocity has its horizontal component.
Therefore kinetic energy of a particle at highest point is
\ W1 = k 2 K H = Kcos2 q = Kcos2 6 0° = K4 .
W2 k1
8. (b) : Let the spring be compressed by x
As W1 > W2 \ k2 > k 1 or k 1 < k 2 Initial kinetic energy of the mass = potential energy of the
spring + work done due to friction
Statement 2 is true.
For the same extension, x 1 = x 2 = x
Work done on spring S1 is
W1 = 1 k1x12 = 12 k1 x2
2
1 12 ´10000 ´
Work done on spring S2 is 2 ´ 2 ´ 42 = x 2 + 15 x
W2 = 1 k2 x22 = 21 k2 x2 or 5000 x 2 + 15x – 16 = 0
2
or x = 0.055 m = 5.5 cm.
\ W1 = k1
W2 k2 9. (b) : Total energy ET = 2 J. It is fixed.
For maximum speed, kinetic energy is maximum
As k 1 < k 2 \ W1 < W 2
Statement 1 is false. The potential energy should therefore be minimum.
3. (b) Q V ( x) = x4 - x 2
4 2
4. (c) : v = u + gt. As the ball is dropped, v = gt when coming
down. v increases, makes collision, the value of v becomes or dV = 4x3 - 2 x = x3 - x = x( x 2 - 1)
dx 4 2
+ve, decreases, comes to zero and increases. The change from
+v to –v is almost instantaneous. Using –ve signs when coming For V to be minimum, dV = 0
dx
down, (c) is correct.
\ x(x 2 – 1) = 0, or x = 0, ± 1
Further h = 1 gt 2 is a parabola. Therefore (c). At x = 0, V(x) = 0
2
1
5. (d) : By the law of conservation of momentum At x = ± 1, V ( x ) = - 4 J
mu = (M + m)v \ (Kinetic energy) max = ET – Vm in
0.50 × 2.00 = (1 + 0.50) v, 11..0500 = v or (Kinetic energy) max = 2 – çæè - 1 ÷öø = 9 J
Initial K.E. = (1/2) × 0.50 × (2.00) 2 = 1.00 J. 4 4
Final K .E. = 1 ´1.50 ´ 1.002 = 13..0000 = 0.33 or 1m vm2 = 9
2 (1.50)2 2 4
\ Loss of energy = 1.00 – 0.33 = 0.67 J. or vm 2 = 9´2 = 9´2 = 9
m´4 1´ 4 2
6. (a) : v = v / 2 is average velocity
s = 100 m, t = 10 s. \ (v/2) = 10 m/s. \ vm = 3 m/s.
2
v average = (v/2) = 10 m/s.
18 JEE MAIN CHAPTERWISE EXPLORER
10. (c) : Kinetic energy at projection point is converted into potential Since p 2 = 2ME
\ p = 2 ME = 2 M ´ KL2 = MK L .
energy of the particle during rise. Potential energy measures
2
the workdone against the force of gravity during rise.
\ (– work done) = Kinetic energy = 1 mv 2 12 mv 2 çèæ1 + k 2 ö
2
16. (b) : mgh = ÷
or (– work done) R 2 ø
= 1 ´ èæç 100 ÷øö ( 5) 2 = 5 ´ 5 = 1.25 J = 1 mv 2 ∙ 75
2 1000 2 ´10 2
\ Work done by force of gravity = – 1.25 J
11. (c) : Linear momentum is conserved 100 m
\ 0 = m 1 v 1 + m2 v 2 = (12 × 4) + (4 × v 2 )
or 4v 2 = – 48 Þ v 2 = – 12 m/s 30 m 20 m
\ Kinetic energy of mass m2 = 12 m2v22
= 12 ´ 4 ´ (–12)2 = 288 J . \ 1 mv2 æ 7 ö = mg ´ 80
2 èç 5 ÷ø
or v2 = 2 × 10 × 80 × 5 = 1600 × 5
7 7
12. (a) : Work done in displacement is equal to gain in potential or v = 34 m/s.
energy of mass
17. (b) : Acceleration a = v1
t1
l cos 45° 45° l \ velocity (v) = 0 + at = v1 t
l sin 45° t1
F \ Power P = Force × velocity = m a v
or P = m æèç v1 ö ´ çæèrvt11 t r÷öø = mv12 t .
(b) t1 ø÷ = F ×r t12
Mg
18. : Work done
Work done = F ´ l sin 45° = Fl
2 or work done = (5iˆ + 3 ˆj + 2kˆ ) × (2iˆ – ˆj )
Gain in potential energy = Mg(l – lcos45°) or work done = 10 – 3 = 7 J.
= Mgl çèæ1 - 1 ö 19. (b) : The centre of mass of the hanging part is at 0.3 m
2 ÷ø
from table
\ Fl = Mgl ( 2 - 1)
2 2 1.4 m
or F = Mg ( 2 – 1) . 0.6 m
13. (c) : Power = Force × velocity
= (ma) (v) = (ma) (at) = ma 2 t
or Power = m æèç v ø÷ö 2 (t ) = mv 2 t mass of hanging part = 4 ´20.6 = 1.2 kg
T T 2 \ W = mgh
= 1.2 × 10 × 0.3
14. (a) : Let v 1 = speed of second mass after collision Momentum = 3.6 J.
is conserved
Along Xaxis, mv 1c osq = mv .......(i) 20. (c) : No work is done when a force of constant magnitude
always acts at right angles to the velocity of a particle when
Along Yaxis, mv 1s inq = mv ....... (ii) the motion of the particle takes place in a plane.
3 Hence kinetic energy of the particle remains constant.
From (i) and (ii)
2
æ mv ö
\ (mv1 c osq) 2 + (mv 1 sinq) 2 = (mv) 2 + çè 3 ÷ø 21. (a) : Given : Retardation µ displacement
or m 2v12 = 4 m 2 v 2 or dv = kx
3 dt
or v1 = 2 v . or æçè dv ø÷ö æèç ddxt ÷öø = kx
3 dx
15. (c) : Elastic energy stored in spring = 1 KL2 or dv (v) = kx dx
2
v 2 x
\ kinetic energy of block E = 21 KL2
or ò vdv = k ò x dx
v1 0
Work, Energy and Power 19
or v 2 - v 12 = k x 2 24. (c) : A system of particles implies that one is discussing
2 total momentum and total energy.
2 2 2 u m m u
1(a ) 1(b )
or mv22 - mv1 2 = mkx2
22 2
or (K2 – K1 ) = mk x2 1(a ) explodes
2
or Loss of kinetic energy is proportional to x2 . Total momentum = 0
( ) But total kinetic energy =
22. (b) : Force constant of spring (k) = F/x 2 1 mu 2
or F = kx 2
\ dW = kxdx
But if total kinetic energy = 0, velocities are zero.
Here A is true, but B is not true.
or ò dW = 0.1 = k2 éë(0.1)2 - (0.05) 2 ùû A does not imply B, but B implies A.
ò kx dx x 2 0.15
0.05 25. (b) : W = ò Fdx = ò kx dx
= 2k ´ [0.01 - 0.0025] x1 0.05
or Workdone = (5 ´ 103 ) (0.0075) = 18.75 N m . W = 0.15 = 8020 éë x2 ùû00..1055
´
2 \ ò 800 xdx
0.05
23. (b) : Power = Work = ForceT×imdiestance = Force× velocity = 400 ëé(0.15)2 – ( 0.05) 2 ù
Time û
\ Force × velocity = constant (K) or W = 8 J.
or (ma) (at) = K 26. (c) : When water is cooled to form ice, its thermal energy
çèæ K ÷øö1 / 2 decreases. By mass energy equivalent, mass should decrease.
mt
or a = (E ) = 1 mu2
2
Q s = 1 at 2 27. (c) : Kinetic energy point of projection
2
At highest point velocity = u cosq
1 æèç K øö÷1/ 2 t2 12 èçæ Km ø÷ö1 / 2 / 2 \ Kinetic energy at highest point
2 mt
\ s = = t 3 = 1 m (u cos q ) 2
2
or s is proportional to t 3/2. = 12 mu2 cos2 45° = E2 .
20 JEE MAIN CHAPTERWISE EXPLORER
CHAPTER ROTATIONAL MOTION
5
1. A hoop of radius r and mass m rotating with an angular velocity x as k(x/L)n where n can be zero or any positive number. If the
w 0 is placed on a rough horizontal surface. The initial velocity position xC M of the centre of mass of the rod is plotted against
of the centre of the hoop is zero. What will be the velocity of n, which of the following graphs best approximates the
the centre of the hoop when it ceases to slip? dependence of x CM on n?
(a) rw 0 (b) rw 0 (c) rw 0 (d) rw 0 x CM x CM
4 3 2 L L
L L
(2013)
(a) 2 (b) 2
2. A pulley of radius 2 m is rotated about its axis by a force
F = (20t – 5t 2) newton (where t is measured in seconds) applied O n
O
tangentially. If the moment of inertia of the pulley about its axis x CM xC M
of rotation is 10 kg m2 , the number of rotations made by the L L
pulley before its direction of motion if reversed, is L
(a) less than 3 (c) 2
(d) 2
(b) more than 3 but less than 6 O O
(c) more than 6 but less than 9
(d) more than 9 (2011) (2008)
3. A mass m hangs with the help of a string wrapped around a 7. Consider a uniform square plate of side a and mass m. The
pulley on a frictionless bearing. The pulley has mass m and
moment of inertia of this plate about an axis perpendicular to
radius R. Assuming pulley to be a perfect uniform circular its plane and passing through one of its corners is
disc, the acceleration of the mass m, if the string does not slip (a) 2 ma 2 (b) 5 ma 2 (c) 1 ma 2 (d) 7 ma 2
on the pulley, is 3 6 12 12
(a) 32 g (b) g (c) 23 g g (2008)
(d) 3
(2011) 8. For the given uniform square lamina ABCD, whose centre is
4. A thin horizontal circular disc is rotating about a vertical axis O,
passing through its centre. An insect is at rest at a point near (a) I AC = 2 IEF F C
the rim of the disc. The insect now moves along a diameter of (b) 2 I AC = IEF D
the disc to reach its other end. During the journey of the insect,
the angular speed of the disc (c) I AD = 3 IEF O
(a) remains unchanged (d) I AC = I EF
(b) continuously decreases A E B
(c) continuously increases
(2007)
(d) first increases and then decreases (2011) 9. Angular momentum of the particle rotating with a central force
5. A thin uniform rod of length l and mass m is swinging freely is constant due to
about a horizontal axis passing through its end. Its maximum (a) constant torque (b) constant force
angular speed is w. Its centre of mass rises to a maximum (c) constant linear momentum
height of (d) zero torque
(a) 1 l 2w 2 (b) 1 l w (c) 1 l 2w 2 (d) 1 l 2w 2 (2007)
3 g 6 g 2 g 6 g
(2009) 10. A round uniform body of radius R, mass M and moment of
6. A thin rod of length L is lying along the xaxis with its ends inertia I rolls down (without slipping) an inclined plane making
at x = 0 and x = L. Its linear density (mass/length) varies with an angle q with the horizontal. Then its acceleration is
Rotational Motion 21
g sin q g sin q 17. A body A of mass M while falling vertically downwards
(a) 1 - MR2 / I (b) 1 + I / MR2
under gravity breaks into two parts; a body B of mass 1
g sin q g sin q M
(c) 1 + MR2 / I (d) 1 - I / MR2 3
and body C of mass 2 M . The center of mass of bodies B
(2007)
3
and C taken together shifts compared to that of body A towards
11. A circular disc of radius R is removed from a bigger circular (a) body C (b) body B
disc of radius 2R such that the circumferences of the discs (c) depends on height of breaking
coincide. The centre of mass of the new disc is a/R from the (d) does not shift (2005)
centre of the bigger disc. The value of a is 18. The moment of inertia of a uniform semicircular disc of mass
(a) 1/4 (b) 1/3 (c) 1/2 (d) 1/6. M and radius r about a line perpendicular to the plane of the
(2007)
disc through the center is
12. Four point masses, each of value m, are placed at the corners (a) Mr 2 (b) 1 Mr 2 (c) 1 Mr 2 (d) 2 Mr2
of a square ABCD of side l. The moment of inertia of this 2 4 5
system about an axis through A and parallel to BD is (2005)
(a) ml 2 (b) 2ml 2 (c) 3 ml 2 (d) 3ml 2 .
(2006) 19. One solid sphere A and another hollow sphere B are of same
mass and same outer radii. Their moment of inertia about
13. A thin circular ring of mass m and radius R is rotating about their diameters are respectively I A and IB such that
its axis with a constant angular velocity w. Two objects each (a) I A = I B (b) IA > IB
of mass M are attached gently to the opposite ends of a (c) IA < I B (d) IA / IB = dA / dB
diameter of the ring. The ring now rotates with an angular where dA and dB are their densities. (2004)
velocity w¢ = 20. A solid sphere is rotating in free space. If the radius of the
wm w(m + 2M ) sphere is increased keeping mass same which one of the
(a) (m + 2M ) (b) m
following will not be affected?
w(m - 2M ) wm
(c) (m + 2M ) (d) (m + M ) . (2006) (a) moment of inertia (b) angular momentum
(c) angular velocity (d) rotational kinetic energy.
14. A force of - Fkˆ acts on O, the origin of the coordinate (2004)
system. The torque about the point (1, –1) is r
21. Let F rr bea ntdh eT r fobrec eth ae cttoinrqgu eo no f at hpisa rftoircclee ahbaovuint gth ep oosriitgiionn.
z vector
(a) -F (iˆ - ˆj )
(b) F (iˆ - ˆj ) (Tah)e nrr ×Tr r r rr × r rr
0 and F ×T rr × Tr 0 and F r×T r= 0
(c) -F (iˆ + ˆj ) O y rr × Tr = r r ¹ 0 (b) T ¹ 0 and F ×T = 0
0 and F ×T (d) =
(d) F (iˆ + ˆj ) . (c) ¹ ¹ 0
x (2006) (2003)
15. Consider a two particle system with particles having masses 22. A particle performing uniform circular motion has angular
m 1 and m2 . If the first particle is pushed towards the centre
of mass through a distance d, by what distance should the momentum L. If its angular frequency is doubled and its
second particle be moved, so as to keep the centre of mass
at the same position? kinetic energy halved, then the new angular momentum is
(a) L /4 (b) 2L (c) 4L (d) L /2.
(2003)
(a) d (b) m2 d (c) m1 d (d) m1 d . 23. A circular disc X of radius R is made from an iron plate of
m1 m1 + m2 m 2
thickness t, and another disc Y of radius 4R is made from an
(2006)
iron plate of thickness t/4. Then the relation between the
16. A T shaped object with dimensions l moment of inertia IX and I Y is
shown in the figure, isr lying on a A
smooth floor. A force F is applied B (a) IY = 32IX (b) I Y = 16IX
at the point P parallel to AB, such rP 2l
that the object has only the F (c) I Y = IX (d) IY = 64I X . (2003)
translational motion without
rotation. Find the location of P with C 24. A particle of mass m L C
respect to C. moves along line PC with
velocity v as shown. What P r
(a) 4 l (b) l (c) 3 l (d) 3 l is the angular momentum
3 4 2 of the particle about P? lO (d) zero.
(a) mvL (b) mvl (2002)
(c) mvr
(2005)
22 JEE MAIN CHAPTERWISE EXPLORER
25. Moment of inertia of a circular wire of mass M and radius R 27. Initial angular velocity of a circular disc of mass M is w 1.
Then two small spheres of mass m are attached gently to two
about its diameter is
(a) MR2 / 2 (b) MR 2 diametrically opposite points on the edge of the disc. What
(c) 2MR 2 (d) MR 2 /4. is the final angular velocity of the disc?
(2002) ( ) (a) M+ m w1 M( ) (b) + m w1
M m
26. A solid sphere, a hollow sphere and a ring are released from
( ) (c) ( ) (d)
top of an inclined plane (frictionless) so that they slide down M w1 M w1 . (2002)
M + 4 m M + 2 m
the plane. Then maximum acceleration down the plane is for 28. Two identical particles move towards each other with velocity
(no rolling) 2v and v respectively. The velocity of centre of mass is
(a) solid sphere (b) hollow sphere (a) v (b) v/3 (c) v/2 (d) zero.
(c) ring (d) all same. (2002) (2002)
Answer Key
1. (d) 2. (b) 3. (c) 4. (d) 5. (d) 6. (b)
7. (a) 8. (d) 9. (d) 10. (b) 11. (*) 12. (d)
13. (a) 14. (d) 15. (d) 16. (a) 17. (d) 18. (b)
19. (c) 20. (b) 21. (d) 22. (a) 23. (d) 24. (d)
25. (a) 26. (d) 27. (c) 28. (c)
Rotational Motion 23
1. (d): a = t ...(ii)
I
Here, t = T × R
According to law of conservation at point of contact, and I = 1 mR2 (For circular disc)
mr 2w 0 = mvr + mr 2w 2 (Using (ii))
( ) = mvr + mr 2 v \ T = mRa
r 2
mRa
mg - 2
mr 2w 0 = mvr + mvr Therefore, a = m (Using (i))
mr 2w 0 = 2mvr
ma ( ) Q a
or v = rw0 ma = mg - 2 a = R
2
2 g
2. (b) : Torque exerted on pulley t = FR \ a = 3
or a= FR a( ) Q = t 4. (d)
I I
5. (d) : The uniform rod of length l and mass m is swinging
Here, F = (20t – 5t2 ), R = 2 m, I = 10 kg m 2 about an axis passing through the end.
(20t - 5t 2 ) ´ 2 When the centre of mass is raised through
10
\ a = h, the increase in potential energy is mgh. q
This is equal to the kinetic energy h
a = (4t – t 2 ) or d w = (4t - t 2 ) dt = 12 Iw 2 .
dt
2h
dw = (4t – t 2 )dt
( ) Þ 1 l 2 l 2 × w2
2t 2 t 3 mgh = 2 m 3 × w 2 . \ h = 6 g .
3
On integrating, w = -
At t = 6 s, w = 0 L0ò çèæ k × xn × dx ÷øöx
Ln
dq t 3 6. (b) : xC .M =
w = dt = 2t 2 - 3 Lò0 Lkn × xn × dx
( ) dq = 2t 2 - t 3 dt L
3
ò xn +1d x
2t 3 - 1t 24 Ln +2 × (n + 1)
On integration, q = 3 Þ xC .M = 0 = n + 2 Ln +1
L
ò xn dx
At, t = 6 s, q = 36 rad
36 0
2 p
2pn = 36 Þ n = < 6 L( n + 1)
(n + 2)
3. (c) : The free body diagram of pulley and mass Þ xC .M =
The variation of the centre of mass with x is given by
R dx = L ïíì (n + 2)1 - (n + 1) ïü L
dn îï (n + 2)2 ý + 2) 2
= (n
ïþ
T If the rod has the same density as at x = 0 i.e., n = 0, therefore
uniform, the centre of mass would have been at L/2. As the
mg m a density increases with length, the centre of mass shifts towards
the right. Therefore it can only be (b).
mg – T = ma mg
7. (a) : For a rectangular sheet moment of inertia passing through
\ a = mg - T ...(i) O, perpendicular to the plate is
m
As per question, pulley to be consider as a circular disc.
\ Angular acceleration of disc
24 JEE MAIN CHAPTERWISE EXPLORER
I0 = M æ a2 + b 2 ö a 12. (d) : AO cos 45° = l
çè 12 ÷ø b O 2
axis
for square plate it is Ma 2 . \ AO ´ 1 = l A B
6 2 2 C
A a/2 B 45° l
2
r= a2 + a2 = a . \ r 2 = a 2 a/2 r or AO = l
4 4 2 2 D C 2 O
I = I D + I B + I C D
l / 2
2 ml 2 æ 2
2 çè l
\ I about B parallel to the axis through O is or I = + m 2 l ö
2 ÷ø
Ma2 Ma2 4 Ma 2
Io + Md 2 = 6 + 2 = 6 I = 2ml2 + 4 ml 2
2 2
I = 23 Ma2 or I = 6 ml 2 = 3ml 2 .
2
8. (d) : By perpendicular axes theorem,
I EF = M a2 + b2 = M ( a2 + a 2 ) = M 2 a 2 13. (a) : Angular momentum is conserved
12 12 12
\ L 1 = L 2
M (2a2) + M (2a 2 ) = Ma 2 . \ mR2 w = (mR2 + 2 MR2 ) w¢ = R2 (m + 2M) w¢
I z = 12 12 3
w¢ = mw .
By perpendicular axes theorem, or m + 2 M
I AC + IBD = I z Þ I AC = I z = Ma 2 14. (d) : TForrq=u-eF kˆtr , = rr ´ r
2 6 F
rr = iˆ – ˆj
By the same theorem IEF = I z = Ma 2
2 6 iˆ ˆj kˆ
r 1 -1 0
\ I AC = I EF. \ rr ´ F =
9. (d) : Central forces passes through axis of rotation so torque 0 0 -F
is zero.
If no external torque is acting on a particle, the angular = iˆF – ˆj(-F ) = F (iˆ + ˆj ).
momentum of a particle is constant. 15. (d) : Let m 2 be moved by x so as to keep the centre of mass
at the same position
10. (b) : Acceleration of a uniform body of radius R and mass M
and moment of inertia I rolls down (without slipping) an \ m1 d + m2 (–x) = 0
inclined plane making an angle q with the horizontal is given or m 1 d = m 2 x or x = m1 d .
m2
by g sin q
a = I 16. (a) : It is a case of translation motion without rotation. The
MR
1 + 2 . force should act at the centre of mass
11. (*) :(M¢ + m) = M = p (2R) 2×s 2R Yc m = (m´ 2l) + (2m ´ l ) = 43 l .
where s = mass per unit area m + 2m
M¢ O m
m = sR2 ×s, M¢ = 3pR 2s O¢ x R 17. (d) : The centre of mass of bodies B and C taken together
does not shift as no external force is applied horizontally.
3p R2s × x+ pR2 s × R = 0
M
(Mass of semicircular disc) ´ r 2
Because for the full disc, the centre of mass is at the centre O. 18. (b) : I = 2
Þ x = - R3 = aR. or I = Mr 2 .
2
\ | a | = -1 . 19. (c) : For solid sphere, I A = 52 MR2
3
The centre of mass is at R/3 to the left on the diameter of the
original disc. For hollow sphere, I B = 23 MR2
The question should be at a distance aR and not I A 2 MR 2
I B 5
a/R. \ = ´ 3 = 3
2 MR2 5
None of the option is correct.
or IA < IB .
Rotational Motion 25
20. (b) : Free space implies that no external torque is operating I X = pR4 st ´ 1 = 1
I Y 2 32 pR4 st 64
on the sphere. Internal changes are responsible for increase \
in radius of sphere. Here the law of conservation of angular
momentum applies to the system. \ IY = 64 I X .
rrr
21. (d) : Q T =r´F 24. (d) : The particle moves with linear velocity v along line
r r PC. The line of motion is through P.
\ r × T = r × (rr ´ F ) = 0 Hence angular momentum is zero.
r r
r × r = r × (rr ´ r ) = 0 . 25. (a) : A circular wire behaves like a ring
Also F T F F M.I. about its diameter = MR2 .
2
22. (a) : Angular momentum L = Iw
26. (d) : The bodies slide along inclined plane. They do not
Rotational kinetic energy ( K ) = 1 I w2 roll. Acceleration for each body down the plane = gsinq.
2 It is the same for each body.
\ L = Iw´2 = 2 Þ L = 2 K
K I w2 w w
or L1 = K1 ´ w2 = 2 ´ 2 = 4 27. (c) : Angular momentum of the system is conserved
L2 K2 w1
1 2 2 mR2w + 21 MR 2 w
\ 2 MR w1 =
L1
\ L2 = 4 = L4 . or Mw1 = (4m + M)w
23. (d) : Mass of disc X = (pR2 t) s where s = density or w = M w1 .
M + 4 m
\ I X = MR2 = (pR2ts) R 2 = pR4 st
2 2 2
m1v1 + m2v 2
(Mass)( 4 R )2 p(4R ) 2 t s ´ 16 R2 28. (c) : vc = m1 + m2
2 2 4
Similarly, IY = = m(2v) + m(-v ) = v .
m + m 2
or vc =
or I Y = 32pR4 t s
26 JEE MAIN CHAPTERWISE EXPLORER
CHAPTER GRAVITATION
6
1. What is the minimum energy required to launch a satellite of 6. A planet in a distant solar system is 10 times more massive
mass m from the surface of a planet of mass M and radius R than the earth and its radius is 10 times smaller. Given that the
in a circular orbit at an altitude of 2R? escape velocity from the earth is 11 km s –1, the escape velocity
GmM 5G mM 2 GmM GmM from the surface of the planet would be
3 R 6 R 3 R 2 R
(a) (b) (c) (d) (a) 0.11 km s –1 (b) 1.1 km s –1
(c) 11 km s –1 (d) 110 km s– 1
(2013) (2008)
7. Average density of the earth
2. The mass of a spaceship is 1000 kg. It is to be launched from (a) is directly proportional to g
the earth’s surface out into free space. The value of g and R (b) is inversely proportional to g
(radius of earth) are 10 m/s2 and 6400 km respectively. The (c) does not depend on g
required energy for this work will be (d) is a complex function of g (2005)
(a) 6.4 × 10 8 Joules (b) 6.4 × 109 Joules 8. A particle of mass 10 g is kept on the surface of a uniform
(c) 6.4 × 10 10 Joules
(d) 6.4 × 101 1 Joules (2012) sphere of mass 100 kg and radius 10 cm. Find the work to be
3. Two bodies of masses m and 4m are placed at a distance r. The done agains t the gravitational force between them to take the
gravitational potential at a point on the line joining them where particle far away from the sphere.
the gravitational field is zero is (you may take G = 6.67 ´ 10– 11 Nm2 /kg 2)
(a) zero (b) - 4Grm (c) - 6Grm (d) - 9Grm (a) 6.67 ´ 10– 9 J (b) 6.67 ´ 10– 10 J (2005)
(c) 13.34 ´ 10 –10 J (d) 3.33 ´ 10– 10 J
(2011) 9. The change in the value of g at a height h above the surface
4. The height at which the acceleration due to gravity becomes of the earth is the same as at a depth d below the surface of
g/9 (where g = the acceleration due to gravity on the surface earth. When both d and h are much smaller than the radius of
of the earth) in terms of R, the radius of the earth is earth, then which of the following is correct?
R (a) d = 2h (b) d = h
(b) 2
(a) 2R (c) R/2 (d) 2R (c) d = h/2 (d) d = 3h/2 (2005)
(2009)
10. Suppose the gravitational force varies inversely as the n th power
5. Directions : The following question contains statement1 and of distance. Then the time period of a planet in circular orbit
statement2. Of the four choices given, choose the one that of radius R around the sun will be proportional to
bes t describes the two statements.
(a) Statement1 is true, statement2 is false. ( ) ( ) (a) ( ) (d)
(b) Statement1 is false, statement2 is true. n +1 (b) n -1 (c) R n R n - 2 .
2
R 2 R 2
(2004)
(c) Statement1 is true, statement2 is true; statement2 is a 11. If g is the acceleration due to gravity on the earth’s surface, the
correct explanation for statement1. gain in the potential energy of an object of mass m raised from
the surface of the earth to a height equal to the radius R of the
(d) Statement1 is true, statement2 is true; earth is
statement2 is not a correct explanation for
statement1. (a) 2 mgR (b) 1 mgR (c) 1 mgR (d) mgR.
2 4 (2004)
Statement1 : For a mass M kept at the centre of a cube of side
a, the flux of gravitational field passing through its sides is
4pGM. 12. The time period of an earth satellite in circular orbit is
independent of
Statement2 : If the direction of a field due to a point source (a) the mass of the satellite
(b) radius of its orbit
is radial and its dependence on the distance r from the source (c) both the mass and radius of the orbit
(d) neither the mass of the satellite nor the radius of its orbit.
is given as 1/r 2, its flux through a closed surface depends only (2004)
on the strength of the source enclosed by the surface and not
on the size or shape of the surface. (2008)
Gravitation 27
13. A satellite of mass m revolves around the earth of radius R 16. The time period of a satellite of earth is 5 hour. If the separation
between the earth and the satellite is increased to 4 times the
at a height x from its surface. If g is the acceleration due to previous value, the new time period will become
gravity on the surface of the earth, the orbital speed of the
satellite is (a) 10 hour (b) 80 hour
(c) 40 hour (d) 20 hour.
gR (2003)
(b) R - x
(a) gx 17. The escape velocity of a body depends upon mass as
(c) gR 2 æ gR 2 ö1/ 2 (a) m 0 (b) m 1 (c) m 2 (d) m3 .
R + x
(d) ç R + x ÷ . (2004) (2002)
è ø
18. The kinetic energy needed to project a body of mass m from
14. The escape velocity for a body projected vertically upwards the earth surface (radius R) to infinityis
from the surface of earth is 11 km/s. If the body is projected
at an angle of 45° with the vertical, the escape velocity will be (a) mgR/2 (b) 2mgR
(c) mgR (d) mgR/4. (2002)
(a) 11 2 km/s (b) 22 km/s 19. Energy required to move a body of mass m from an orbit of
(c) 11 km/s (d) 11 / 2 m/s. (2003) radius 2R to 3R is
(a) GMm/12R2 (b) GMm/3R2
15. Two spherical bodies of mass M and 5M and radii R and 2R (c) GMm/8R (d) GMm/6R. (2002)
respectively are released in free space with initial separation 20. If suddenly the gravitational force of attraction between Earth
between their centres equal to 12R. If they attract each other and a satellite revolving around it becomes zero, then the
due to gravitational force only, then the distance covered by satellite will
the smaller body just before collision is (a) continue to move in its orbit with same velocity
(a) 2.5R (b) 4.5R (c) 7.5R (d) 1.5R. (b) move tangentially to the original orbit in the same velocity
(2003) (c) become stationary in its orbit
(d) move towards the earth. (2002)
Answer Key
1. (b) 2. (c) 3. (d) 4. (a) 5. (c) 6. (d)
7. (a) 8. (b) 9. (a) 10. (a) 11. (b) 12. (a)
13. (d) 14. (c) 15. (c) 16. (c) 17. (a) 18. (c)
19. (d) 20. (b)
28 JEE MAIN CHAPTERWISE EXPLORER
1. (b) : Energy of the satellite on the surface of the planet is 5. (c) : Let A be the Gaussian surface enclosing a spherical
( ) Ei = KE + PE = 0 + cEErrh a×=4rg p4erp 2eQQ 0= ..rr eQ20
- GMm = - GMm Flux f = E × 4p r B
R R A
If v is the velocity of the satellite at a distance 2R from the Q +
r
surface of the planet, then total energy of the satellite is
( ) E f = 12 mv 2 +
- GMm 2 = Q
(R + 2R) e0
= 12 m çèæ GM ö 2 GMm Every line passing through A, has B
(R + 2R) ø÷ 3 R
- to pass through B, whether B is a A
= 1 GMm - GMm = - GMm cube or any surface. It is only for M
2 3R 3R 6 R Gaussian surface, the lines of field r
should be normal. Assuming the
\ Minimum energy required to launch the satellite is
mass is a point mass.
gr ,
( ) DE = Ef
- Ei = - GMm - - GMR m gravitational field = - GM
6 R r 2
= - GMm + GMm = 5G Mm Flux fg =| gr × 4pr 2 | = 4p r 2 × GM = 4p GM .
6R R 6 R r 2
GMm Here B is a cube. As explained earlier, whatever be the shape,
R
2. (c) : Energy required = all the lines passing through A are passing through B, although
m ( ) Q GM all the lines are not normal.
R R 2
= gR2 ´ g = Statement2 is correct because when the shape of the earth
= mgR is spherical, area of the Gaussian surface is 4pr 2. This ensures
= 1000 × 10 × 6400 × 10 3 inverse square law.
= 64 × 10 9 J = 6.4 × 10 10 J 6. (d) : ve scape = 2G M for the earth
R
3. (d) : Let x be the distance of the point P from the mass m
where gravitational field is zero. ve = 11 kms –1
Mass of the planet = 10 Me , Radius of the planet = R/10.
\ ve = 2GM ´10 = 10 ´ 11 = 110 km s -1
R /10
7. (a) : Density = Mass of earth
Volume of earth
G(4m ) 2
(r - x) 2
( ) \ Gm = or (r x = 1 r= M = 3 M ... (i)
x2 - x) 4 (4 / 3)pR3 4 p R3
or x = r ...(i) g = GRM2 ... (ii)
3
Gravitational potential at a point P is R 2
GM
= - Gm - G(4m ) \ r = 3M ´ = 3 or r = 3 g
x (r - x) g 4 p R3 4 pRG 4 pRG
Gm G(4m ) \ Average density is directly proportional to g.
( ) ( ) =-r - (Using (i))
3 r
r - 3 8. (b) : Gravitational force F = Gm1m2
R2
3G m - 3G2(4 rm ) - 9 Grm
= - r = \ dW = FdR = G m1m2 dR
R 2
4. (a) : The acceleration due to gravity at a height h from
W ¥ dR Gm1m2 êéë- 1 ù¥ = Gm1m2
the ground is given as g/9. M \ = Gm1m2 R2 = R ûúR R
ò dW ò
GM GM × 91
r2 R2 0 R
= (6.67 ´10–11) ´ (100) ´ (10 ´10–3 )
\ r = 3R r \ Workdone = 10 ´ 10–2
The height above the ground = 6.67 × 10 –10 J.
is 2R.
Gravitation 29
9. (a) : At height, gh = g çæè1 - 2R h ÷øö where h << R or v0 = gR 2 .
R + x
or g – g h = 2 hg or D g h = 2 hg ... (i) 14. (c) : The escape velocity of a body does not depend on the
R R .... (ii) angle of projection from earth.
It is 11 km/sec.
At depth, gd = g çæè1 - d ÷øö where d << R
R
or g – g d = dg 15. (c) : Let the spheres collide after time t, when the smaller
R
sphere covered distance x 1 and bigger sphere covered
or D g d = dg distance x 2 .
R The gravitational force acting between two spheres depends
on the distance which is a variable quantity.
From (i) and (ii), when Dgh = D g d
2hg = dg or d = 2h. The gravitational force, F(x ) = GM ´ 5 M
RR (12R - x) 2
10. (a) : For motion of a planet in circular orbit, Acceleration of smaller body, a1 (x ) = G ´ 5 M
(12R - x) 2
Centripetal force = Gravitational force
\ mR w2 = GRMnm or w = GM Acceleration of bigger body, a2 (x ) = GM
R n +1 (12R - x) 2
( ) \ From equation of motion,
T = 2p = 2 p R n +1 = n +1
2 p x1 = 1 a1 ( x ) t 2 and x2 = 1 a 2 ( x) t 2
R 2 2 2
w GM GM
n +1 Þ x1 = a1 ( x ) = 5 Þ x1 = 5 x2
x2 a 2 (x)
( ) \ T is proportional to R 2 .
11. (b) : Force on object = GMm at x from centre of earth. x1 x2 2R
x 2
R 9R
\ Work done = GxM2 m dx
We know that x1 + x2 = 9 R
\ Work done = 2 R dx
ò x 2 x 1 45 R = 7.5 R
GMm ò 5 \ x1 = 6
x1 + = 9 R
R
\ Potential energy gained Therefore the two spheres collide when the smaller sphere
= GMm éêë- 1 2 R = GMm ´1 covered the distance of 7.5R.
x 2 R
ù 16. (c) : According to Kepler's law T 2 µ r3
ûúR
\ Gain in P.E. 2 3 3
= 1 mR æ GM ÷öø = 12 mgR éêëQ g = GM ù . \ æ T1 ö = æ r1 ö = çèæ 1 ÷ö = 1 or T1 = 81
2 èç R2 R2 úû èç T2 ÷ø èç r2 ø÷ 4 ø 64 T2
12. (a) : For a satellite or T 2 = 8T1 = 8 × 5 = 40 hour.
Centripetal force = Gravitational force 17. (a) : Escape velocity = 2 gR = 2G M e
R
\ m R w2 = G mM e where R = r e + h
R 2
Escape velocity does not depend on mass of body which
GM e GM e escapes or it depends on m0 .
R3 (re + h) 3
or w= = 18. (c) : Escape velocity ve = 2 gR
\ Kinetic energy
2 p (re + h ) 3
\ T = w = 2 p GM e = 1 mve 2 = 1 m ´ 2 gR = mgR .
2 2
\ T is independent of mass (m) of satellite.
13. (d) : For a satellite 19. (d) : Energy = (P.E.) 3R – (P.E.) 2R
centripetal force = Gravitational force
= - GmM - çèæ - GmM ÷öø = + G6m RM .
3R 2 R
\ mv0 2 = GMm 20. (b) : The centripetal and centrifugal forces disappear, the
( R + x ) ( R + x) 2
satellite has the tangential velocity and it will move in a
v0 2 GM gR 2 ëêéQ GM ù straight line.
(R + x) (R + x) R2 úû
or = = g = Compare Lorentzian force on charges in the cyclotron.
30 JEE MAIN CHAPTERWISE EXPLORER
CHAPTER PROPERTIES OF SOLIDS
AND LIQUIDS
7
1. Assume that a drop of liquid evaporates by decrease in its (c) log e( q – q 0 )
surface energy, so that its temperature remains unchanged. log e( q – q 0 )
What should be the minimum radius of the drop for this to be (d)
possible? The surface tension is T, density of liquid is r and
L is its latent heat of vaporization.
2T (b) rL (c) T (d) T t 0
(a) r L T r L rL t
(2012)
(2013) 5. A wooden wheel of radius R is made of two
2. A uniform cylinder of length L and mass M having cross semicircular parts (see figure). The two parts
sectional area A is suspended, with its length vertical, from a are held together by a ring made of a metal R
strip of cross sectional area S and length L.
fixed point by a massless spring, such that it is half submerged
in a liquid of density s at equilibrium position. The extension L is slightly less than 2pR. To fit the ring on
x0 of the spring when it is in equilibrium is the wheel, it is heated so that its temperature
Mg ( ) (a) LAs Mg rises by DT and it just steps over the wheel.
k 1 + M (b) k
Mg ( ) (c) LAs ( ) (d)Mg LAs As it cools down to surrounding temperature, it presses the
k M k 2 M
1 - 1 - (2013) semicircular parts together. If the coefficient of linear expansion
of the metal is a, and its Young’s modulus is Y, the force that
3. If a piece of metal is heated to temperature q and then allowed one part of the wheel applies on the other part is
to cool in a room which is at temperature q0 , the graph between (a) SYaDT (b) pSYaDT
the temperature T of the metal and time t will be closed to (c) 2SYaDT (d) 2pSYaDT (2012)
(a) (b) 6. A thin liquid film formed between a FILM
(c) (d) Ushaped wire and a light slider supports
a weight of 1.5 × 10 –2 N (see figure). w
The length of the slider is 30 cm and its
weight negligible. The surface tension (2012)
of the liquid film is
(a) 0.1 Nm– 1
(b) 0.05 Nm– 1
(c) 0.025 Nm– 1
(d) 0.0125 Nm –1
7. Water is flowing continuously from a tap having an internal
(2013) diameter 8 × 10– 3 m. The water velocity as it leaves the tap is
0.4 m s –1 . The diameter of the water stream at a distance
4. A liquid in a beaker has temperature q(t) at time t and q 0 is 2 × 10 –1 m below the tap is close to
temperature of surroundings, then according to Newton’s law (a) 5.0 × 10– 3 m (b) 7.5 × 10– 3 m
of cooling the correct graph between loge (q – q0 ) and t is (c) 9.6 × 10– 3 m (d) 3.6 × 10– 3 m
log e( q – q 0 ) (2011)
log e( q – q 0 )
(a) (b) 8. Work done in increasing the size of a soap bubble from a
radius of 3 cm to 5 cm is nearly (Surface tension of soap
t t solution = 0.03 N m– 1 )
(a) 4p mJ (b) 0.2p mJ (c) 2p mJ (d) 0.4p mJ
(2011)
Properties of Solids and Liquids 31
9. The potential energy function for the force between two atoms A B
B
in a diatomic molecule is approximately given by (a) B
U (x ) = a - b , where a and b are constants and x is the A
x12 x6
(b)
distance between the atoms. If the dissociation energy of the
A
molecule is D = [U(x = ¥) – Ua t equilibrium] , D is
(c)
(a) b 2 (b) b 2 (c) b 2 (d) b 2
6 a 2a 12 a 4 a
(2010)
10. A ball is made of a material of density r where r oil < r < r water
with ro il and rw ater representing the densities of oil and water,
respectively. The oil and water are immiscible. If the above
ball is in equilibrium in a mixture of this oil and water, which
of the following pictures represents its equilibrium position?
Oil Water A B
(a) (b) (d)
Water Oil
Oil Water (2008)
(c) (d) (2010) 14. A spherical solid ball of volume V is made of a material of
density r1 . It is falling through a liquid of density r2 (r2 < r1 ).
Water Oil Assume that the liquid applies a viscous force on the ball that
is proportional to the square of its speed v, i.e., Fv iscous = –kv 2
11. Two wires are made of the same material and have the same (k > 0). The terminal speed of the ball is
volume. However wire 1 has crosssectional area A and wire
2 has crosssectional area 3A. If the length of wire 1 increases
by Dx on applying force F, how much force is needed to (a) Vg (r1 - r 2 ) (b) Vg (r1 - r 2 )
k k
stretch wire 2 by the same amount?
(a) F (b) 4F (c) 6F (d) 9F (c) Vg r1 (d) Vg r1
k k
(2009) (2008)
12. A long metallic bar is carrying heat from one of its ends to 15. A jar is filled with two nonmixing liquids 1 and 2 having
the other end under steadystate. The variation of temperature
q along the length x of the bar from its hot end is best described densities r 1 and r2 respectively. A solid ball, made of a material
by which of the following figures? of density r3 , is dropped in the jar. It comes to equilibrium
in the position shown in the figure. Which of the following
q q
is true for r1 , r 2 and r3 ?
(a) (b) (a) r 1 < r 3 < r2 r3
(b) r3 < r1 < r2
x x (c) r 1 > r 3 > r2
q q (d) r1 < r 2 < r3 (2008)
(c) (d) 16. One end of a thermally insulated rod is kept at a temperature
x x T1 and the other at T 2. The rod is composed of two sections
of lengths l1 and l2 and thermal conductivities K 1 and K2
(2009) respectively. The temperature at the interface of the two
13. A capillary tube (A) is dipped in water. Another identical tube sections is
(B) is dipped in a soapwater solution. Which of the following T 1 l 1 l 2 T 2
shows the relative nature of the liquid columns in the two
tubes? K 1 K 2
32 JEE MAIN CHAPTERWISE EXPLORER
(K1l1T1 + K2l2T2 ) (K2l2T1 + K1l1T2 ) 23. If two soap bubbles of different radii are connected by a tube,
(K1l1 + K2l2 ) (K1l1 + K2l2 )
(a) (b) (a) air flows from the bigger bubble to the smaller bubble
till the sizes become equal
(c) (K2l1T1 + K1l2T2 ) (d) (K1l2T1 + K2l1T2 ) (2007) (b) air flows from the bigger bubble to the smaller bubble
(K2l1 + K1l2 ) (K1l2 + K2l1 )
till the sizes are interchanged
17. A wire elongates by l mm when a load W is hanged from it. (c) air flows from the smaller bubble to the bigger
If the wire goes over a pulley and two weights W each are (d) there is no flow of air. (2004)
hung at the two ends, the elongation of the wire will be 24. Spherical balls of radius R are falling in a viscous fluid of
(in mm) viscosity h with a velocity v. The retarding viscous force
(a) l/2 (b) l (c) 2l (d) zero. acting on the spherical ball is
(2006) (a) directly proportional to R but inversely proportional to v
18. If the terminal speed of a sphere of gold (density = 19.5 kg/m3 ) (b) directly proportional to both radius R and velocity v
is 0.2 m/s in a viscous liquid (density = 1.5 kg/m3 ) find the (c) inversely proportional to both radius R and velocity v
terminal speed of a sphere of silver (density 10.5 kg/m3 ) of the (d) inversely proportional to R but directly proportional to
same size in the same liquid velocity v. (2004)
(a) 0.2 m/s (b) 0.4 m/s
(c) 0.133 m/s (d) 0.1 m/s. (2006) 25. A wire fixed at the upper end stretches by length l by applying
19. Assuming the sun to be a spherical body of radius R at a a force F. The work done in stretching is
temperature of T K, evaluate the total radiant power, incident
on earth, at a distance r from the sun. (a) F/2l (b) Fl (c) 2Fl (d) Fl/2.
(2004)
(a) R2s T 4 (b) 4 pr02 R2sT 4 26. The temperature of the two outer surfaces of a composite
r 2 r 2
slab, consisting of two materials having coefficients of thermal
pr02 R2s T 4 r02 R 2sT 4 conductivity K and 2K and thickness x and 4x, respectively
r 2 4 pr 2
(c) (d) . are T 2 and T 1 (T2 > T 1 ). The rate of heat transfer through the
slab, in a steady state is æ A(T2 - T1 ) K ÷ö f , with f equal to
çè x ø
where r0 is the radius of the earth and s is Stefan's constant.
(2006) x 4x
20. If S is stress and Y is Young’s modulus of material of a wire,
the energy stored in the wire per unit volume is
(a) 2Y/S (b) S/2Y (c) 2S 2 Y S 2 T 2 K 2K T1
(d)
2Y
(2005)
21. A 20 cm long capillary tube is dipped in water. The water (a) 1 (b) 1/2
(c) 2/3 (d) 1/3.
rises up to 8 cm. If the entire arrangement is put in a freely (2004)
falling elevator the length of water column in the capillary 27. A radiation of energy E falls normally on a perfectly reflecting
tube will be surface. The momentum transferred to the surface is
(a) 4 cm (b) 20 cm (c) 8 cm (d) 10 cm
(2005) (a) E/c (b) 2E/c
22. The figure shows a system of two concentric spheres of radii (c) Ec (d) E/c2 . (2004)
r1 and r 2 and kept at temperatures T 1 and T 2 , respectively. The 28. If the temperature of the sun were to increase from T to 2T
radial rate of flow of heat in a substance between the two
concentric spheres is proportional to and its radius from R to 2R, then the ratio of the radiant
energy received on earth to what it was previously will be
(a) r1r2 (a) 4 (b) 16
(r2 - r1 )
r1 T1 (c) 32 (d) 64. (2004)
r 2
(b) (r 2 – r1 ) T 2 29. A wire suspended vertically from one of its ends is stretched
(r2 - r1 ) by attaching a weight of 200 N to the lower end. The weight
r1r2
(c) stretches the wire by 1 mm. Then the elastic energy stored in
the wire is
æ r2 ö (a) 0.2 J (b) 10 J (c) 20 J (d) 0.1 J.
ln ç r1 ÷
(d) ø (2005) (2003)
è
Properties of Solids and Liquids 33
30. According to Newton's law of cooling, the rate of cooling of on the side wall of the cylinder near its bottom is
a body is proportional to (Dq) n , where Dq is the difference of (a) 10 (b) 20 (c) 25.5 (d) 5.
the temperature of the body and the surroundings, and n is (2002)
equal to 33. Two spheres of the same material have radii 1 m and 4 m and
(a) two (b) three temperatures 4000 K and 2000 K respectively. The ratio of
(c) four (d) one. (2003)
the energy radiated per second by the first sphere to that by
31. The earth radiates in the infrared region of the spectrum. the second is
The wavelength of the maximum intensity of the spectrum is (a) 1 : 1 (b) 16 : 1 (c) 4 : 1 (d) 1 : 9.
correctly given by (2002)
(a) Rayleigh Jeans law
(b) Planck's law of radiation 34. Which of the following is more close to a black body?
(c) Stefan's law of radiation (a) black board paint
(d) Wien's law. (2003) (b) green leaves
32. A cylinder of height 20 m is completely filled with water. (c) black holes
The velocity of efflux of water (in ms– 1) through a small hole
(d) red roses. (2002)
Answer Key
1. (a) 2. (d) 3. (d) 4. (d) 5. (c) 6. (c)
7. (d) 8. (d) 9. (d) 10. (c) 11. (d) 12. (b)
13. (d) 14. (b) 15. (a) 16. (d) 17. (b) 18. (d)
19. (c) 20. (d) 21. (b) 22. (a) 23. (c) 24. (b)
25. (d) 26. (d) 27. (b) 28. (d) 29. (d) 30. (d)
31. (d) 32. (b) 33. (a) 34. (a)
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