49 6.4 CONSERVATION OF ANGULAR MOMENTUM 1. a) A figure skater increases her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3.0 rev s-1. Her initial moment of inertia was 4.6 kg m-2. i. Calculate her final moment of inertia. ii. How does she physically accomplish this change? b) A bicycle wheel of radius 0.32 m has a moment of inertia of 0.035 kg m2 about its axle. When the wheel is lifted from the ground and set spinning at a frequency of 10 Hz it comes to rest after 30 min. What is i. the angular deceleration of the wheel? ii. the average frictional torque acting on it? [Ans: 0.77 kg m2, -0.35 rad s-2, -0.012 N m] 3. A solid cylinder of mass 2.8 kg rotates with a speed of 1500 revolutions per minute. If diameter of cross sectional area of the cylinder is 32.0 cm, a) Calculate the angular momentum of the cylinder. b) How much torque is required to bring the cylinder to rest in 0.7 s. [Ans: 22.45 kg m2 s -1, -3.21N m] SECTION C 1. FIGURE 6.7 FIGURE 6.7 shows in an emergency, a person with a broken forearm ties a strap from his hand to clip on his shoulder. His 1.60-kg forearm remains in a horizontal position and the strap makes an angle of u 5 50.0° with the horizontal. Assume the forearm is uniform, has a length of, 5 0.320 m, assume the biceps muscle is relaxed, and ignore the mass and length of the hand. Find a) the tension in the strap b) the components of the reaction force exerted by the humerus on the forearm.
50 2. a) A pulley system is used to lift a heavy object. If more pulleys are added to the system, will the torque increase, decrease, or stay the same? Explain your answer. b) A screwdriver is used to turn a screw. If more force are applied to the screwdriver, will the torque increase, decrease, or stay the same? Explain your answer. 3. a) Why is it more difficult to balance a pencil on its point than on its eraser? b) A gymnast is performing a somersault. If she tucks her body into a ball, will her moment of inertia increase, decrease, or stay the same? 4. Explain how rotational kinematics is applied to motors, turbines and fylwheels in real-world systems.
51 TOPIC 7 : SIMPLE HARMONIC MOTION AND WAVES 7.1 Kinematics of Simple Harmonic Motion 7.2 Graph of Simple Harmonic Motion 7.3 Period of Simple Harmonic Motion 7.4 Properties of Waves 7.5 Superposition of Waves 7.6 Application of Standing Waves 7.7 Doppler Effect At the end of this chapter, students should be able to: 7.1 KINEMATICS OF SIMPLE HARMONIC MOTION a) Explain SHM b) Apply SHM displacement equation y = Asint c) Use equations : (i) velocity, = = ± √2 − 2 (ii) acceleration, = − 2 = − 2 (iii) kinetic energy, K = 1 2 2 ( 2 − 2 ) (iv) potential energy, U = 1 2 2 2 d) Emphasise the relationship between total SHM energy and amplitude. e) Apply equation of velocity, acceleration, kinetic energy and potential energy for SHM 7.2 GRAPH OF SIMPLE HARMONIC MOTION a) Analyse the following graphs: (i) displacement - time (ii) velocity - time (iii) acceleration - time (iv) energy - displacement 7.3 PERIOD OF SIMPLE HARMONIC MOTION a) Use expression for period of SHM, T for simple pendulum and mass-spring system
52 (i) Simple pendulum oscillation: 2 l T g = (ii) Mass spring oscillation: k m T = 2 7.4 PROPERTIES OF WAVES a) Define wavelength. b) Define and use wave number, 2 k = c) Solve problems related to equation of progressive wave, y x t A t kx ( , ) sins( ) = d) Distinguish between particle vibration velocity and wave propagation velocity. e) Use particle vibration velocity, v A t kx = + cos( ) f) Use wave propagation velocity, v f = g) Analyse the graphs of : (i) displacement-time, y-t. (ii) displacement-distance, y-x. 7.5 SUPERPOSITION OF WAVES a) State the principle of superposition of waves for the constructive and destructive interferences. b) Use the standing wave equation, y A kx t = 2 cos sin c) Compare progressive waves and standing waves. 7.6 APPLICATION OF STANDING WAVES (a) Solve problems related to the fundamental and overtone frequencies for : (i) stretched string, 2 n nv f L = (ii) air columns (open end, 2 n nv f L = and closed end, 4 n nv f L = ). (b) Use wave speed in a stretched string, T v =
53 7.7 DOPPLER EFFECT (a) State Doppler Effect for sound waves. (b) Apply Doppler Effect equation, o o s v v f f v v = for relative motion between source and observer. Limit to stationary observer and moving source, and vice versa.
54 KOLEJ MATRIKULASI KEJURUTERAAN JOHOR KEMENTERIAN PENDIDIKAN MALAYSIA TUTORIAL 7 (SIMPLE HARMONIC MOTION AND WAVES) SECTION A 1. In simple harmonic motion, velocity at equilibrium position is __________. A. Minimum B. Constant C. Maximum D. Zero 2. Which one of the following statements is TRUE when an object performs simple harmonic motion about a central point O? A. The acceleration is always away from O B. The acceleration and velocity are always in opposite directions C. The acceleration and the displacement from O are always in the same direction D. The acceleration is always directed towards O 3. Which one of the following statements concerning the acceleration of an object moving with simple harmonic motion is CORRECT? A. It is constant B. It is at a maximum when the object moves through the centre of the oscillation C. It is zero when the object moves through the centre of the oscillation D. It is zero when the object is at the extremity of the oscillation 4. The frequency of a body moving with simple harmonic motion is doubled. If the amplitude remains the same, which one of the following is also doubled? A. The period B. The total energy C. The maximum velocity D. The maximum acceleration 5. A particle oscillating in simple harmonic motion is A. never in equilibrium because it is in motion B. never in equilibrium because there is always a force C. in equilibrium at the end of its path because its velocity is zero there D. in equilibrium at the centre of its path because the acceleration is zero there
55 6. A simple pendulum and a mass-spring system are taken to the Moon, where the gravitational field strength is less than on Earth. Which line, A to D, CORRECTLY describes the change, if any, in the period when compared with its value on Earth? Period of Pendulum Period of Mass-Spring System A. decrease decrease B. increase increase C no change decrease D increase no change 7. When a mass suspended on a spring is displaced, the system oscillates with simple harmonic motion. Which one of the following statements regarding the energy of the system is INCORRECT? A. The potential energy has a minimum value when the spring is fully compressed or fully extended. B. The kinetic energy has a maximum value at the equilibrium position. C. The sum of the kinetic and potential energies at any time is constant. D. The potential energy has zero value when the mass is at rest. 8. Waves transmit …………… from one place to another. A. Energy B. Mass C. Both D. None 9. The distance between any two consecutive crests or troughs is called …………….. A. Frequency B. Period C. Wavelength D. Phase difference 10. A stationary wave is set up in the air column of a closed pipe. At the closes end of the pipe ………….. A. Always a node informed B. Always an antinode is formed C. Neither node nor antinode is formed D. Sometimes a node and sometimes an antinode is formed 11. The distance between two consecutive nodes is ………… A. λ/2 B. λ/4 C. λ D. 2λ
56 12. The distance between a consecutive node and antinode is A. Λ B. λ/2 C. 2λ D. λ/4 SECTION B 7.1 KINEMATICS OF SIMPLE HARMONIC MOTION 1. The equation of motion for a particle oscillating in SHM is given as, y = 3 sin 2t where y is the displacement in meter and t is the time in second. Determine a) amplitude b) frequency of oscillation c) the position of particle at t = 0.2s (ANS: A = 3 m, f = 0.32 Hz, y = 1.168 m) 2. The displacement, y of a particle varies with time, t is given by y = 4 sin 2πt where y is in cm and t in s. Calculate the a) frequency of the motion. b) velocity of the particle at t = 0.2s c) acceleration of the particle at t= 0. 2s (ANS: f = 1 Hz, v = 7.77 cms-1, a = 150.2 cms-2) 3. An object of mass 3.0 kg executes linear SHM on a smooth horizontal surface at frequency 10 Hz & with amplitude 5.0 cm. Neglect all resistance forces. Determine a) The energy of the system b) The potential & kinetic energy when the displacement of the object is 3.0cm (ANS: E = 14.8 J, U = 5.33 J, K = 9.47 J) 4. A 0.5 kg block connected to a light spring for which the force constant is 20.0 Nm-1 oscillates on a horizontal, frictionless track. a) Calculate the total energy of the system and the maximum speed of the block if the amplitude of the motion is 3.0 cm. b) the kinetic & potential energies of the system when displacement is 2.0 cm. (ANS: E = 9x10-3 J, v = 0.19 ms-1, K = 5x10-3 J, U = 4x10-3 J)
57 7.2 GRAPH OF SIMPLE HARMONIC MOTION 1. The displacement of a particle oscillating in simple harmonic motion is given as, x = 4sin10t where x is measured in cm and t in second. Determine a) the maximum velocity of the particle b) the velocity of the particle when x = 3 cm. c) Sketch a graph of velocity, v against time, t for this particle. (ANS: v = 125.66 cms-1, v = 83.12 cms-1) 2. The equation of motion for a particle oscillating in SHM is given as, y = 5 sin (3t -π/2) where y is the displacement in cm and t in s. Determine a) its amplitude b) period of oscillation c) its displacement at time t = 2s d) maximum velocity e) maximum acceleration f) velocity when its displacement is 4 cm. g) Sketch the graph of displacement against time (ANS: A = 5 cm, T = 2.09 s, y = -4.8 cm, v = 0.15 ms-1, a = 0.45 ms-2, v = 9 cms-1) 7.3 PERIOD OF SIMPLE HARMONIC MOTION 1. A block of a mass 4kg is attached to a spring and undergoes SHM with period of T = 0.25s. The total energy of the system is 3.5 J. a) Calculate the force constant of the spring b) Determine the amplitude of the motion (ANS: k = 2.53x103 Nm-1, A = 0.053 m) 2. A student uses a simple pendulum of length 80.0 cm to determine the gravitational acceleration. If there are 20 oscillations in 35.9 s, find: a) the value of g. b) the period of oscillation if the experiment is done in the moon, where its gravitational field strength is only 1/6 of that of the earth (ANS: g = 9.80 ms-2, T = 4.4 s) 3. A 0.2kg block is attached to a light spring of force constant of 11 Nm-1 on a horizontal frictionless surface. If the block is displaced a distance of 8 cm from its equilibrium position, find a) the amplitude, the angular frequency, the period and the frequency of motion when the block is released. b) the maximum force exerted on the block c) the total mechanical energy of the system
58 d) the maximum speed and maximum acceleration of the block e) the velocity of the block when its displacement is 2cm f) the acceleration of the block when its displacement is 3cm. (ANS: A = 8 cm, ω = 7.42 rads-1, T = 0.85 s, f = 1.17 Hz, F = 0.88 N, E = 0.035 J, v = 0.593 ms-1, a = -4.40 ms-2, v = 0.575 ms-1, a = -1.65 ms-2) 4. A 175g mass is suspended from a vertical spring. When the mass is pulled down a distance of 76 mm and released, the time taken for 25 oscillations is 23 s. Calculate a) the period of oscillation b) the maximum speed of the mass c) total energy of the system. (ANS: T = 0.92 s, v = 0.52 ms-1, E = 0.024 J) 7.4 PROPERTIES OF WAVES 1. Ocean waves with a crest-to-crest distance of 10.0 m can be described by the wave function y(x, t) = (0.800) sin [0.628(vt – x)] where y is in meters and t is in seconds and v = 1.20 m s-1. a) Sketch y-x graph at t = 0. b) Sketch y-x graph at t = 2.00 s. 2. A sinusoidal wave traveling from right to left has an amplitude of 0.20 m, a wavelength of 0.35 m and a frequency of 12.0 Hz. The transverse position of an element of the medium at t = 0, x = 0 is y = – 0.030 m and the element has a positive velocity. a) Sketch y-x graph for the wave at t = 0. b) Calculate i. wave number ii. period iii. angular frequency iv. speed c) Write an expression for the progressive wave function y(x,t). (ANS: k = 17.95 radm-1, T = 83.3 ms, ω = 24π rads-1, v = 4.2 ms-1) 3. A transverse traveling wave on a taut wire has an amplitude of 0.200 mm and a frequency of 500 Hz. It travels with a speed of 196 m s-1. The mass per unit length of this wire is 4.10 g m-1. a) Write an equation of the form y = A sin (t – kx) for this wave, where y in meters and t in seconds. b) Calculate the tension in the wire. (ANS: T = 157.5 N)
59 7.5 SUPERPOSITION OF WAVES 1. Two progressive waves, y1 = 0.075 sin (4πt – πx), and y2 = 0.075 sin (4πt + πx) form a standing wave where y1 and y2 is in meters and t is in seconds. a) Write the expression representing the new wave. b) Calculate the i. smallest value of x that corresponds to a node. ii. distance between two consecutive nodes. (ANS: x1 = 0.5 m, x2 = 1.0 m) 2. Two sinusoidal waves traveling in opposite directions interfere to produce a standing wave where y and x in meters and t in seconds. Calculate the a) wavelength, b) frequency c) speed. (ANS: λ = 15.7 m, f = 31.8 Hz, v = 500 ms-1) 7.6 APPLICATION OF STANDING WAVES 1. The overall length of a piccolo is 32.0 cm. The resonating air column vibrates as in a pipe open at both ends. Calculate the frequency of the lowest note a piccolo can play. (ANS: fo = 517.2 Hz) 2. A steel wire in a piano has a length of 0.70 m and a mass of 4.30×10–3 kg. To what tension must this wire be stretched in order that the fundamental vibration correspond to middle C (fC = 261.6 Hz on the chromatic musical scale)? (ANS: T = 824 N) 3. A variable length air column is placed just below a vibrating wire that is fixed at both ends. The length of the air column open at one end is gradually increased until the first position of resonance is observed at 34.0 cm. The wire is 120 cm long and is vibrating in its third harmonic. Calculate the speed of transverse waves in the wire. (ANS: v = 20 ms-1 ) 4. An organ pipe of length 33 cm is open at one end and closed at the other. By assuming that end correction is negligible, calculate the a) frequencies of the fundamental note and first overtone. b) length of a pipe open at both ends and having fundamental frequency that is equal to the different between the two frequencies calculated in (a).
60 c) A closed pipe can produce a fundamental note at 300 Hz with end correction of 8.00 mm. Calculate the length of the pipe. (ANS: fo = 250.8 Hz, f1 = 752.3 Hz, L = 33 cm, L = 28 cm) 7.7 DOPPLER EFFECT 1. A car is traveling at 50 ms-1 towards a bus which sounds its horn at a frequency of 1 kHz. Calculate the apparent frequency heard by the driver of the car if the bus is stationary. (ANS: fa = 1.15 kHz) 2. A sound source and an observer are located on the same horizontal straight line. The observer hears sound of frequency 510 Hz when the sound source moves towards him. When the sound source passes him and moves away, he hears a sound of frequency 450 Hz. Determine the speed of the sound source. (ANS: vs = 20.7 ms-1) 3. Hearing the siren of an approaching truck, moving with velocity of 80 km h-1 a man pulls over to the side of the road and stop. As the truck approaching, he hears a tone of 460 Hz. Calculate the a) actual frequency sound by the truck. b) apparent frequency heard by the man when the truck passed him. (ANS: f = 429.1 Hz, fa = 402.1 Hz) SECTION C 1. A particle is in simple harmonic motion of amplitude 5 cm and period of 2.4 s. At time t = 0, it is at y = -5 cm. a) Write the equation to represent he variation of displacement y with time t. b) Find the value of t i. when y = -2.5 cm ii. when y = 2.5 cm c) Sketch the graph of y against t and shows the values of t calculated in (b) above. d) Use the graph to find the time taken for the particle to move i. from y = -2.5 cm to y = 2.5 cm ii. from y = 2.5 cm and back but in the opposite direction. (ANS: t = 0.40 s, t = 0.80 s OR t = 1.60 s)
61 2. FIGURE 7.1 A string that is 150 cm long has mass of 1.92 g. As seen in FIGURE 7.1 above, a block of mass m, stretches the string as it runs over a support at Q while being attached to a sinusoidal oscillator at P. There are 3000 oscillations are performed by the oscillator per minute. P and Q are separated, L by 120 cm. In order to assure that this kind of standing wave might be formed, determine the mass of the block, m. (ANS: m = 117 g)
62 TOPIC 8 : PHYSICS OF MATTERS 8.1 STRESS AND STRAIN 8.2 YOUNG’S MODULUS 8.3 HYDROSTATIC PRESSURE 8.4 FLUID DYNAMICS 8.5 VISCOSITY 8.6 HEAT CONDUCTION 8.7 THERMAL EXPANSION At the end of this topic, students should be able to: 8.1 STRESS AND STRAIN a) Distinguish between stress and strain for tensile and compression force. b) Analyse the graph of stress-strain for a metal under tension. c) Explain elastic and plastic deformations. d) Analyse graph of force-elongation for brittle and ductile materials. 8.2 YOUNG’S MODULUS a) Define and use Young’s Modulus. b) Apply strain energy from force-elongation graph. c) Apply strain energy per unit volume from stress-strain graph. 8.3 HYDROSTATIC PRESSURE a) Express and use atmospheric pressure, gauge pressure and absolute pressure. 8.4 FLUID DYNAMICS a) Illustrate fluid flow (remarks: laminar flow only). b) Explain continuity principle and Bernoulli principle. c) Use continuity and Bernoulli’s equations. 8.5 VISCOSITY a) Explain viscosity. b) State and use Stoke’s law. c) Explain the terminal velocity in fluid using graph of velocity-time. 8.6 HEAT CONDUCTION a) Define heat conduction. b) Solve problems related to rate of heat transfer through a cross-sectional area (*maximum two insulated objects in series).
63 c) Analyse graphs of temperature-distance for heat conduction through insulated and non-insulated rods (*maximum two rods in series). 8.7 THERMAL EXPANSION a) Define coefficient of linear expansion, area expansion and volume expansion. b) Solve problems related to thermal expansion of linear, area and volume (*include expansion of liquid in a container).
64 KOLEJ MATRIKULASI KEJURUTERAAN JOHOR KEMENTERIAN PENDIDIKAN MALAYSIA TUTORIAL 8 (PHYSICS OF MATTERS) SECTION A 1. A wire is stretched until it undergoes plastic deformation. Which of the following statements is TRUE of plastic deformation? A. The bonds between atoms are all broken. B. The extension is directly proportional to force. C. The atomic planes in the wire slide over each other. D. The wire returns to its original length when the force is removed. 2. Two copper wires have different length. Which of the following statements is TRUE about the Young’s Modulus of these wires? A. The longer copper wire has greater Young’s Modulus. B. The shorter copper wire has greater Young’s Modulus. C. Both wires have the same Young’s Modulus. D. Both wires have no Young’s Modulus at all. 3. Several cans of different sizes and shapes are all filled with the same liquid to the same depth. Which of the following statements is CORRECT? A. The weight of the liquid in each can is the same. B. The pressure on the bottom of each can is the same. C. The least pressure is at the bottom of the can with the largest bottom area. D. The least pressure is at the bottom of the can with the smallest bottom area. 4. A strip is made of two metals P and Q of the same length and cross-sectional area as shown in FIGURE 8.1. The linear expansion of metal P is greater than Q. What will happen to the strip when it is heated? FIGURE 8.1 A The strip expands vertically. B The strip does not expand at all. C The strip bends to the left side. D The strip bends to the right side. P Q
65 5. Energy transfer in the same substance from one point to the other is defined as A. Work B. Temperature C. Heat D. Power 6. A small steel ball of temperature 120oC is touched to a bigger copper ball of temperature 70oC. Which of the following statements is CORRECT? A. Heat will be transferred from the steel ball to the copper ball. B. Heat will be transferred from the copper ball to the steel ball. C. Heat will be transferred each other with the same rate. D. No heat will be transferred at all. SECTION B 8.1 STRESS AND STRAIN 8.2 YOUNG’S MODULUS 1. A metal rod that is 5.0 m long and 0.54 cm2 in cross sectional area is found to stretch 0.12 cm under a tension of 4700 N. What is Young’s Modulus of this metal? (Ans :3.63x1011 Nm-2) 2. A wire P produces a strain of 6 x10-4 when a load of 4kg is suspended at its free end. Another wire Q, made from the same metal and having the same length, has a diameter twice the diameter of P. Find the strain in Q if it is suspended with a 4kg load. (Ans :1.5x 10-4) 3. A 30.0kg hammer with speed 20.0 m s1strikes a steel spike 2.30cm in diameter. The hammer rebounds with speed 10.0 m s-1 after 0.110 s. What is the average strain in the spike during the impact? [ Y steel = 2.00 x1011 N m-2] (Ans :9.86x10-5) 4. FIGURE 8.2
66 The force F against elongation e graphs for wire X and wire Y as shown in FIGURE 8.2. The wires have the same original length and same cross-sectional area of 4.0 mm2. i. Calculate the work done to extend wire X and wire Y by 1.0 mm. State which wire is more rigid ii. From the graphs above, calculate the ratio of the Young’ modulus of wire X to the Young’ modulus of wire Y (Ans : Wx=0.1J, Wy=0.0625J ,1.6) 5. FIGURE 8.3 A 2.0m long steel wire with a diameter of 4.00 mm is placed over a light frictionless pulley as shown in Figure 1, with one end of the wire connected to a 5.0kg object and the other end connected to a 3.00kg object as shown in FIGURE 8.3. By how much does the wire stretch when the objects are in motion? [Y steel=2.0 x 1011 Nm-2] (Ans :2.92x10-5m) 6. A thin steel wire initially 1.5 m long and of diameter 0.50 mm is suspended from a rigid support. A mass of 3 kg is attached to the lower end of the wire. Calculate a) the extension of the wire, b) the energy stored in the wire. (Young’s modulus for steel = 2.0 x1011 N m-2) (Ans: 21.13x10-3 m, 0.017J) 8.3 HYDROSTATIC PRESSURE 1. The maximum pressure most organisms can survive is about 1000 times atmospheric pressure. Only small, simple organisms such as tadpoles and bacteria can survive such high pressures. What then is the maximum depth at which these organisms can live under the sea (assuming that the density of seawater is 1025 kg/m3) (Ans :10km) 5 kg 3 kg
67 2. FIGURE 8.4 A U-shape tube is filled with water (water = 1000 kgm-3) at one end and a small amount of cooking oil (oil = 850 kgm-3) at the other end as shown in FIGURE 8.4. Based on the figure, if the length of the oil is 12.0 cm determine : a) the length h. b) the difference level, h. (Ans : 10.2 cm, 1.8 cm) 3. FIGURE 8.5 A U-shaped tube as shown in FIGURE 8.5 is partly filled with water and partly filled with a liquid that does not mix with water. Both sides of the tube are open to the atmosphere. What is the density of the liquid (in g/cm3)? (Ans: 0.83 g/cm3) 12.0 cm h h oil water B C A
68 8.4 FLUID DYNAMICS 1. FIGURE 8.6 A liquid of density 740 kgm-3 is flowing through a venturi meter as ashown in FIGURE 8.6 below If the radius of r1 = 40.0 cm and r2 = 15.0 cm and velocity v2= 0.3 ms-1, determine : a) the velocity v1. b) the pressure difference, p between the two point. (Ans: 4.2 cms-1, 32.65Pa) 2. FIGURE 8.7 The Venturi tube shown in FIGURE 8.7 may be used as a fluid flowmeter. Suppose the device is used at a service station to measure the flow rate of gasoline 7.00 x 102 kg/m3 through a hose having an outlet radius of 1.0 cm. If the difference in pressure is measured to be P1-P2 = 21.0 kPa and the radius of the inlet tube to the meter is 2.0 cm, find a) the speed of the gasoline as it leaves the hose b) the fluid flow rate in cubic meters per second. (Ans : 2ms-1 , 2.51x10-3 m3s -1)
69 3. FIGURE 8.8 A large pipe with a cross-sectional area of 1.0m2 descends 5.0 m and narrows to 0.5m2, where it terminates in a valve at point 1 (FIGURE 8.8). If the pressure at point 2 is atmospheric pressure, and the valve is opened wide and water allowed to flow freely, find the speed of the water leaving the pipe. (Ans : 5.72ms-1) 8.5 VISCOCITY 1. A steel ball of diameter 2.0 cm and density 7900 kgm-3 falls vertically in a cup of liquid of density 850 kgm-3 and depth 20.0 cm in 3.3 s. By using a suitable figure, determine: a) the terminal velocity of the ball. b) the viscous (frictional) force due to the liquid on the ball. c) the viscosity of the liquid. (Ans :0.06ms-1, 0.29N, 25.2 Pa s) 2. A sphere of radius 1.0 cm is dropped into a glass cylinder filled with a viscous liquid. The mass of the sphere is 12.0 g and the density of the liquid is 1200 kgm-3. The sphere reaches a terminal speed of 0.15 m/s. What is the viscosity of the liquid? (Ans: 2.4Pas) 3. A steel ball of diameter 5.00 mm and density 8.0 × 103 kg m-3 falls vertically with its terminal velocity in oil of density 9.0 × 102 kg m-3. The time taken for the ball to fall a distance of 20.0 cm is 0.52 s. Calculate a) the terminal velocity of the ball b) the drag force on the ball c) the viscosity of the oil (Ans: 0.385 ms-1, 4.56x10-3 N, 0.251Pa s)
70 8.6 HEAT CONDUCTION 1. FIGURE 8.9 A composite rod is made by joining a copper rod of diameter 4 cm with an iron rod of similar diameter. The rod is insulated and its ends are kept at two different temperatures as shown in FIGURE 8.9. The coefficient of thermal conductivity of copper and iron are 385 Wm-1 K-1 and 80 Wm-1 K-1 respectively. Determine the temperature at the joint. (Ans :130.06°C ) 2. A gold rod is in contact with a silver rod. The gold end and the silver end of the compound rod is at 90°C and 30°C respectively. The silver rod has thermal conductivity 427 Wm-1 K-1, length 2.5 cm and the cross-sectional area 7.85 ×10-5 m2. If 341.3 J heat flows through the gold rod in 10 s, calculate the temperature at the contact surface. (Ans :55.5°C ) 3. A perfectly insulated aluminium rod has length 50 cm and cross-sectional area 3.0 cm2. At the steady state, the temperatures at 0 cm and 50 cm ends are 150°C and 50°C. (Thermal conductivity of aluminium is 210 Wm-1 K-1) a) Sketch a labelled graph of temperature against distance b) Calculate the temperature gradient along the rod c) Calculate the rate of heat flow in the rod (Ans :-200°Cm-1, 12.6J/s) 4. Heat is conducted through a brick wall of area 2m x 5m and of thickness 10cm for one minute. The temperature on both sides of the wall is 10°C and 30°C respectively. The thermal conductivity of the brick is 0.13Wm-1°K-1. Find the amount of heat flowing through the wall. (Ans : 1.56x104 J) 5. Two identical rods respectively are made of copper and iron. The length of the rods is 20.0cm and cross-sectional area of 3.5cm2. One end of both rods is maintained at temperature of 80oC. The thermal conductivity of copper is 400 Wm1K-1 and of iron is 70 Wm-1K-1. If the other end of the both rods is maintained at 27 oC, by using a suitable figure, determine: a) the heat flows through the copper rod in 1 minute. b) the heat flows through the iron rod in 1 minute. What conclusion can be made? (Ans : 2226 kJ , 390 J)
71 8.7 THERMAL EXPANSION 1. A 100cm rod A expands by 8.0 mm when heated from 0 °C to 100 °C. Rod B of the same length expands by 3.5 mm when heated in the same temperature range. Calculate the coefficient of linear expansion for rod A and rod B. (Ans: 3.5x10-5 K-1) 2. A steel rod with initial length of 75.0cm has a final length of 75.04 cm after its temperature is increased by 50 °C. Determine the coefficient of linear expansion for steel rod. (Ans: 1.07x10-5 K-1) 3. Two metal rods A and B are at 20 °C. Compare the expansion of both metals if metal A is heated to 100 °C and B is heated to 140 °C. Both have the same initial length and αA=3αB. (Ans: 2) 4. A glass beaker of volume 200cm3 is completely filled with olive oil at 30°C. When the beaker and the oil are heated to 70°C, some of the oil overflows. The beaker and the oil are then cooled down to 20°C. find the volume of the oil that overflowed at 70°C. (The coefficient of volume expansion for glass and olive oil is 2.7x10-5 K-1 and 6.8x10-4 K-1 respectively) (Ans: 5.22x10-6 m3) 5. A copper kettle contains full of water at 27oC. When the water is heated to its boiling point, the volume of the kettle expands by 1.2 x10-5 m3. If the coefficient of volume of copper is 1.5 x10-6 oC-1 and of water is 207 x10-6 oC-1, determine: a) the volume of the kettle at 27oC. b) the volume of the water displaced out from the kettle at the boiling point (Ans: 0.1096 m3, 1.66 x10-3 m3) 6. FIGURE 8.10
72 FIGURE 8.10 show two metal rods, one aluminum and one brass, are each clamped at one end. At 0.0°C, the rods are each 50.0 cm long and are separated by 0.024 cm at their unfastened ends. At what temperature will the rods just come into contact? (Assume that the base to which the rods are clamped undergoes a negligibly small thermal expansion.) (Given α for brass is 19x10-6 K-1, α for aluminium is 23x10-6 K-1) (Ans: 11.43°C) 7. A hollow copper cylinder is filled to the brim with water at 20.0°C. If the water and the container are heated to a temperature of 91°C, what percentage of the water spills over the top of the container? (Given α for copper is 16 x10-6 °C-1, ᵞ for water is 207x10-6°C-1) (Ans: 1.13%)
72 TOPIC 9 : KINETIC THEORY OF GASES AND THERMODYNAMICS 9.1 Kinetic theory of gases 9.2 Molecular kinetic energy and internal energy 9.3 Molar specific heat 9.4 First Law of Thermodynamics 9.5 Thermodynamics processes 9.6 Thermodynamics work At the end of this topic, students should be able to: 9.1 KINETIC THEORY OF GASES a) State the assumptions of kinetic theory of gases. b) Describe root mean square (rms) speed of gas molecules, = <V> c) Solve problems related to root mean square (rms) speed of gas molecules. 9.2 MOLECULAR KINETIC ENERGY AND INTERNAL ENERGY a) Discuss translational kinetic energy of a molecule, b) Discuss internal energy of gas. c) Solve problems related to internal energy, 9.3 MOLAR SPECIFIC HEATS a) Define molar specific heat at constant pressure, and volume, b) Use equation, − = and = Cp / Cv 9.4 FIRST LAW OF THERMODYNAMICS a) State the First Law of Thermodynamics, ΔU = Q – W b) Solve problem related to First Law of Thermodynamics. 9.5 THERMODYNAMIC PROCESSES a) Define the following thermodynamic processes: i. isothermal; ii. isochoric; iii. isobaric; and iv. adiabatic. b) Analyse p-V graph for all the thermodynamic processes. c) Determine the initial and final state for adiabatic process: PVγ = constant and TVγ-1
73 9.6 THERMODYNAMICS WORK a) State work done in isothermal, isochoric and isobaric processes. b) Solve problem related to work done in:
74 KOLEJ MATRIKULASI KEJURUTERAAN JOHOR KEMENTERIAN PENDIDIKAN MALAYSIA TOPIC 9 (KINETIC THEORY OF GASES AND THERMODYNAMICS) SECTION A 1. The following statements are the kinetic theory of gases EXCEPT A. there is intermolecular forces during collisions. B. the gas molecules move haphazardly and randomly. C. molecular collisions are elastic. D. the molecular motion was affected by its mass and velocity. 2. The first law of Thermodynamics may be written in the form of equation Q =ΔU + W where Q is the energy supplied to a gas, ΔU is the increase in its internal energy and W is the work done by expansion. When a real gas undergoes a change at constant pressure which of the following statements is TRUE? A. Q is necessarily zero B. ΔU is necessarily zero C. W is necessarily zero D. None Q, ΔU or W is necessarily zero 3. Which the following statements is/are false? I. For an isolated system, energy is always constant II. The First Law of Thermodynamics based on the conservation of mass III. For an ideal gas, the internal energy is a function of temperature only A. Only I B. Only II C. Both I and II D. Only I and III 4. The heat given to an ideal gas in isothermal conditions is used to A. Increase temperature B. Do external work C. Increase temperature and in doing external work D. Increase internal energy 5. In an isobaric expansion of a gas A. Volume and temperature both increases B. Volume increases and temperature decreases C. Volume decreases and temperature increases D. Volume and temperature both decreases
75 6. In which thermodynamic process is there no flow of heat between the system and the surrounding? A. Isobaric B. Isochoric C. Adiabatic D. Isothermal 7. For an isothermal process and adiabatic process, if the initial condition and final volume is the same, then the work done will be A. More in adiabatic process B. More in isothermal process C. Equal in both adiabatic and isothermal D. All above is wrong 8. The kinetic energy of an ideal gas depends on the A. Volume of the gas B. Pressure of the gas C. Temperature of the gas D. All of the above 9. Which of the following best represents the relationship between internal energy, U of an ideal gas and its absolute temperature, T? U T U T U T U T A B C D
76 10. In equilibrium, the total energy is equally distributed in all possible energy modes for a molecule, with each mode having an average energy equal to? A. kBT/2 B. kBT C. 2kBT D. kBT/4 SECTION B 9.1 KINETIC THEORY OF GASES 1. The above table shows the distribution of the speeds of 20 particles, find the Speed/ms-1 10 20 30 40 50 60 Number of Particles 1 3 8 5 2 1 Square Speed/m2s -2 100 400 900 1600 2500 3600 TABLE 9.1 a) most probable speed b) average speed c) r.m.s speed (Ans: 30 ms-1, 34 ms-1, 35.4 ms-1) 2. a) Find the ratio of r.m.s speed of O2 and H2 at equal temperature. b) The r.m.s speed of nitrogen molecules at 127oC is 600 ms-1. Calculate the r.m.s speed at 1127oC. c) Calculate the temperature at which the r.m.s speed of oxygen molecules is twice as great as their r.m.s speed at 27oC. (Ans: 4:1, 1787.36 ms-1, 927 °C) 3. The root mean square speed of a gas molecule is 300 ms-1. What will be the root mean square speed of the molecules if the atomic mass is doubled and absolute temperature is halved? (Ans: 150 ms-1) 9.2 MOLECULAR KINETIC ENERGY AND INTERNAL ENERGY 1. a) Determine the average value of the translational kinetic energy of the molecules of an ideal gas at i. 0.000 oC ii. 1000 oC. b) What is the translational kinetic energy per mole of an ideal gas at i. 0.000 oC ii. 1000 oC? (Ans: 5.65 10-21 J, 7.72 10-21 J, 3.40 103 J, 4.70 103 J)
77 2. The average translational energy and the r.m.s speeds of molecules in a sample of oxygen gas at 300 K are 6.21 10-21 J and 484 ms -1 respectively. The corresponding values at 600 K are nearly? (12.42 10-21 J, 684 ms -1) 3. A vessel of volume 30 liters contains 2.5 moles of gas at 30 ºC. a. Calculate the gas pressure b. If the gas is neon, calculate the mass of the gas in the vessel. [Molar gas constant, R = 8.31 J mol-1 K-1; relative atomic mass of neon = 40] (Ans:2.098 105 Pa, 100 g) 4. The mass of nitrogen, N2 gas in a container at STP is 120 g. If the gas is compressed at pressure 10 a.t.m, the temperature of the gas increases to 200°C. Calculate the initial and final volume of the gas. (Mass of 1 mole of N2 gas is 28 g) (Ans: 97.2310-3 m3; 16.6310-3 m3) 5. a) An ideal gas is compressed at a constant temperature, will its internal energy increase or decrease? b) An ideal diatomic gas is cooled from 26oC to 10oC at constant pressure of 1.0 atm. If there are 1200 moles of the ideal gas, what is the change in internal energy of the gas? (Ans: 398.88 kJ) c) A chamber is filled with 25.0 moles of helium gas at 200oC. i. Calculate the internal energy of the system ii. If the helium gas is replaced with 10 moles of nitrogen (N2) gas, determine the difference between the internal energy of helium and nitrogen gases. ( 23 6.02 10− NA = molecules, 23 1 1.38 10− − kB = JK ) (Ans: 147.40 kJ, - 49.13 kJ) 6. a) Which type of motion of the molecules is responsible for internal energy of a monoatomic gas? b) A 200 joule of work is done on a gas to reduce its volume. If this change is done under adiabatic conditions, find out the change in internal energy of the gas and also the amount of heat absorbed by the gas? (Ans: +200 J; 0) 9.3 MOLAR SPECIFIC HEAT 1. The quantity of heat absorbed by 2.0 moles of an ideal gas is 416 J. The gas expands at constant pressure & the gas temperature rises by 10 K. Determine the quantity of heat to be absorbed by the gas so that the temperature of the gas can be increase by 5 K at constant volume. (Ans: +124.9 J)
78 2. Two cylinders A and B fitted with pistons contain equal amounts of an ideal diatomic gas at 300 K. The piston of A is free to move, while that of B is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in A is 30 K, find the rise in temperature of the gas in B. (Given = 1.40 v P C C ) (Ans: 42 K) 3. 292.88 J of heat is required to raise the temperature of 2 mole of an ideal diatomic gas at constant pressure from 30oCto 35oC. Calculate the amount of heat required to raise the temperature of the same gas through the same range at constant volume. (Ans: 209.78 J) 4. The initial and final temperatures of 2 mol of helium gas are 300 K and 600 K respectively. Find the heat absorbed by the gas at (a) constant volume. (b) constant pressure. [ = 2.98 J mol-1 K-1, = 4.17 J mol-1 K-1] (Ans: +1788 J; +2502 J) 9.4 FIRST LAW OF THERMODYNAMICS 1. (a) In thermodynamic process pressure of a fixed mass of gas is changed in such a manner that the gas release 30 joule of heat and 18 joule of work was done on the gas. It the initial internal energy of the gas was 60 joule, then, the final internal energy will be? (b) A polyatomic gas with six degrees of freedom does 25 J work when it is expanded at constant pressure. The heat given to the gas is? (Ans: 48 J, 100 J) 2. FIGURE 9.1
79 FIGURE 9.1 shows a P-V graph for the compression of a gas from A to B. a) Calculate the work done in this process. b) If the internal energy of the gas decreases by 900 J during this compression, calculate the amount of heat dissipated from the gas. (Ans:-4.2 kJ, -5.1 kJ) 3. An ideal gas absorbs 1.5 kJ of heat & expands from its initial volume of 20 liters to 35 liters under constant pressure of 30 kPa. Determine the change in internal energy of the gas. (Ans: 1050 J) 9.5 THERMODYNAMICS PROCESSES 9.6 THERMODYNAMICS WORK 1. In an isobaric process, an unknown gas expands from an initial volume of 4 L to a final volume of 18 L at a constant pressure of 10 kPa. a) How much work is done by the gas? b) What is the change in internal energy of the gas if 10 J of heat lost during the process? c) Sketch the P-V graph for the process. (Ans: (+140 J, - 150 J) 2. FIGURE 9.2 The p-V diagram in FIGURE 9.2 above applies to a gas undergoing a cyclic change in a piston-cylinder arrangement. Calculate the work done by the gas in a) AB path b) BC path c) CD path d) DA path e) ABCDA path (Ans: +1.0 J, 0J, -0.5 J, 0J, +0.5 J) 4.0 2.0 1.5 4.0 V /cm3 A D B C p/ × 105 Pa
80 3. FIGURE 9.3 FIGURE 9.3 shows a system undergoing a change from A to B following a path of ACB. 90 J heat flows into the system. 70 J of work is done by the system. a) Calculate the heat flows into the system that follows the path of ADB when the work done by the system has a value of 15 J. b) When a system reversing its path to A by a curve path, the amount of work done by the system is 45 J, decide whether the system gains or loses heat and determine its value. c) When UA = 0 and UD = 8 J, evaluate the amount of heat gains during the process A → D and D → B. (Ans: 35 J, 23 J; 12 J, -65 J) 4. A container contains 0.4 ml of gas at 30oC and 2 atm. The gas is compressed isothermally to half its original volume and then the gas is allowed to expand isobarically back to its original volume. a) With the same axes, sketch and label p- V graphs for these two processes. b) Determine the pressure of the gas after the isothermal compression. c) Determine the final temperature of the gas. d) What is the work done for the whole processes? (Ans: 4.0 atm, 606 K, 2.50 104 J) 5. A fixed mass of gas initially at 7°C and a pressure of 1.00×105 Nm-2 is compressed isothermally to one-third of its original volume. It is then expanded adiabatically to its original volume. Calculate the final temperature and pressure, assuming γ = 1.40. (Ans: 180 K; 64.44 kPa) 6. FIGURE 9.4
81 One mole of a monatomic gas is taken through a cycle ABCDA as shown in the FIGURE 9.4. column II give the characteristics involved in the cycle. Match them with each of the processes given in column I. 7. An ideal gas of fixed mass initially of volume 3.0×10-3 m3 at 17°C undergoes these following changes: • The temperature increases to 27°C at constant pressure, and 20.5 J heat is supplied. • The temperature is reduced to the initial temperature of 17°C at constant volume, and 14.6 J heat is released. • The gas is then compressed adiabatically slowly until the volume is 2.0×10-3 m3. a) Show these processes in a graph of pressure p against volume V. b) Calculate the gas temperature in the final process (iii) (Ans: 346 K) SECTION C Two identical containers, X and Y has moveable piston to allow gas expansion. Both containers contain nitrogen at 30°C. Gas in container X expands adiabatically while gas in container Y expands isothermally to twice their original volumes. Molecular mass of nitrogen is 28 g mol -1 and its molar specific heat ratio γ = 1.40. Assume nitrogen behaves like an ideal gas. a) State the difference in speed of the piston in containers X and Y during expansion. b) Using the same axes, sketch and label the P-V graphs of the expansion experienced by the gas in containers X and Y. Assume that the initial state of the process is P1, V1 and T1. c) (i) Calculate the final temperature of the gas in X. (ii) Calculate the rms speed of the gas molecule in X and Y after the expansion. d) There is a difference in the r.m.s speed of gas in X and Y. Explain. e) If container Y contains 0.03 mole nitrogen gas, calculate the work done during the expansion. (Ans: 229.74 K, 452.27 ms-1, 519.53 ms-1, 52.36 J) Column I Column II Process A→B (a) Internal energy decreases Process B→C (b) Internal energy increases Process C→D (c) Heat is lost Process D→A (d) Heat is gained (e) Work is done on the gas