Physics

Appendix B

The Scientific Process

Science and Its Scope

Science is a specific way of looking at and understanding the world around us. The scope of
science encompasses a search for understanding natural and physical phenomena. For
example, biologists explore how living things function in their environment. Geologists
examine how Earth’s structures and materials have changed over time. Chemists investigate
the nature of matter and the changes it undergoes. Physicists search for an understanding
of the interactions between matter and energy.

Often, the areas that scientists investigate overlap. As a result, there are biochemists who
study the chemistry of living things, geophysicists who investigate the physical properties of
Earth, and physical chemists who apply physical laws to chemical reactions. Moreover, the
scope of science is not limited to investigating phenomena on Earth. In fact, the scope of
science extends throughout the universe.

Science and Its Limitations

Science is limited to investigating phenomena that can be examined usefully in a scientific
way. Some questions are outside the realm of science because they deal with phenomena
that are not scientifically testable. In other words, scientists must be able to use scientific
processes in their search for an answer or a solution. For example, they may need to design
a controlled experiment, analyze the results in a logical way, or develop scientific models to
explain data. As a result, the scope of science does not extend to issues of morals, values, or
the supernatural. In effect, the scope of science is limited to answering the question “how,”
not “why.”

Sometimes technology is a limitation for scientists. For example, scientists who studied
space were once limited to observing only what they could see with their eyes. In the 1600s,
Galileo used his telescope to observe things the eye could not see, such as the large moons
of Jupiter. Since Galileo, scientists have developed instruments that have allowed them to
see even farther and fainter objects. They have even put telescopes into space, such as the
Hubble Space Telescope, and sent probes to the edges of our solar system. However, there is
still a limit to what our current technology can detect. One of the deepest mysteries in space
involves dark energy, which scientists think is responsible for the expansion of the universe.
To detect this dark energy, scientists will need to build a space telescope that is able to
make the large number and specific types of observations that are needed.

Science and Its Methods

All scientists use certain processes in their search for explanations as to how the natural
world operates. These processes include making observations, asking questions, forming a
reasonable answer, evaluating the validity of the explanation, and communicating the
results. Taken together, the processes constitute the scientific method. The scientific
method is not a series of exact steps, but rather a strategy for drawing sound conclusions.
The scientific method also includes procedures that a scientist follows, such as conducting
an experiment in a laboratory or using a computer to analyze data. A scientist chooses the
procedures to use depending on the nature of the investigation.

The Scientific Process R17

There is no one correct scientific procedure. One scientist might use a field study to
investigate a geologic formation, while another might do a chemical analysis on a rock
sample. Another scientist might develop an experimental procedure to determine how to
slow down the division of cells. Still another scientist might use a computer to create a
model of a molecule.

Sometimes different groups come to the same scientific conclusions through two
different approaches. For example, in 1964, Arno Penzias and Robert Wilson were using a
supersensitive antenna in their research laboratory to pick up certain radio waves coming
from space. However, there was a background noise that they were picking up everywhere
they pointed the antenna. Penzias and Wilson could not eliminate this steady noise. They
checked their equipment and found nothing unusual. The scientists even cleaned the
antenna, but the noise still persisted. They concluded that this radiation must actually be
coming from space.

While these two scientists were working with their antenna, another team of scientists
just 60 km away at Princeton University was about to start a search for the same cosmic
radiation that Penzias and Wilson had discovered. The members of the Princeton team had
reasoned that when the universe was formed, a tremendous blast of radiation must have
been released into space. The Princeton team had been planning to make observations
designed to find and measure this radiation. When another scientist became aware of the
coincidence, he put Penzias and Wilson in touch with the Princeton team. Penzias and
Wilson had observed the radiation the Princeton team had predicted.

Science and Its Investigators

Sometimes, scientists investigating the same phenomenon might interpret the results
quite differently. One scientist might have one explanation, while the other scientist has a
completely different explanation. One example is the behavior of light. Some scientists
explained the behavior of light in terms of waves. Others explained the same behavior of
light in terms of particles. Today, scientists recognize that light has a dual nature—its
behavior resembles that of both waves and particles. In this case, both explanations were
logically consistent. Moreover, these explanations were tested by other scientists who
confirmed the results.

Science and Its Evidence

Any explanation proposed by a scientist must abide by the rules of evidence. The data
must support the conclusion. If they don’t, then the explanation must be modified or
even discarded. For example, observational evidence of the orbit of Mercury could not
be explained by Newton’s law of gravity. Albert Einstein proposed a new way of thinking
about gravity that explained the change in Mercury’s orbit. When scientists were able to
directly observe some of the results that Einstein predicted, they accepted his theory of
general relativity.

Scientists also expect that all results can be replicated by other scientists working under
the same conditions. For instance, in 1989, two scientists reported that they had performed
“cold fusion.” In effect, the scientists claimed to have carried out nuclear fusion at room
temperature in a container on a countertop. People were at first hopeful that this discovery
would lead to cheap and plentiful energy sources. However, a group of scientists organized
that same year by the U.S. Department of Energy found no evidence to support “cold
fusion.” Other groups were unable to obtain the same results of the original experimenters.
If no one can replicate a scientific result, then that result is usually not accepted as valid.

R18 Appendix B

Science and Its Theories

Scientists hope to arrive at a conclusion about the phenomenon they investigate. They start
by making observations and asking questions. Then they suggest a reasonable explanation
for what they observe. This explanation is known as a hypothesis. A hypothesis is a rational
explanation of a single event or phenomenon based upon what is observed, but which has
not been proven.

A hypothesis usually develops from observations about the natural world. Sometimes
the results of observations are expected, but sometimes they are not. Unexpected results
can lead to new hypotheses. For example, in 1820, a Danish scientist named Hans Christian
Oersted discovered a relationship between magnetism and electricity. While working with
equipment for a lecture demonstration, Oersted placed a compass near a wire connected to
an apparatus that generated an electrical current. Oersted noticed that the needle on the
compass jumped and pointed toward the wire. He formed a hypothesis that electricity and
magnetism were related.

A hypothesis is a testable explanation. One way to test a hypothesis is by carrying out
experiments that test the predictions made by a hypothesis. Oersted conducted further
experiments and found that he could control the direction in which the compass needle
pointed by moving the wire. His new observations supported his hypothesis.

A good scientist recognizes that there is a chance that a test can fail to support the
hypothesis. If Oersted’s compass needle had jumped for some other reason, he would have
gotten different results. When this occurs, the scientist needs to rethink the hypothesis and
construct another explanation for the event or phenomenon.

Unlike a hypothesis, a theory is a well-established and highly reliable explanation
accepted by scientists. A theory is an explanation of a set of related observations or events
based upon proven hypotheses, and it is verified multiple times by different groups of
scientists. A theory may also develop from a collection of hypotheses that have been tested
and verified. For example, during the nineteenth century, various scientists developed
hypotheses to account for observations that linked electricity and magnetism. In 1873,
James Maxwell published his book Treatise on Electricity and Magnetism. The theory of
electromagnetism is now a well-established part of science.

Science and Its Laws

In science, a law is a descriptive statement that reliably predicts events under certain
conditions. A law can sometimes be expressed in terms of a single mathematical equation.
Some scientific laws include Newton’s laws of motion, the laws of thermodynamics, the
ideal gas laws, and the laws of conservation of mass and energy. It is important to know the
conditions under which a law is valid before using it. Laws can be valid over a wide range of
circumstances, or over a very limited range of circumstances.

A law is not the same as a theory. A law describes what is observed in nature under
certain conditions. A theory is a system of ideas that explains many related observations
and is supported by a large body of evidence acquired through scientific investigation. For
example, Newton’s law of gravitation predicts the size of the gravitational force between
masses. It says nothing of what causes this force. Einstein’s theory of gravity, however,
explains that motion due to gravity is due to the bending of space-time caused by mass.

Laws and theories do, however, share certain features—both are supported by observa-
tional evidence, both are widely accepted by scientists, and both may need to be modified
or abandoned if conflicting evidence is discovered. They are both tools that help scientists
to answer questions about the world around them.

The Scientific Process R19

Appendix C

Symbols

Diagram Symbols

Mechanics waves and electromagnetism

Symbol Meaning Symbol Meaning
displacement vector,
displacement component ray (light or sound)

velocity vector, + positive charge
velocity component – negative charge

acceleration vector electric field lines
force vector,
force component electric field vector
electric current
momentum vector
gravitational field vector

angle marking

rotational motion

magnetic field lines

Symbol thermodynamics

Meaning magnetic field vector
energy transferred as heat (into page, out of
page)
energy transferred as work

cycle or process

R20 Appendix C

Mathematical Symbols

Symbol Meaning Symbol Meaning
Δ
∑ (Greek delta) change in some quantity ≤ less than or equal to
θ
= (Greek sigma) sum of quantities ∝ is proportional to
>
≥ (Greek theta) any angle ≈ is approximately equal to
<
equal to | n | absolute value or magnitude of

greater than sin sine

greater than or equal to cos cosine

less than tan tangent

Quantity Symbols Used Throughout

Symbols that are boldfaced refer to vector quantities that have both a magnitude and a
direction. Symbols that are italicized refer to quantities with only a magnitude. Symbols
that are neither are usually units.

Symbol Quantity
A area
D diameter
F, F force
m mass
M total mass
R radius (of a spherical body, a curved mirror, or a curved lens)
r radius (of sphere, shell, or disk)
t time
V volume

Symbols R21

Translational Mechanics Symbols Used in This Book

Symbols that are boldfaced refer to vector quantities that have both a magnitude and a
direction. Symbols that are italicized refer to quantities with only a magnitude. Symbols
that are neither are usually units.

Symbol Quantity
acceleration
a, a free-fall acceleration (acceleration due to gravity)
​a g​ ​ displacement
d, d impulse
gravitational force
FΔt force of kinetic friction
​F g​ ​, F​ g ​​ normal force
​F k​ ,​ F​ k ​ ​ net force
F​  n​ ​,​ F n​ ​  force of air resistance
​F n​ et​, ​Fn ​ et​  force of static friction
​F R​ ​, F​ R ​ ​  maximum force of static friction
F​  s​ ,​ F​ s ​​  height
F​  ​s,max​, F​ s ​,max​  spring constant
h kinetic energy
translational kinetic energy
k mechanical advantage
mechanical energy (sum of all kinetic and potential energy)
KE (Greek mu) coefficient of kinetic friction
K​ E t​rans​  (Greek mu) coefficient of static friction
MA power
momentum
ME potential energy
​μ k​ ​ elastic potential energy
​μs ​​ gravitational potential energy
P separation between point masses
velocity or speed
p, p work
work done by a frictional force (or work required to overcome a frictional force)
PE net work done
​PEe ​lastic​ displacement in the x direction
P​ E g​​ displacement in the y direction
r

v, v

W
​W f​riction​
​W n​ et​ 
Δx, Δx

Δy, Δy

R22 Appendix C

Rotational Mechanics Fluid Dynamics and
Symbols Used in This Book Thermodynamics Symbols Used
in This Book
Symbols that are boldfaced refer to vector quanti-
ties that have both a magnitude and a direction. Symbols that are boldfaced refer to vector quanti-
Symbols that are italicized refer to quantities with ties that have both a magnitude and a direction.
only a magnitude. Symbols that are neither are Symbols that are italicized refer to quantities with
usually units. only a magnitude. Symbols that are neither are
usually units.

Symbol Quantity Symbol Quantity
at tangential acceleration cp specific heat capacity
ac centripetal acceleration eff
α (Greek alpha) angular acceleration efficiency of a simple machine,
d sin θ lever arm (for torque calculations) FB, FB thermal efficiency of a heat engine
Fc, Fc centripetal force L buoyant force
I moment of inertia Lf latent heat
KErot rotational kinetic energy Lv latent heat of fusion
L angular momentum N latent heat of vaporization
length of a rotating rod P number of gas particles or nuclei
ℓ arc length P0 pressure
s (Greek tau) torque Pnet initial pressure, atmospheric pressure
τ (Greek tau) net torque ρ net pressure
τnet (Greek theta) angle of rotation Q (Greek rho) mass density
θ ( Greek delta and theta) angular Qc
∆θ displacement (in radians) heat
tangential speed Qh e nergy transferred as heat to or from
vt (Greek omega) angular speed a low-temperature (cold) substance
ω Qnet e nergy transferred as heat to or from
a high-temperature (hot) substance
T n et amount of energy transferred as
TC heat to or from a system
Tc temperature (absolute)
temperature in degrees Celsius
TF temperature of a low-temperature
Th (cool) substance
temperature in degrees Fahrenheit
U temperature of a high-temperature
(hot) substance
internal energy

Symbols R23

Vibrations, Waves, and Optics Electromagnetism Symbols
Symbols Used in This Book Used in This Book

Symbols that are boldfaced refer to vector quantities Symbols that are boldfaced refer to vector quantities
that have both a magnitude and a direction. Symbols that have both a magnitude and a direction. Symbols
that are italicized refer to quantities with only a that are italicized refer to quantities with only a
magnitude. Symbols that are neither are usually units. magnitude. Symbols that are neither are usually units.

Symbol Quantity Symbol Quantity
C center of curvature for spherical mirror B, B magnetic field
d slit separation in double-slit interference C capacitance
of light d separation of plates in a capacitor
d sin θ path difference for interfering light waves E, E electric field
Felastic, spring force emf emf (potential difference) produced by a
Felastic battery or electromagnetic induction
F focal point Felectric, electric force
f focal length Felectric
f frequency Fmagnetic, magnetic force
fn nth harmonic frequency Fmagnetic
h I electric current
h’ object height i instantaneous current (ac circuit)
k image height Imax maximum current (ac circuit)
L spring constant Irms root-mean-square current (ac circuit)
length of a pendulum, vibrating string, or L self-inductance
ℓ vibrating column of air ℓ length of an electrical conductor in a
λ path length of light wave magnetic field
m (Greek lambda) wavelength M mutual inductance
M order number for interference fringes N number of turns in a current-carrying
n magnification of image loop or a transformer coil
n harmonic number (sound) PEelectric electrical potential energy
p index of refraction Q large charge or charge on a fully
q object distance charged capacitor
T image distance q charge
period of a pendulum (simple R resistance
θ harmonic motion) r separation between charges
(Greek theta) angle of incidence of a Req equivalent resistance
θ beam of light (reflection) V
(Greek theta) angle of fringe separation ∆V electric potential
θ’ from center of interference pattern ∆v potential difference
θc (Greek theta) angle of reflection instantaneous potential difference
θi (Greek theta) critical angle of refraction ∆Vmax (ac circuit)
(Greek theta) angle of incidence of a ∆Vrms maximum potential difference (ac circuit)
θr beam of light (refraction) root-mean-square potential difference
(Greek theta) angle of refraction ω (ac circuit)
(Greek omega) angular frequency

R24 Appendix C

Particle and Electronic Symbols Used in This Book

For this part of the book, two tables are given because some symbols refer to quantities
and others refer to specific particles. The symbol’s context should make clear which table
should be consulted.

Symbol Quantity
mass number
A (Greek beta) current or potential difference gain of an amplifier
β photon energy
rest energy
E threshold frequency (photoelectric effect)
ER work function (photoelectric effect)
ft maximum energy of ejected photoelectron
hft (Greek lambda) decay constant
KEmax decay rate (activity)
λ neutron number, number of decayed particles
λN energy quantum number
N half-life
n atomic number
T1/2
Z

Symbol Particle
alpha particle
α bottom quark, antiquark
b, b​̶   ​  (Greek beta) positron (beta particle)
(Greek beta) electron (beta particle)
β+ charmed quark, antiquark
down quark, antiquark
β- positron
c, c​̶   ​  electron
d, d​̶   ​  (Greek gamma) photon (gamma rays)
e+,​  +​    01  e​ alpha particle (helium-4 nucleus)
e−,​  -​    01 ​e  (Greek mu) muon
γ
​ ​24  H ​ e neutron
µ proton
strange quark, antiquark
​ 01​  n​  top quark, antiquark
​ 11​   ​p up quark, antiquark
s, ​s̶   ​  (Greek tau) tauon
t, ​t̶   ​  (Greek nu) neutrino, antineutrino
u, u​̶   ​  boson (weak force)
boson (weak force)
τ
v, v​̶​ 

W+, W−

Z

Symbols R25

Appendix D

Equations

Motion in One Dimension Δx = xf − xi

Displacement

Average velocity vavg = _​Δ Δxt   ​= ​_x tff −− xtii   ​

Average Speed average speed = ​_d  itsitmanecoef_ttrra av veel  el d​

Average acceleration aavg = _​Δ Δvt  ​= ​_v  tff −− vtii   ​

Displacement Δx = _​12 _ ​(vi + vf)Δt
These equations are valid only for constantly Δx = vi Δt + _​21 _ a​ (Δt)2
accelerated, straight-line motion. vf = vi + aΔt
Final Velocity vf2 = vi2 + 2aΔx
These equations are valid only for constantly
accelerated, straight-line motion.

Two-Dimensional Motion and Vectors

Pythagorean Theorem c2 = a2 + b2
This equation is valid only for right triangles.

Tangent, Sine, and Cosine Functions tan θ = _​o apdpj  ​   sin θ = _​o hpypp  ​   tan θ = _​h aydpj   ​
These equations are valid only for right triangles.

Vertical motion of a projectile that vy , f = ayΔt
falls from rest vy , f 2 = 2ayΔy
These equations assume that air resistance is Δy = _​21   a ​ y(Δt)2
negligible, and apply only when the initial
vertical velocity is zero. On Earth’s surface, vx = vx,i = constant
ay = −g = −9.81 m/s2. Δx = vx Δt

Horizontal motion of a projectile
These equations assume that air resistance is
negligible.

R26 Appendix D

Projectiles launched at an angle vx = vi cos θ = constant
These equations assume that air resistance Δx = (vi cos θ)Δt
is negligible. On Earth’s surface, vy,f = vi sin θ + ayΔt
ay = −g = −9.81 m/s2. vy,f 2 = vi2(sin θ)2 + 2ayΔy
Δy = (vi sin θ)Δt + _​21   a ​ y(Δt)2
Relative Velocity
vac = vab + vbc

Forces and the Laws of Motion

Newton’s First Law An object at rest remains at rest, and an object in
motion continues in motion with constant
Newton’s Second Law velocity (that is, constant speed in a straight line)
∑F is the vector sum of all external forces acting unless the object experiences a net external force.
on the object.
Newton’s third Law ∑F = ma

Weight If two objects interact, the magnitude of the force
On Earth’s surface, ag = g = 9.81 m/s2. exerted on object 1 by object 2 is equal to the
Coefficient of Static Friction magnitude of the force exerted on object 2 by
object 1, and these two forces are opposite in
direction.

Fg = mag

µs =_​F  sF,mna x​  

Coefficient of Kinetic Friction µk =_​F Fnk   ​
The coefficient of kinetic friction varies with speed, Ff = µFn
but we neglect any such variations here.

force of Friction

Equations R27

Work and Energy W​ n ​ et​= F​  n​ et​d cos θ
KE = ​_12    ​mv 2
Net Work W​ n ​ et​= ΔKE
This equation applies only when the force PE​ g ​​= mgh
is constant. P​ Ee ​lastic​= ​_12    ​kx2
Kinetic Energy ME = KE + ΣPE

WORK-KINETIC ENERGY THEOREM

GRAVITATIONAL POTENTIAL ENERGY

ELASTIC POTENTIAL ENERGY

MECHANICAL ENERGY

CONSERVATION OF MECHANICAL ENERGY ME​ i ​​= M​ E f​​
This equation is valid only if nonmechanical forms P = ​_Δ Wt   ​= Fv
of energy (such as friction) are disregarded.

POWER

Momentum and Collisions p = mv

MOMENTUM

IMPULSE-MOMENTUM THEOREM FΔt = Δp = mv​  f​​− m​v i​​
This equation is valid only when the force ​pi ​​= p​ f ​​
is constant. m1v​  1​ ,i +​ m2​v 2​ ,i​= m1v​  1​ ,f​+ m2​v 2​ ,f​

CONSERVATION OF MOMENTUM
These equations are valid for a closed system, that
is, when no external forces act on the system
during the collision. When such external forces are
either negligibly small or act for too short a time to
make a significant change in the momentum, these
equations represent a good approximation. The
second equation is valid for two-body collisions.

R28 Appendix D

CONSERVATION OF MOMENTUM FOR m1​v 1​ ,i​+ m2​v 2​ ,i​= (m1 + m2) v​  f​​
A PERFECTLY INELASTIC COLLISION _ ​21      ​m ​1 _21v​   1 ​m​​ ,i​21 ​ v​+​ 1 ​ ,f_​21 ​ 2  m​​+2 ​ v​_​21​  2 ​  m​​,i​22 ​=v​​ 2 ​ ,f​ 2
This is a simplified version of the conservation
of momentum equation valid only for perfectly
inelastic collisions between two bodies.

CONSERVATION OF KINETIC ENERGY FOR
AN ELASTIC COLLISION
No collision is perfectly elastic; some kinetic energy
is always converted to other forms of energy. But if
these losses are minimal, this equation can provide
a good approximation.

Circular Motion and Gravitation ac = ​_v​ ​ r t​​ 2 ​ ​ ​

CENTRIPETAL ACCELERATION

CENTRIPETAL FORCE ​Fc ​​= _​m  r​v​   t​​2 ​​​  

NEWTON’S LAW OF UNIVERSAL GRAVITATION ​Fg ​​= G ​_​m  ​r 1​ m​2​​ ​ 2 ​​​  
The constant of universal gravitation (G) equals
6.673 × ​10 −​ 11​N•m2/kg2. First Law:  Each planet travels in an elliptical
KEPLER’S LAWS OF PLANETARY MOTION orbit around the sun, and the sun is at one of
the focal points.
Period and Speed of an Object in Second Law:  An imaginary line drawn from
Circular Orbit the sun to any planet sweeps out equal areas in
The constant of universal gravitation (G) equals equal time intervals.
6.673 × 1​ 0 −​ 11​N•m2/kg2. Third Law:  The square of a planet’s orbital
Torque period (T 2  ) is proportional to the cube of the
average distance (r 3) between the planet and
the sun, or T 2  ∝ r 3  .

T = 2π ​ ​_ Grm3  ​   ​

vt = ​ G  _​m  r ​   ​

τ = Fd sin θ

Equations R29

Mechanical Advantage MA = ​_F​  ​Fo ​i ​unt​​  ​ = _​d​ d​ o ​i ​unt ​ ​ ​
This equation disregards friction. eff = _​​W W​ o ​i ​unt​  ​​ 

Efficiency
This equation accounts for friction.

Fluid Mechanics ρ = _​m V   ​

Mass Density

Buoyant Force F​ B ​ ​= F​  g​ (​ displaced fluid) = m​  f​​ g
The first equation is for an object that is completely F​ B ​ ​= ​F g​​(object) = mg
or partially submerged. The second equation is for P = ​_A F  ​
a floating object.

Pressure

Pascal’s Principle Pressure applied to a fluid in a closed container is
Hydraulic Lift Equation transmitted equally to every point of the fluid and
to the walls of the container.

​F2 ​ ​= ​_​A A​ 1 ​ 2​​​   F​​  1​ ​

Fluid Pressure as a Function of Depth P = ​P 0​ ​+ ρgh

Continuity Equation ​A1 ​ v​​ 1 ​ ​= A​  2​ v​​  2​ ​

Bernoulli’s Principle The pressure in a fluid decreases as the fluid’s
velocity increases.

Heat T​ F ​ ​= _​59    ​​TC ​ ​+ 32.0
T = ​TC ​ ​+ 273.15
Temperature Conversions

R30 Appendix D

Conservation of Energy ΔPE + ΔKE + ΔU = 0

Specific Heat Capacity ​cp ​ ​= _​m  QΔ T  ​

Calorimetry Q​ w ​ ​= −Q​  x​​
These equations assume that the energy trans- ​cp ​ ,w​​m w​ Δ​ T​ w ​ ​= −c​  p​ ,x​m​  x​ ​ΔT​ x ​​
ferred to the surrounding container is negligible.
Latent Heat Q = mL

Thermodynamics W = PAd = PΔV

Work Done by a Gas ΔU = Q − W
This equation is valid only when the pressure is ΔU​ n ​ et​= 0 and Q​  n​ et​= W​  n​ et​
constant. When the work done by the gas (W) eff = ​_W​  ​Q n​h ​e ​t​​  =​_  ​Qh ​​Q​− h​ ​ ​Q​ c​​  = 1 − _​​Q Q​ h ​c ​ ​​  ​
is negative, positive work is done on the gas.

The First Law of Thermodynamics
Q represents the energy added to the system
as heat and W represents the work done by
the system.

Cyclic Processes

Efficiency of a Heat Engine

Vibrations and Waves F​ e ​lastic​= –kx

Hooke’s Law

Period of a Simple Pendulum in Simple T = 2π ​​_a​ L g​ ​ ​  ​
Harmonic Motion
This equation is valid only when the amplitude T = 2π ​_​ mk ​  ​
is small (less than about 15°).

Period of a Mass-Spring System in
Simple Harmonic Motion

Speed of a Wave v = f λ 

Equations R31

Sound intensity = _​4  πPr 2  ​
f​n ​ ​= n​_ 2vL   ​   n = 1, 2, 3, . . .
Intensity of a Spherical Wave
This equation assumes that there is no absorption
in the medium.

Harmonic Series of a Vibrating String
or a Pipe Open at Both Ends

Harmonic Series of a Pipe Closed ​fn ​ ​= n ​_4  vL   ​   n = 1, 3, 5, . . .
at one End

Beats frequency difference = number of
beats per second

Light and Reflection c = f λ
angle of incidence (θ) = angle of reflection (θ')
Speed of Electromagnetic Waves
This book uses the value c = 3.00 × 108 m/s for
the speed of EM waves in a vacuum or in air.

Law of Reflection

Mirror Equation _​p 1  ​+ _​q 1  ​= _​1 f  ​
This equation is derived assuming that the rays
incident on the mirror are very close to the _ M = ​ hh'  ​= −​_pq    ​
principal axis of the mirror.

Magnification of a Curved Mirror

Refraction n = _​v c  ​
n​ i ​​sin θ​  i​​= ​n r​​sin θ​  r​​
Index of Refraction
For any material other than a vacuum, the index
of refraction varies with the wavelength of light.

Snell’s Law

R32 Appendix D

Thin-Lens Equation _​p 1  ​+ _​q 1  ​= _​1 f  ​
This equation is derived assuming that the thick-
ness of the lens is much less than the focal length _ M = ​ hh'  ​= −​_pq    ​( for ​ni ​​> ​n r​​)
of the lens.
sin θ​  c​​= ​_​n ​n ri ​​​  ​ (​ for ​ni ​​> n​  r​​)
Magnification of a Lens
This equation can be used only when the index of
refraction of the first medium (n​ i ​)​ is greater than
the index of refraction of the second medium (n​  r​​).

Critical Angle
This equation can be used only when the index of
refraction of the first medium (n​ i ​​) is greater than
the index of refraction of the second medium (n​  r​)​ .

Interference and Diffraction Constructive Interference:
d sin θ = ±mλ
Constructive and Destructive     m = 0, 1, 2, 3, . . .
Interference
The grating spacing multiplied by the sine of the Destructive Interference:
angle of deviation is the path difference between ​_12 .  .) ​ λ.
two waves. To observe interference effects, the d sin θ = ±(m +
sources must be coherent and have identical     m = 0, 1, 2, 3,
wavelengths.
See the equation above for constructive
Diffraction Grating interference.

Limiting Angle of Resolution θ = 1.22 _​D λ  ​
This equation gives the angle θ in radians and
applies only to circular apertures.

Electric Forces and Fields ( )​Fe ​lectric​= k​ C ​ ​​ ​_​q  r​ 1​ q​2​​  ​2 ​​​  ​

Coulomb’s Law E = k​  C​ ​​_r q2   ​
This equation assumes either point charges
or spherical distributions of charge.

Electric Field Strength Due to
a Point Charge

Equations R33

Electrical Energy and Current ​PEe ​lectric​= −qEd
ΔV = _​Δ  ​PEqe ​l ectr​ic​  = −EΔd
Electrical Potential Energy ΔV = k​  C​ ​​_q r  ​
The displacement, d, is from the reference point C = ​_Δ  QV   ​
and is parallel to the field. This equation is valid
only for a uniform electric field.

Potential Difference
The second half of this equation is valid only for
a uniform electric field, and Δd is parallel to the
field.

Potential Difference Between a
Point at Infinity and a Point Near
a Point Charge

Capacitance

Capacitance for a Parallel-Plate C = ε0 ​_A d  ​
​PEe ​lectric​= ​_12    Q​ ΔV = _​21   C ​ (ΔV)2 = ​_2 QC2   ​
Capacitor in a Vacuum
I = ​_Δ ΔQt  ​ 
The permittivity in a vacuum (ε0 ) equals
8.85 × ​10 −​ 12​C2/(N•m2).

ELECTRICAL POTENTIAL ENERGY STORED
IN A CHARGED CAPACITOR
There is a limit to the maximum energy (or charge)
that can be stored in a capacitor because electrical
breakdown ultimately occurs between the plates
of the capacitor for a sufficiently large potential
difference.

ELECTRIC CURRENT

RESISTANCE R = ​_Δ  IV ​  

OHM’S LAW _​Δ  IV ​  = constant
Ohm’s law is not universal, but it does apply to P = IΔV = I2R = ​_(  ΔRV )2​  
many materials over a wide range of applied
potential differences.

Electric Power

R34 Appendix D

Circuits and Circuit Elements ​Re ​q​= ​R 1​ ​+ ​R 2​ ​+ ​R 3​ ​. . .
The current in each resistor is the same and
Resistors in Series: Equivalent is equal to the total current.
Resistance and Current ​_R​  1e ​ q ​ ​= ​_​R  11 ​  ​ +​ ​_​R  12 ​  ​ +​ ​_​R  1 3​  ​ ​. . .
The sum of the current in each resistor equals
Resistors in Parallel: Equivalent the total current.
Resistance and Current
Φ​ M ​ ​= AB cos θ
Magnetism

Magnetic Flux

MAGNITUDE OF A MAGNETIC FIELD B = _​F​  m ​ aqgvn eti​c ​ 
The direction of ​Fm ​ agnetic​is always perpendicular to F​ m ​ agnetic​= BIℓ
both B and v, and can be found with the right-hand
rule.

FORCE ON A CURRENT-CARRYING
CONDUCTOR PERPENDICULAR TO
A MAGNETIC FIELD
This equation can be used only when the current
and the magnetic field are at right angles to each
other.

Electromagnetic Induction emf = −N _​Δ  ΔΦ​ tM ​ ​ ​ 

FARADAY’S LAW OF MAGNETIC INDUCTION
N is assumed to be a whole number.

EMF PRODUCED BY A GENERATOR emf = NABω sin ωt
N is assumed to be a whole number. maximum emf = NABω

FARADAY’S LAW FOR MUTUAL INDUCTANCE emf = −M _​Δ ΔIt   ​

Equations R35

RMS Current and Potential I​ r ​ms​= _​I √ ​m  2 a x  ​ ​ = 0.707 ​I m​ ax​
Difference Δ​V r​ms​= _​Δ  √​ V  2 m  a ​ x​  = 0.707 ΔV
Δ​V2 ​ ​= _​N​​N  21  ​​ ​  ​ ​ΔV​  1​ ​
Transformers
N is assumed to be a whole number.

Atomic Physics E = hf

Energy of a Light Quantum ​KEm ​ ax​= hf − ​hf t​​
λ = _​hp    ​= ​_m  hv   ​
MAXIMUM KINETIC ENERGY OF f = ​_E h  ​
A PHOTOELECTRON
WAVELENGTH AND FREQUENCY OF ​ER ​ ​= m​c 2​ ​
MATTER WAVES ​Eb ​ ind​= Δmc2
Planck’s constant (h) equals 6.63 × ​10 −​ 34​J•s. Δm = Z (atomic mass of H)
     + N​m n​ ​− atomic mass
Subatomic Physics activity = −​_Δ ΔNt  ​ = λN
T​ 1 ​ /2​= ​_0  .6λ9 3​  
RELATIONSHIP BETWEEN REST ENERGY
AND MASS
BINDING ENERGY OF A NUCLEUS

Mass Defect

ACTIVITY (DECAY RATE)

HALF-LIFE

R36 Appendix D

Take It Further Topics θ (rad) = _​1  8π0 °  θ​ (deg)

CONVERSION BETWEEN RADIANS AND
DEGREES

ANGULAR DISPLACEMENT Δθ = _​Δ  r s​  
This equation gives Δθ in radians. ​ωa ​ vg​= _​Δ Δθt   ​

AVERAGE ANGULAR VELOCITY

AVERAGE ANGULAR Acceleration ​αa ​ vg​= _​Δ  Δωt  ​ 

ROTATIONAL KINEMATICS ωf = ω​ i ​​+ αΔt
These equations apply only when the angular Δθ = ​ω i​​Δt + ​_12    ​α(Δt)2
acceleration is constant. The symbol ω represents ω​ f  ​2​= ​ω ​i2​ ​+ 2α(Δθ)
instantaneous rather than average angular velocity. Δθ = ​_12   ( ​ ​ωi ​​+ ​ω f​​ )Δt

TANGENTIAL SPEED ​v t​​= rω
For this equation to be valid, ω must be in rad/s.

TANGENTIAL ACCELERATION a​ t ​​= rα
For this equation to be valid, α must be in rad/s2.

NEWTON’S SECOND LAW FOR ROTATING τ = Iα
OBJECTS

ANGULAR MOMENTUM L = Iω

ROTATIONAL KINETIC ENERGY ​KEr ​ot​= ​_12    ​Iω​  2​ ​

IDEAL GAS LAW PV = Nk​ B ​ T​
Boltzmann’s constant (k​  B​ ​) equals 1.38 × ​10 −​ 23​J/K.

BERNOULLI’S EQUATION P + ​_12    ​ρv​  2​ ​+ ρgh = constant

Equations R37

Appendix E

SI Units

Si base units used in thIs book Si prefixes

Symbol Name Quantity Symbol Name Numerical equivalent
a
A ampere current f atto 10−18
p
K kelvin absolute temperature n femto 10−15
μ
kg kilogram mass m pico 10−12
c
m meter length d nano 10−9
k
s second time M micro 10−6
G
T milli 10−3
P
E centi 10−2

deci 10−1

kilo 103

mega 106

giga 109

tera 1012

peta 1015

exa 1018

Other commonly used units

Symbol Name Quantity Conversions
atm standard atmosphere pressure 1.013 250 × 105 Pa
Btu British thermal unit energy 1.055 × 103 J
Cal food calorie energy = 1 kcal = 4.186 × 103 J
cal calorie energy 4.186 J
Ci curie decay rate or activity 3.7 × 1010 s−1
°F degree Fahrenheit temperature 0.5556°C

ft foot length 0.3048 m

ft•lb foot-pound work and energy 1.356 J
g gram mass 0.001 kg
gal gallon volume 3.785 × 10−3 m3
hp horsepower power 746 W
in inch length 2.54 × 10−2 m
kcal kilocalorie energy 4.186 × 103 J
lb pound force 4.45 N
mi mile length 1.609 × 103 m
rev revolution angular displacement 2π rad

° degrees angular displacement ( )= ​ _​3 22π6   ​  ​rad = 1.745 × 10−2 rad

R38 Appendix E

Other units acceptable with si

Symbol Name Quantity Conversion
Bq
C becquerel decay rate or activity ​_1s     ​
°C
dB coulomb electric charge 1 A•s
eV
F degree Celsius temperature 1K

H decibel relative intensity (sound) (unitless)
h
Hz electron volt energy 1.60 × 10−19 J

J farad capacitance 1 ​_k Ag2••ms42   ​= 1 _​C V   ​
kW•h
L henry inductance 1 ​_k Ag2••ms22  ​ = 1 _​A J2   ​
min 3.600 × 103 s
N hour time ​_1s     ​
hertz frequency
Pa 1 ​_k  gs•2m 2​  = 1 N•m
rad joule work and energy 3.60 × 106 J
T 10−3 m3
u kilowatt-hour energy 6.0 × 101 s
V liter volume
minute time
W
newton force 1 ​_k  gs•2m ​  
Ω
pascal pressure 1 ​_m  k•gs 2  ​= 1 _​m N2   ​
radian angular displacement (unitless)

tesla magnetic field strength 1 ​_A  k•gs 3  ​= 1 ​_A  N•m   ​= 1 ​_V m•2s   ​
unified mass unit 1.660 538 782 × 10−27 kg
volt mass (atomic masses) 1 ​_k  Ag••sm32 ​  = 1 _​C J   ​
electric potential
difference

watt power 1 ​_k  gs•3m 2​  = 1 _​Js    ​

ohm resistance 1 _​k Ag2••ms32  ​ = 1 _​AV    ​

SI Units R39

Appendix F

Reference Tables

Symbol Quantity Fundamental constants Value used for
calculations in this book
c Established value 3.00 × 108 m/s
e− 1.60 × 10−19 C
e1 speed of light in a vacuum 299 792 458 m/s 2.72
1.602 176 487 × 10−19 C 8.85 × 10−12 C2/(N•m2)
ε0 elementary charge 2.718 2818 28
8.854 187 817 × 10−12 C2/ 6.673 × 10−11 N•m2/kg2
G base of natural logarithms (N•m2) 9.81 m/s2
g 6.672 59 × 10−11 N•m2/kg2
(Greek epsilon) permittivity of 9.806 65 m/s2 6.63 × 10−34 J•s
h a vacuum 1.38 × 10−23 J/K
kB 6.626 068 96 × 10−34 J•s 8.99 × 109 N•m2/C2
kC constant of universal gravitation 1.380 6504 × 10−23 J/K 8.31 J/(mol•K)
R 8.987 551 787 × 109 N•m2/C2 calculator value
π free-fall acceleration at Earth’s 8.314 472 J/(mol•K)
surface 3.141 592 654

Planck’s constant

Boltzmann’s constant (R/​NA ​ ​)
Coulomb constant

molar (universal) gas constant

(Greek pi) ratio of the circum-
ference to the diameter of a circle

coefficients of friction (approximate values)

steel on steel µs µk waxed wood on wet snow µs µk
0.74 0.57 0.14 0.1
0.04
aluminum on steel 0.61 0.47 waxed wood on dry snow – 0.06
0.03
rubber on dry concrete 1.0 0.8 metal on metal (lubricated) 0.15 0.04
0.003
rubber on wet concrete − 0.5 ice on ice 0.1

wood on wood 0.4 0.2 Teflon on Teflon 0.04

glass on glass 0.9 0.4 synovial joints in humans 0.01

useful astronomical data

Symbol Quantity Value used for calculations in this book
IE
ME moment of inertia of Earth 8.03 × 1037 kg•m2
RE
mass of Earth 5.97 × 1024 kg
yr
radius of Earth 6.38 × 106 m

Average Earth–moon distance 3.84 × 108 m

Average Earth–sun distance 1.50 × 1011 m

mass of the moon 7.35 × 1022 kg

mass of the sun 1.99 × 1030 kg

period of Earth’s orbit 3.16 × 107 s

R40 Appendix F

the moment of inertia the moment of inertia
for a few shapes for a few shapes

Shape Moment of inertia Shape Moment of
inertia

R tshyimn mhoeotrpy about MR2 ℓ tpheinrpreonddaicbuolaurt axis _​1  12   ​Mℓ2
axis through center

PH9H 9RPWE-•C0H8o-l 0tR0P2h-y0s1ic2sadth-iAainmheotoerp about _​21   M ​ R2 ℓ thin rod about _​13   M ​ ℓ2
HRW • Holt Physicsperpendicular axis
PH99PE-C08-002-012dth-Arough end

R MR2 PH 9H9RPWE-•C0H 8oR-l0t 0P2h-y0s1idsc2soiea-limAd estpehr ere about ​_52    ​MR2
point mass

HRW • Holt Physicsabout axis

PH99PE-C08-002-012h-A

PH9H 9RPWE-•C0H8o-l 0t 0PR2h-y0s1ic2sbad-biAsokuot rscyymlimndeetrry hp06se_apx00j030a.eps ​_32    ​MR2
axis N anda Pat eRl/ LLCoothpeinr spherical shell
_​21   M ​ R2 about diameter
9/10/04

2nd pass

PH9H9RPWE-•C0H8o-l0t 0P2hc-y0s1ioc2dscm-eAmnsoimtiessubosftsaonmcees* hNpa0n6dsaeP_aaptexl0/ s0LjL0pC31oeao.pceeprisfic heat capacities

Substance ρ(kg/m3) 9/1S0u/0b4stance cp(J/kg•˚C)
2nadlpuamssinum 8.99 × 102

hydrogen 0.0899 copper 3.87 × 102

helium 0.179 glass 8.37 × 102

steam (100°C) 0.598 gold 1.29 × 102

air 1.29 ice 2.09 × 103

oxygen 1.43 iron 4.48 × 102

carbon dioxide 1.98 lead 1.28 × 102

ethanol 0.806 × 103 mercury 1.38 × 102

ice 0.917 × 103 silver 2.34 × 102

fresh water (4°C) 1.00 × 103 steam 2.01 × 103

sea water (15°C) 1.025 × 103 water 4.186 × 103

glycerine 1.26 × 103

aluminum 2.70 × 103

iron 7.86 × 103

copper 8.92 × 103

silver 10.5 × 103

lead 11.3 × 103

mercury 13.6 × 103

gold 19.3 × 103

*A ll densities are measured at 0°C and 1 atm
unless otherwise noted.

Reference Tables R41

latent heats of fusion and vaporization at standard pressure

Substance Melting point (˚C) Lf(J/kg) Boiling point (˚C) Lv(J/kg)
nitrogen −209.97 2.55 × 104 −195.81 2.01 × 105

oxygen −218.79 1.38 × 104 −182.97 2.13 × 105

ethyl alcohol −114 1.04 × 105 78 8.54 × 105

water 0.00 3.33 × 105 100.00 2.26 × 106

lead 327.3 2.45 × 104 1745 8.70 × 105

aluminum 660.4 3.97 × 105 2467 1.14 × 107

speed of sound in various media

Medium v(m/s) Medium v(m/s) Medium v(m/s)
Gases
air (0˚C) Liquids at 25˚C 1140 Solids 5100
air (25˚C) 331 methyl alcohol aluminum 3560
air (100˚C) 5130
helium (0˚C) 346 sea water 1530 copper 1320
hydrogen (0˚C) 54
oxygen (0˚C) 366 water 1490 iron

972 lead

1290 vulcanized rubber

317

conversion of intensity to decibel level

Intensity (W/m2) Decibel level (dB) Examples
1.0 × 10−12
1.0 × 10−11    0 threshold of hearing
1.0 × 10−10
1.0 × 10−9   10 rustling leaves
1.0 × 10−8
1.0 × 10−7   20 quiet whisper
1.0 × 10−6
1.0 × 10−5   30 whisper
1.0 × 10−4
1.0 × 10−3   40 mosquito buzzing
1.0 × 10−2
1.0 × 10−1   50 normal conversation
1.0 × 100
1.0 × 101   60 air conditioning at 6 m
1.0 × 103
  70 vacuum cleaner

  80 busy traffic, alarm clock

  90 lawn mower

100 subway, power motor

110 auto horn at 1 m

120 threshold of pain

130 thunderclap, machine gun

150 nearby jet airplane

R42 Appendix F

indices of refraction for various substances*

Solids at 20˚C n Liquids at 20˚C n Gases at 0˚C, 1 atm n
1.000 293
cubic zirconia 2.20 benzene 1.501 air 1.000 450

diamond 2.419 carbon disulfide 1.628 carbon dioxide

fluorite 1.434 carbon tetrachloride 1.461

fused quartz 1.458 ethyl alcohol 1.361

glass, crown 1.52 glycerine 1.473

glass, flint 1.66 water 1.333

ice (at 0˚C) 1.309

polystyrene 1.49

sodium chloride 1.544

zircon 1.923

*measured with light of vacuum wavelength = 589 nm

Symbol Quantity useful atomic data Value used for calculations in
me mass of electron this book
mn mass of neutron Established value
mp mass of proton 9.109 × 10−31 kg
9.109 382 15 × 10−31 kg 5.49 × 10−4 u
5.485 799 0943 × 10−4 u 5.110 × 10−1 MeV
0.510 998 910 MeV
1.675 × 10−27 kg
1.674 927 211 × 10−27 kg 1.008 665 u
1.008 664 915 97 u 9.396 × 102 MeV
939.565 346 MeV
1.673 × 10−27 kg
1.672 621 637 × 10−27 kg 1.007 276 u
1.007 276 466 77 u 9.383 × 102 MeV
938.272 013 MeV

Reference Tables R43

MC_CNLESE586632_643A Appendix G

final

3-2-11 Periodic Table of the Elements
LKell

1 Group 2 Atomic number Key:

H1 Hydrogen 4 Symbol 13
Name
1.008 Be Al
1s1 Average atomic mass
Beryllium Electron configuration Aluminum
Group 1 9.012 182 26.981 5386
[Ne]3s 23p1
3 [He]2s2

2 Li
Lithium
6.94
[He]2 s 1

Period 11 12 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9

Na3 Sodium Mg 21 22 23 24 25 26 27

22.989 769 28 Magnesium Sc Ti V Cr Mn Fe Co
[Ne]3s1 24.3050
[Ne]3s2 Scandium Titanium Vanadium Chromium Manganese Iron Cobalt
19 44.955 912 47.867 50.9415 51.9961 54.938 045 55.845 58.933 195
20 [Ar]3d 14s2 [Ar]3d 24s2 [Ar]3d 34s2 [Ar]3d 54s1 [Ar]3d 54s2 [Ar]3d 64s2 [Ar]3d 74s2
4K
Potassium Ca 40 41 42 43 44 45
39.0983
[Ar]4s1 Calcium Zr Nb Mo Tc Ru Rh
40.078
[Ar]4s2 Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium
91.224 92.906 38 95.94 (98) 101.07 102.905 50
37 38 39 [Kr]4d 25s2 [Kr]4d 45 s1 [Kr]4d 85s1
[Kr]4d 55s1 [Kr]4d 65s1 [Kr]4d 75s1
Rb5 Rubidium Sr Y 72 73 77
74 75 76
85.4678 Strontium Yttrium Hf Ta Ir
[Kr]5s1 87.62 88.905 85 W Re Os
[Kr]5 s 2 [Kr]4d 15s2 Hafnium Tantalum Iridium
178.49 180.947 88 Tungsten Rhenium Osmium 192.217
55 56 57 [Xe]4f145d 26s2 [Xe]4f145d 36s2 183.84 186.207 190.23 [Xe]4f145d 76s2
[Xe]4f145d 46s2 [Xe]4f145d 56s2 [Xe]4f145d 66s2
6 Cs Ba La 104 105 109
Cesium 106 107 108
132.905 4519 Barium Lanthanum Rf Db Mt
[Xe]6s1 Sg Bh Hs
137.327 138.905 47 Rutherfordium Dubnium Meitnerium
[Xe]6s2 [Xe]5d 16 s2 (261) (262) Seaborgium Bohrium Hassium (268)
(266) (264) (277)
87 88 89 [Rn]5f146d 27s2 [Rn]5f146d 37s2 [Rn]5f146d 77s2
[Rn]5f146d 47s2 [Rn]5f146d 57s2 [Rn]5f146d 67s2
7 Fr Ra Ac 62
Francium
(223) Radium Actinium Sm
[Rn]7s1
(226) (227) Samarium
[Rn]7s2 [Rn]6d 17s2 150.36
[Xe]4f 66s2
* The systematic names and symbols 58 59 60 61
94
for elements greater than 112 will Ce Pr Nd Pm
be used until the approval of trivial Pu
names by IUPAC. Cerium Praseodymium Neodymium Promethium
140.116 140.907 65 144.242 (145) Plutonium
Elements whose average atomic masses appear bolded [Xe]4 f15d16s2 [Xe]4f 36s2 [Xe]4f 46s2 (244)
and italicized are recognized by the International Union [Xe]4f 56s2
of Pure and Applied Chemistry (IUPAC) to have several 90 91 92 [Rn]5f 67s2
stable isotopes. Thus, the average atomic mass for 93
each of these elements is officially expressed as a Th Pa U
range of values. A range of values expresses that the Np
average atomic mass of a sample of one of these Thorium Protactinium Uranium
elements is not a constant in nature but varies 232.038 06 231.035 88 238.028 91 Neptunium
depending on the physical, chemical, and nuclear [Rn]6d 27s2 [Rn]5f 26d 17s2 [Rn]5f 36d 17s2 (237)
history of the material in which the sample is found.
However, the values in this table are appropriate for [Rn]5f 46d 17s2
everyday calculations. A value given in parentheses is
not an average atomic mass but is the mass number of
that element's most stable or most common isotope.

R44 Appendix G

Hydrogen Group 13 Group 14 Group 15 Group 16 Group 17 Group 18
Semiconductors
5 6 7 8 9 2
(also known as metalloids)
B C N O F He
Metals
Alkali metals Boron Carbon Nitrogen Oxygen Fluorine Helium
Alkaline-earth metals 10.81 12.01 1 14.007 15.999 18.998 4032 4.002 602
Transition metals [He]2s 22p1 [He]2s 22p2 [He]2s 22p3 [He]2s 22p4 [He]2s 22p5
Other metals 1s 2
13 14 15 16 17
Nonmetals 10
Halogens Al Si P S Cl
Noble gases Ne
Other nonmetals Aluminum Silicon Phosphorus Sulfur Chlorine
26.981 5386 28.085 30.973 762 32.06 35.45 Neon
Group 10 Group 11 Group 12 [Ne]3s 23p1 [Ne]3s 23p2 [Ne]3s 23p3 [Ne]3s 23p4 [Ne]3s 23p5 20.1797
[He]2s 22p6
28 29 30 31 32 33 34 35
18
Ni Cu Zn Ga Ge As Se Br
Ar
Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine
58.6934 63.546 65.409 69.723 72.63 74.921 60 78.96 79.904 Argon
[Ar]3d 84s2 [Ar]3d 104s1 [Ar]3d 104s2 [Ar]3d 104s24p1 [Ar]3d 104s24p3 [Ar]3d 104s24p5 39.948
[Ar]3d 104s24p2 [Ar]3d 104s24p4 [Ne]3s 23p6
46 47 48 49 51 53
50 52 36
Pd Ag Cd In Sb I
Sn Te Kr
Palladium Silver Cadmium Indium Tin Antimony Iodine
106.42 107.8682 112.411 114.818 118.710 121.760 Tellurium 126.904 47 Krypton
[Kr]4d 10 [Kr]4d 105s1 [Kr]4d 105s2 [Kr]4d 105s25p1 [Kr]4d 105s25p2 [Kr]4d 105s25p3 127.60 [Kr]4d 105s25p5 83.798
[Kr]4d 105s25p4 [Ar]3d 104s24p6

54

Xe

Xenon
131.293
[Kr]4d 105s25p6

78 79 80 81 82 83 84 85 86

Pt Au Hg Tl Pb Bi Po At Rn

Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon
204.38
195.084 196.966 569 200.59 [Xe]4f 145d106s26p1 207.2 208.980 40 (209) (210) (222)
[Xe]4f 145d96s1 [Xe]4f 145d106s1 [Xe]4f 145d106s2 [Xe]4f 145d106s26p2 [Xe]4f 145d106s26p3 [Xe]4f 145d106s26p4 [Xe]4f 145d106s26p5 [Xe]4f 145d106s26p6

110 111 112 113 114 115 116 117 118

Ds Rg Cn Uut* Uuq* Uup* Uuh* Uus* Uuo*

Darmstadtium Roentgenium Copernicium Ununtrium Ununquadium Ununpentium Ununhexium Ununseptium Ununoctium
(285) (292) (294) (294)
(271) (272) (284) (289) (288)
[Rn]5f 146d 97s1 [Rn]5f 146d 107s1 [Rn]5f 146d 107s2 [Rn]5f 146d 107s27p1 [Rn]5f 146d 107s27p2 [Rn]5f 146d 107s27p3 [Rn]5f 1464d 107s27p [Rn]5f 1465d 107s27p [Rn]5f 1466d 107s27p

The discoveries of elements with atomic numbers 113–118 have been reported but not fully confirmed.

63 64 65 66 67 68 69 70 71

Eu Gd Tb Dy Ho Er Tm Yb Lu

Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium
151.964 157.25 158.925 35 162.500 164.930 32 167.259 168.934 21 173.04 174.967
[Xe]4f 76s2 [Xe]4f 96s2 [Xe]4f 106s2 [Xe]4f 116s2 [Xe]4f 126s2 [Xe]4f 136s2 [Xe]4f 146s2 [Xe]4f 145d16s2
[Xe]4f 75d16s2
95 97 98 99 100 101 102 103
96
Am Bk Cf Es Fm Md No Lr
Cm
Americium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium
(243) Curium (247) (251) (252) (257) (258) (259) (262)
(247)
[Rn]5f 77s2 [Rn]5f 76d17s2 [Rn]5f 97s2 [Rn]5f 107s2 [Rn]5f 117s2 [Rn]5f 127s2 [Rn]5f 137s2 [Rn]5f 147s2 [Rn]5f 146d17s2

Periodic Table of the Elements R45

Appendix H

Abbreviated Table of Isotopes and
Atomic Masses

Fundamental Constants

Z Element Symbol Average Atomic Mass Number Atomic Percent Half-life
0 (Neutron) Mass (u) (*indicates Mass (u) Abundance (if radioactive)
1 Hydrogen radioactivity) A T1/2

Dueterium n 1* 1.008 665 10.4 m
Tritium H 1.0079
2 Helium D 1 1.007 825 99.985
3 Lithium T 2 2.014 102 0.015
4 Beryllium He 4.002 60 3* 3.016 049
Li 6.941 12.33 y
5 Boron Be 9.0122 3 3.016 029
6 Carbon 4 4.002 602 0.000 14
B 10.81 6* 6.018 886 99.999 86
7 Nitrogen C 12.011
6 6.015 121 0.81 s
8 Oxygen N 14.0067 7 7.016 003
7.5
9 Fluorine O 15.9994 7* 7.016 928 92.5
10 Neon 8* 8.005 305
F 18.998 40 9.012 174 100 53.3 d
11 Sodium Ne 20.180 9 10.013 584 6.7 × 10–17 s
12 Magnesium 10* 19.9 1.5 × 106 y
Na 22.989 87 10.012 936 80.1
R46 Appendix H Mg 24.305 10 11.009 305
11 19.3 s
10.016 854
10* 11.011 433 20.4 m
11* 12.000 000 98.9
13.003 355
12 14.003 242 1.10
13
14* 13.005 738 5715 y
14.003 074
13* 15.000 108 99.63 996 m
14 16.006 100 0.37 7.13 s
15
15.003 065 99.761 122 s
16* 15.994 915 0.039 26.9 s
16.999 132 0.200
15* 17.999 160
16 19.003 577 109.8 m
17 100
18 18.000 937
18.998 404 11.0 s
19* 19.999 982
90.48 17.2 s
18* 19.001 880 0.27
19 19.992 435 9.25
20.993 841
20* 21.991 383 2.61 y
100
19* 21.994 434
20 22.989 767 14.96 h
21 23.990 961
22 78.99 11.3 s
22.994 124 10.00
22* 23.985 042 11.01
23 24.985 838
25.982 594
24*

23*
24
25
26

Z Element Symbol Average Atomic Mass Number Atomic Percent Half-life
13 Aluminum Mass (u) (*indicates Mass (u) Abundance (if radioactive)
14 Silicon radioactivity) A T1/2
15 Phosphorus 25.986 892 100 7.4 × 105 y
16 Sulfur Al 26.981 54 26* 26.981 534 92.23
Si 28.086 27 4.67 2.50 m
17 Chlorine P 30.973 76 27.976 927 3.10
18 Argon S 32.066 28 28.976 495
29 29.973 770 100 14.263 d
19 Potassium Cl 35.453 30 95.02
20 Calcium Ar 39.948 29.978 307 0.75 87.5 d
30* 30.973 762 4.21 3.0 × 105 y
21 Scandium K 39.0983 31 31.973 907 35.04 d
22 Titanium Ca 40.08 75.77 269 y
23 Vanadium 32* 31.972 071 1.28 × 109 y
24 Chromium Sc 44.9559 32.971 459 24.23 1.0 × 105 y
25 Manganese Ti 47.88 32 33.967 867 0.337
26 Iron V 50.9415 33 34.969 033 0.596 s
27 Cobalt Cr 51.996 34 0.063 60 y
28 Nickel Mn 54.938 05 35* 34.968 853 1.5 × 1017 y
29 Copper Fe 55.847 35.968 307 99.600 21.6 h
Co 58.933 20 35 36.975 893 93.2581 312.1 d
Ni 58.793 36* 0.0117 2.7 y
Cu 63.54 35.967 547 6.7302 5.27 y
37 36.966 776 96.941 7.5 × 104 y
37.962 732
36 38.964 314 0.647
37* 39.962 384 0.135
2.086
38 38.963 708
39* 39.964 000 100
40.961 827
40 7.3
39.962 591 73.8
39 40.962 279 0.25
40* 41.958 618 99.75
42.958 767
41 43.955 481 83.79
9.50
40 40.969 250
41* 44.955 911 100
5.9
42 43.959 691
43 46.951 765 91.72
44 47.947 947 100

41* 49.947 161 68.077
45 50.943 962
26.223
44* 47.954 033 69.17
47 51.940 511 30.83
48 52.940 652

50* 53.940 361
51 54.938 048

48* 53.939 613
52 54.938 297
53 55.934 940

54* 58.933 198
55 59.933 820

54 57.935 345
55* 58.934 350
56 59.930 789

59 62.929 599
60* 64.927 791

58
59*

60

63
65

Table of Isotopes and Atomic Masses R47

Z Element Symbol Average Atomic Mass Number Atomic Percent Half-life
30 Zinc Mass (u) (*indicates Mass (u) Abundance (if radioactive)
31 Gallium radioactivity) A T1/2
32 Germanium 63.929 144 48.6
Zn 65.39 64 65.926 035 27.9
33 Arsenic Ga 69.723 66 66.927 129 4.1
34 Selenium Ge 72.61 67 67.924 845 18.8
68
35 Bromine As 74.9216 68.925 580 60.108
36 Krypton 69 70.924 703 39.892
71
37 Rubidium 69.924 250 21.23
38 Strontium 70 71.922 079 27.66
39 Yttrium 72 72.923 462 7.73
40 Zirconium 73 73.921 177 35.94
74 75.921 402 7.44
41 Niobium 76
42 Molybdenum 74.921 594 100
75
43 Technetium 75.919 212
Se 78.96 76 76.919 913 9.36 1.4 × 1020 y
77 77.917 397 7.63 2.1 × 105 y
Br 79.904 78 79.916 519 23.78 10.76 y
Kr 83.80 80 81.916 697 49.61 4.75 × 1010 y
82* 8.73 29.1 y
Rb 85.468 78.918 336
Sr 87.62 79 80.916 287 50.69
Y 88.9058 81 49.31
80.916 589
81* 81.913 481 11.6
82 82.914 136 11.4
83 83.911 508 57.0
84 84.912 531
85.910 615 17.3
85*
86 84.911 793 72.17
86.909 186 27.83
85
87* 85.909 266 9.86
86.908 883 7.00
86 87.905 618 82.58
87 89.907 737
88 100
90* 88.905 847

89 89.904 702
90.905 643
Zr 91.224 90 91.905 038 51.45 1.5 × 106 y
Nb 92.9064 91 92.906 473 11.22 2 × 104 y
Mo 95.94 92 93.906 314 17.15 3.5 × 103 y
93*
Tc 94 92.906 376 17.38 2.6 × 106 y
93.907 280 4.2 × 106 y
93 100 2.1 × 105 y
94* 91.906 807
92.906 811 14.84
92 93.905 085
93* 94.905 841 9.25
95.904 678 15.92
94 96.906 020 16.68
95 97.905 407 9.55
96 99.907 476 24.13
97 9.63
98 96.906 363
100 97.907 215
98.906 254
97*
98*
99*

R48 Appendix H

Z Element Symbol Average Atomic Mass Number Atomic Percent Half-life
44 Ruthenium Mass (u) (*indicates Mass (u) Abundance (if radioactive)
radioactivity) A T1/2
45 Rhodium 98.905 939 12.7
46 Palladium Ru 101.07 99 99.904 219 12.6
Rh 102.9055 100 100.905 558 17.1
47 Silver 101 101.904 348 31.6
48 Cadmium 102 103.905 558 18.6
104
49 Indium 102.905 502 100
50 Tin 103
103.904 033
51 Antimony Pd 106.42 104 104.905 082 11.14
52 Tellurium 105 105.903 481 22.33
Ag 107.868 106 107.903 898 27.33
53 Iodine Cd 112.41 108 109.905 158 26.46
54 Xenon 110 11.72
In 114.82 106.905 091
55 Cesium Sn 118.71 107 108.904 754 51.84
56 Barium 109 48.16
57 Lanthanum Sb 121.76 108.904 984
58 Cerium Te 127.60 109* 109.903 004 12.49 462 d
59 Praseodymium I 126.9045 110 110.904 182 12.80 9.3 × 1015 y
Xe 131.29 111 111.902 760 24.13
112 112.904 401 12.22
Cs 132.9054 113.903 359 28.73
Ba 137.33 113*
La 138.905 114 112.904 060 4.3 4.4 × 1014 y
Ce 140.12 114.903 876 95.7 55 y
Pr 140.9076 113
115* 115.901 743 14.53
116.902 953 7.58
116 117.901 605 24.22
117 118.903 308 8.58
118 119.902 197 32.59
119 120.904 237
120 57.36
121* 120.903 820 42.64
122.904 215
121 7.12 > 8 × 1024 y
123 124.904 429 18.93 < 1.25 × 1021 y
125.903 309 31.79
125 127.904 468 33.87
126 129.906 228
128* 100
130* 126.904 474 1.6 × 107 y
128.904 984
127 26.4 > 2.36 × 1021 y
129* 128.904 779 21.2
130.905 069 26.9
129 131.904 141 10.4
131 133.905 394 8.9
132 135.907 214
134 100
136* 132.905 436 2 × 106 y
134.905 891 30 y
133 136.907 078
135* 11.23 10.5 y
137* 132.905 990 71.70 1.05 × 1011 y
136.905 816
133* 137.905 236 0.0902
137 99.9098
138 137.907 105
138.906 346 0.25 > 5 × 1016 y
138* 88.43
139 137.905 986 11.13
139.905 434
138 141.909 241 100
140
142* 140.907 647

141

Table of Isotopes and Atomic Masses R49

Z Element Symbol Average Atomic Mass Number Atomic Percent Half-life
60 Neodymium Mass (u) (*indicates Mass (u) Abundance (if radioactive)
radioactivity) A T1/2
61 Promethium 141.907 718
62 Samarium Nd 144.24 142 142.909 809 27.13 2.3× 1015 y
143 143.910 082 12.18
63 Europium Pm 144* 144.912 568 23.80
64 Gadolinium Sm 150.36 145 145.913 113 8.30
146 17.19
65 Terbium Eu 151.96 144.912 745
66 Dysprosium Gd 157.25 145* 145.914 968 15.0 17.7 y
146* 11.3 5.5 y
67 Holmium Tb 158.9253 146.914 894 13.8
68 Erbium 147* 147.914 819 7.4 1.06 × 1011 y
148* 148.917 180 26.7 7 × 1015 y
69 Thulium 149* 149.917 273 22.7 > 2 × 1015 y
70 Ytterbium 151.919 728
150 153.922 206 47.8
71 Lutetium 152 13.5 y
72 Hafnium 154 150.919 846
151.921 740 52.2
73 Tantalum 151 152.921 226
74 Tungsten 152* 14.80
154.922 618 20.47
75 Rhenium 153 155.922 119 15.65
156.923 957 24.84
R50 Appendix H 155 157.924 099 21.86
156 159.927 050
157 100
158 158.925 345
160
160.926 930
159 161.926 796
162.928 729
Dy 162.5 161 163.929 172 18.9
Ho 164.9303 162 25.5
163 164.930 316 24.9
164 28.2
165.930 292
165 166.932 047 100
167.932 369
Er 167.26 166 169.935 462 33.6
167 22.95
Tm 168.9342 168 168.934 213 27.8
Yb 173.04 170 170.936 428 14.9

Lu 174.967 169 170.936 324 100
Hf 178.49 171* 171.936 379 1.92 y
172.938 209
Ta 180.9479 171 173.938 861 14.3
172 175.942 564 21.9
173 16.12
174 174.940 772 31.8
176 175.942 679 12.7

175 176.943 218 97.41 3.78 × 1010 y
176* 177.943 697 2.59
178.945 813
177 179.946 547 18.606
178 27.297
179 180.947 993 13.029
180 35.100
181.948 202
181 182.950 221 99.988
183.950 929
W 183.85 182 185.954 358 26.3
Re 186.207 183 14.28
184 184.952 951 30.7
186 186.955 746 28.6

185 37.40 4.4 × 1010 y
187* 62.60

Z Element Symbol Average Atomic Mass Number Atomic Percent Half-life
76 Osmium Mass (u) (*indicates Mass (u) Abundance (if radioactive)
radioactivity) A T1/2
77 Iridium 187.955832 13.3
78 Platinum Os 190.2 188 188.958 139 16.1
79 Gold 189 189.958 439 26.4
80 Mercury Ir 192.2 190 191.961 468 41.0
Pt 195.08 192
81 Thallium Au 196.9665 190.960 585 37.3
Hg 200.59 191 192.962 916 62.7
82 Lead 193
Tl 204.383 193.962 655 32.9
83 Bismuth 194 194.964 765 33.8
84 Polonium Pb 207.2 195 195.964 926 25.3
85 Astatine 196
86 Radon Bi 208.9803 196.966 543 100
87 Francium Po 197
88 Radium At 197.966 743 9.97
89 Actinium Rn 198 198.968 253 16.87
90 Thorium Fr 199 199.968 299 23.10
91 Protactinium 200 200.970 276 13.10
92 Uranium 201 201.970 617 29.86
93 Neptunium 202
94 Plutonium 202.972 320 29.524 3.78 y
203 203.073 839 70.476 3.053 m
204* 204.974 400
207.981 992 24.1
205 22.1
208* 205.974 440 52.4
206.974 871
206 207.976 627 10.64 h
207 211.991 872
208 100
212* 208.980 374 60.6 m
211.991 259
209 102 y
212* 208.982 405 0.30 µs
211.988 842 0.145 s
209* 216.001 889
212* 1.6 s
216* 218.008 685 0.9 m
219.011 294
218* 55.6 s
219* 220.011 369 3.823 d
222.017 571
220* 22 m
222* 223.019 733

223* 224.020 187
226.025 402
Ra 224* 228.031 064 3.66 d
226* 1.6 × 103 y
228* 227.027 701 5.75 y
228.028 716
Ac 227* 18.72 y
228* 232.038 051 1.913 y
234.043 593
Th 232* 100 1.40 × 1010 y
234* 231.035 880 24.1 d
234.043 300
Pa 231* 0.0055 32.760 y
234* 234.040 946 0.720 6.7 h
235.043 924 99.2745
U 234* 238.050 784 2.46 × 105 y
235* 7.04 × 108 y
238* 236.046 560 4.47 × 109 y
237.048 168
Np 236* 1.15 × 105 y
237* 239.052 157 2.14 × 106 y
244.064 200 2.412 × 105 y
Pu 239* 8.1 × 107 y
244*

Table of Isotopes and Atomic Masses R51

Appendix I

Additional Problems

The Science of Physics 16. A small rocket launched from rest travels 12.4 m
upward in 2.0 s. What is the rocket’s net acceleration?
1. Mt. Waialeale in Hawaii gets 1.168 × 103 cm of rainfall
per year. Express this quantity in meters. 17. A jet slows uniformly from 153 km/h to 0 km/h over
42.0 m. What is the jet’s acceleration?
2. An acre is equal to about 4.0469 × 103 m2. Express this
area in square kilometers. 18. A softball thrown straight up at 17.5 m/s is caught 3.60 s
later. How high does the ball rise?
3. A group drinks about 6.4 × 104 cm3 of water per person
per year. Express this in cubic meters. 19. A child, starting from rest, sleds down a snow-covered
slope in 5.50 s. If the child’s final speed is 14.0 m/s, what
4. The largest stone jar on the Plain of Jars in Laos has a the length of the slope?
mass of 6.0 × 103 kg. Express this mass in milligrams.
20. A sky diver opens her parachute and drifts down for
5. Half of a sample of the radioactive isotope beryllium-8 34.0 s with a constant velocity of 6.50 m/s. What is the
decays in 6.7 × ​10 −​ 17​s. Express this time in sky diver’s displacement?
picoseconds.
21. In a race, a tortoise runs at 10.0 cm/s and a hare runs at
Motion in One Dimension 200.0 cm/s. Both start at the same time, but the hare
stops to rest for 2.00 min. The tortoise wins by 20.0 cm.
6. The fastest airplane is the Lockheed SR-71. If an SR-71 At what time does the tortoise cross the finish line?
flies 15.0 km west in 15.3 s, what is its average velocity
in kilometers per hour? 22. What is the length of the race in problem 21?

7. Except for a 22.0 min rest stop, Emily drives with a 23. The cable pulling an elevator upward at 12.5 m/s
constant velocity of 89.5 km/h, north. How long does breaks. How long does it take for the elevator to come
the trip take if Emily’s average velocity is 77.8 km/h, to rest?
north?
24. A disk is uniformly accelerated from rest for 0.910 s over
8. A spaceship accelerates uniformly for 1220 km. How 7.19 km. What is its final speed?
much time is required for the spaceship to increase its
speed from 11.1 km/s to 11.7 km/s? 25. A tiger accelerates 3.0 m/s2 for 4.1 s to reach a final
speed of 55.0 km/h. What was its initial speed in
9. A polar bear initially running at 4.0 m/s accelerates kilometers per hour?
uniformly for 18 s. If the bear travels 135 m in this time,
what is its maximum speed? 26. A shark accelerates uniformly from 2.8 km/h to
32.0 km/h in 1.5 s. How large is its acceleration?
10. A walrus accelerates from 7.0 km/h to 34.5 km/h over a
distance of 95 m. What is the magnitude of the walrus’s 27. The 1903 Wright flyer was accelerated at 4.88 m/s2
acceleration? along a track that was 18.3 m long. How long did it take
to accelerate the flyer from rest?
11. A snail can move about 4.0 m in 5.0 min. What is the
average speed of the snail? 28. A drag racer starts at rest and reaches a speed of 386.0
km/h with an average acceleration of 16.5 m/s2. How
12. A crate is accelerated at 0.035 m/s2 for 28.0 s along a long does this acceleration take?
conveyor belt. If the crate’s initial speed is 0.76 m/s,
what is its final speed? 29. A hummingbird accelerates at −9.20 m/s2 such that its
velocity changes from +50.0 km/h to 0 km/h. What is
13. A person throws a ball vertically and catches it after its displacement?
5.10 s. What is the ball’s initial velocity?
30. A train backs up from an initial velocity of −4.0 m/s and
14. A bicyclist accelerates –0.870 m/s2 during a 3.80 s an average acceleration of −0.27 m/s2. What is the
interval. What is the change in the velocity of the train’s velocity after 17 s?
bicyclist and bicycle?
31. A cross-country skier skiing with an initial velocity of
15. A hockey puck slides 55.0 m in 1.25 s with a uniform +4.42 m/s slows uniformly at −0.75 m/s2. How long
acceleration. If the puck’s final speed is 43.2 m/s, what does it take the skier to stop?
was its initial speed?

R52 Appendix I

32. What is the skier’s displacement in problem 31? 47. Find the displacement direction in problem 46.

33. A speedboat uniformly increases its speed from 25 m/s 48. A train travels 478 km southwest along a straight
west to 35 m/s west. How long does it take the boat to stretch. If the train is displaced south by 42 km, what is
travel 250 m west? the train’s displacement to the west?

34. A ship accelerates at −7.6 × 10−2 m/s2 so that it comes 49. Find the displacement direction in problem 48.
to rest at the dock 255 m away in 82.0 s. What is the
ship’s initial speed? 50. A ship’s total displacement is 7400 km at 26° south of
west. If the ship sails 3200 km south, what is the western
35. A student skates downhill with an average acceleration component of its journey?
of 0.85 m/s2. Her initial speed is 4.5 m/s, and her final
speed is 10.8 m/s. How long does she take to skate 51. The distance from an observer on a plain to the top of a
down the hill? nearby mountain is 5.3 km at 8.4° above the horizontal.
How tall is the mountain?
36. A wrench dropped from a tall building is caught in a
safety net when the wrench has a velocity of −49.5 m/s. 52. A skyrocket travels 113 m at an angle of 82.4° with
How far did it fall? respect to the ground and toward the south. What is the
rocket’s horizontal displacement?
37. A rocket sled comes to a complete stop from a speed of
320 km/h in 0.18 s. What is the sled’s average 53. A hot-air balloon descends with a velocity of 55 km/h at
acceleration? an angle of 37° below the horizontal. What is the
vertical velocity of the balloon?
38. A racehorse uniformly accelerates 7.56 m/s2, reaching
its final speed after running 19.0 m. If the horse starts at 54. A stretch of road extends 55 km at 37° north of east,
rest, what is its final speed? then continues for 66 km due east. What is a driver’s
resultant displacement along this road?
39. An arrow is shot upward at a speed of 85.1 m/s. How
long does the archer have to move from the launching 55. A driver travels 4.1 km west, 17.3 km north, and finally
spot before the arrow returns to Earth? 1.2 km at an angle of 24.6° west of north. What is the
driver’s displacement?
40. A handball strikes a wall with a forward speed of
13.7 m/s and bounces back with a speed of 11.5 m/s. 56. A tornado picks up a car and hurls it horizontally 125 m
If the ball changes velocity in 0.021 s, what is the with a speed of 90.0 m/s. How long does it take the car
handball’s average acceleration? to reach the ground?

41. A ball accelerates at 6.1 m/s2 from 1.8 m/s to 9.4 m/s. 57. A squirrel knocks a nut horizontally at a speed of 10.0
How far does the ball travel? cm/s. If the nut lands at a horizontal distance of 18.6
cm, how high up is the squirrel?
42. A small sandbag is dropped from rest from a hovering
hot-air balloon. After 2.0 s, what is the sandbag’s 58. A flare is fired at an angle of 35° to the ground at an
displacement below the balloon? initial speed of 250 m/s. How long does it take for the
flare to reach its maximum altitude?
43. A hippopotamus accelerates at 0.678 m/s2 until it
reaches a speed of 8.33 m/s. If the hippopotamus runs 59. A football kicked with an initial speed of 23.1 m/s
46.3 m, what was its initial speed? reaches a maximum height of 16.9 m. At what angle was
the ball kicked?
44. A ball is hit upward with a speed of 7.5 m/s. How long
does the ball take to reach maximum height? 60. A bird flies north at 58.0 km/h relative to the wind. The
wind is blowing at 55.0 km/h south relative to Earth.
45. A surface probe on the planet Mercury falls 17.6 m How long will it take the bird to fly 1.4 km relative to
downward from a ledge. If free-fall acceleration near Earth?
Mercury is −3.70 m/s2, what is the probe’s velocity
when it reaches the ground? 61. A racecar moving at 286 km/h is 0.750 km behind a car
moving at 252 km/h. How long will it take the faster car
Two-Dimensional Motion to catch up to the slower car?
and Vectors
62. A helicopter flies 165 m horizontally and then moves
46. A plane moves 599 m northeast along a runway. If the downward to land 45 m below. What is the helicopter’s
northern component of this displacement is 89 m, how resultant displacement?
large is the eastern component?
63. A toy parachute floats 13.0 m downward. If the
parachute travels 9.0 m horizontally, what is the
resultant displacement?

Additional Problems R53

64. A billiard ball travels 2.7 m at an angle of 13° with 79. What is the range of an arrow shot horizontally at
respect to the long side of the table. What are the 85.3 m/s from 1.50 m above the ground?
components of the ball’s displacement?
80. A drop of water in a fountain takes 0.50 s to travel 1.5 m
65. A golf ball has a velocity of 1.20 m/s at 14.0° east of horizontally. The water is projected upward at an angle
north. What are the velocity components? of 33°. What is the drop’s initial speed?

66. A tiger leaps with an initial velocity of 55.0 km/h at an 81. A golf ball is hit up a 41.0° ramp to travel 4.46 m
angle of 13.0° with respect to the horizontal. What are horizontally and 0.35 m below the edge of the ramp.
the components of the tiger’s velocity? What is the ball’s initial speed?

67. A tramway extends 3.88 km up a mountain from a 82. A flare is fired with a velocity of 87 km/h west from a car
station 0.8 km above sea level. If the horizontal traveling 145 km/h north. With respect to Earth, what is
displacement is 3.45 km, how far above sea level is the the flare’s resultant displacement 0.45 s after being
mountain peak? launched?

68. A bullet travels 850 m, ricochets, and moves another 83. A sailboat travels south at 12.0 km/h with respect to the
640 m at an angle of 36° from its previous forward water against a current 15.0° south of east at 4.0 km/h.
motion. What is the bullet’s resulta­ nt displacement? What is the boat’s velocity?

69. A bird flies 46 km at 15° south of east, then 22 km at 13° Forces and the Laws of Motion
east of south, and finally 14 km at 14° west of south.
What is the bird’s displacement? 84. A boat exerts a 9.5 × 104 N force 15.0° north of west on a
barge. Another exerts a 7.5 × 104 N force north. What
70. A ball is kicked with a horizontal speed of 9.37 m/s direction is the barge moved?
off the top of a mountain. The ball moves 85.0 m
horizontally before hitting the ground. How tall is the 85. A shopper exerts a force on a cart of 76 N at an angle of
mountain? 40.0° below the horizontal. How much force pushes the
cart in the forward direction?
71. A ball is kicked with a horizontal speed of 1.50 m/s
from a height of 2.50 × 102 m. What is its horizontal 86. How much force pushes the cart in problem 85 against
displacement when it hits the ground? the floor?

72. What is the velocity of the ball in problem 71 when it 87. What are the magnitudes of the largest and smallest net
reaches the ground? forces that can be produced by combining a force of
6.0 N and a force of 8.0 N?
73. A shingle slides off a roof at a speed of 2.0 m/s and an
angle of 30.0° below the horizontal. How long does it 88. A buoyant force of 790 N lifts a 214 kg sinking boat.
take the shingle to fall 45 m? What is the boat’s net acceleration?

74. A ball is thrown with an initial speed of 10.0 m/s and an 89. A house is lifted by a net force of 2850 N and moves
angle of 37.0° above the horizontal. What are the from rest to an upward speed of 15 cm/s in 5.0 s. What
vertical and horizontal components of the ball’s is the mass of the house?
displacement after 2.5 s?
90. An 8.0 kg bag is lifted 20.0 cm in 0.50 s. If it is initially at
75. A rocket moves north at 55.0 km/h with respect to the rest, what is the net force on the bag?
air. It encounters a wind from 17.0° north of west at
40.0 km/h with respect to Earth. What is the rocket’s 91. A 90.0 kg skier glides at constant speed down a 17.0°
velocity with respect to Earth? slope. Find the frictional force on the skier.

76. How far to the north and west does the rocket in 92. A snowboarder slides down a 5.0° slope at a constant
problem 75 travel after 15.0 min? speed. What is the coefficient of kinetic friction
between the snow and the board?
77. A cable car travels 2.00 × 102 m on level ground, then
3.00 × 102 m at an incline of 3.0°, and then 2.00 × 102 m 93. A 2.00 kg block is in equilibrium on a 36.0° incline.
at an incline of 8.8°. What is the final displacement of What is the normal force on the block?
the cable car?
94. A 1.8 × 103 kg car is parked on a hill on a 15.0° incline.
78. A hurricane moves 790 km at 18° north of west, then A 1.25 × 104 N frictional force holds the car in place.
due west for 150 km, then north for 470 km, and finally Find the coefficient of static friction.
15° east of north for 240 km. What is the hurricane’s
resultant displacement?

R54 Appendix I

95. The coefficient of kinetic friction between a jar slid 1 09. A traffic signal is supported by two cables, each of
across a table and the table is 0.20. What is the which makes an angle of 40.0° with the vertical. If each
magnitude of the jar’s acceleration? cable can exert a maximum force of 7.50 × 102 N, what
is the largest weight they can support?
96. A force of 5.0 N to the left causes a 1.35 kg book to have
a net acceleration of 0.76 m/s2 to the left. What is the 1 10. A certain cable of an elevator is designed to exert a force
frictional force on the book? of 4.5 × 104 N. If the maximum acceleration that a
loaded car can withstand is 3.5 m/s2, what is the
97. A child pulls a toy by exerting a force of 15.0 N at an combined mass of the car and its contents?
angle of 55.0° with respect to the floor. What are the
components of the force? 111. A frictional force of 2400 N keeps a crate of machine
parts from sliding down a ramp with an incline of 30.0°.
98. A car is pulled by three forces: 600.0 N to the north, The coefficient of static friction between the box and
750.0 N to the east, and 675 N at 30.0° south of east. the ramp is 0.20. What is the normal force of the ramp
What direction does the car move? on the box?

99. Suppose a catcher exerts a force of −65.0 N to stop a 112. Find the mass of the crate in problem 111.
baseball with a mass of 0.145 kg. What is the ball’s net
acceleration as it is being caught? 1 13. A 5.1 × 102 kg bundle of bricks is pulled up a ramp at an
incline of 14° to a construction site. The force needed
1 00. A 2.0 kg fish pulled upward by a fisherman rises 1.9 m to move the bricks up the ramp is 4.1 × 103 N. What is
in 2.4 s, starting from rest. What is the net force on the the coefficient of static friction between the bricks and
fish during this interval? the ramp?

101. An 18.0 N force pulls a cart against a 15.0 N frictional Work and Energy
force. The speed of the cart increases 1.0 m/s every
5.0 s. What is the cart’s mass? 114. If 2.13 × 106 J of work must be done on a roller-coaster
car to move it 3.00 × 102 m, how large is the net force
1 02. A 47 kg sled carries a 33 kg load. The coefficient of acting on the car?
kinetic friction between the sled and snow is 0.075.
What is the magnitude of the frictional force on the 115. A force of 715 N is applied to a roller-coaster car to push
sled as it moves up a hill with a 15° incline? it horizontally. If 2.72 × 104 J of work is done on the car,
how far has it been pushed?
1 03. Ice blocks slide with an acceleration of 1.22 m/s2 down
a chute at an angle of 12.0° below the horiz­­ ontal. What 116. In 0.181 s, through a distance of 8.05 m, a test pilot’s
is the coefficient of kinetic friction between the ice and speed decreases from 88.9 m/s to 0 m/s. If the pilot’s
chute? mass is 70.0 kg, how much work is done against his
body?
1 04. A 1760 N force pulls a 266 kg load up a 17° incline. What
is the coefficient of static friction between the load and 1 17. What is the kinetic energy of a disk with a mass of 0.20 g
the incline? and a speed of 15.8 km/s?

105. A 4.26 × 107 N force pulls a ship at a constant speed 1 18. A 9.00 × 102 kg walrus is swimming at a speed of 35.0
along a dry dock. The coefficient of kin­ etic friction km/h. What is its kinetic energy?
between the ship and dry dock is 0.25. Find the normal
force exerted on the ship. 119. A golf ball with a mass of 47.0 g has a kinetic energy of
1433 J. What is the ball’s speed?
1 06. If the incline of the dry dock in problem 105 is 10.0°,
what is the ship’s mass? 1 20. A turtle, swimming at 9.78 m/s, has a kinetic energy of
6.08 × 104 J. What is the turtle’s mass?
1 07. A 65.0 kg skier is pulled up an 18.0° slope by a force of
2.50 × 102 N. If the net acceleration uphill is 0.44 m/s2, 121. A 50.0 kg parachutist is falling at a speed of 47.00 m/s
what is the frictional force between the skis and the when her parachute opens. Her speed upon landing is
snow? 5.00 m/s. How much work is done by the air to reduce
the parachutist’s speed?
1 08. Four forces are acting on a hot-air balloon:
F1 = 2280.0 N up, F2 = 2250.0 N down, F3 = 85.0 N 1 22. An 1100 kg car accelerates from 48.0 km/h to 59.0 km/h
west, and F4 = 12.0 N east. What is the direction of the over 100.0 m. What was the magnitude of the net force
net external force on the balloon? acting on it?

Additional Problems R55

1 23. What is the gravitational potential energy of a 64.0 kg 139. A ball falls 3.0 m down a vertical pipe, the end of which
person at 5334 m above sea level? bends horizontally. How fast does the ball leave the
pipe if no energy is lost to friction?
124. A spring has a force constant of 550 N/m. What is the
elastic potential energy stored in the spring when the 140. A spacecraft’s engines do 1.4 × 1013 J of work in 8.5 min.
spring is compressed 1.2 cm? What is the power output of these engines?

125. What is the kinetic energy of a 0.500 g raindrop that falls 141. A runner exerts a force of 334 N against the ground
0.250 km? Ignore air resistance. while using 2100 W of power. How long does it take him
to run a distance of 50.0 m?
1 26. A 50.0 g projectile is fired upward at 3.00 × 102 m/s and
lands at 89.0 m/s. How much mechanical energy is lost 142. A high-speed boat has four 300.0 kW motors. How
to air resistance? much work is done in 25 s by the motors?

1 27. How long does it take for 4.5 × 106 J of work to be done 1 43. A 92 N force pushes an 18 kg box of books, initially at
by a 380.3 kW engine? rest, 7.6 m across a floor. The coefficient of kinetic
friction between the floor and the box is 0.35. What is
128. A ship’s engine has a power output of 13.0 MW. How the final kinetic energy of the box of books?
much work can it do in 15.0 min?
1 44. A guardrail can be bent by 5.00 cm and then restore its
1 29. A catcher picks up a baseball from the ground with a shape. What is its force constant if struck by a car with
net upward force of 7.25 × 10−2 N so that 4.35 × 10−2 J 1.09 × 104 J of kinetic energy?
of net work is done. How far is the ball lifted?
1 45. A 25.0 kg trunk strikes the ground with a speed of
130. A crane does 1.31 × 103 J of net work when lifting 12.5 m/s. If no energy is lost from air resistance, what is
cement 76.2 m. How large is the net force doing this the height from which the trunk fell?
work?
1 46. Sliding a 5.0 kg stone up a frictionless ramp with a
131. A girl exerts a force of 35.0 N at an angle of 20.0° to the 25.0° incline increases its gravitational potential energy
horizontal to move a wagon 15.0 m along a level path. by 2.4 × 102 J. How long is the ramp?
What is the net work done on it if a frictional force of
24.0 N is present? 147. A constant 4.00 × 102 N force moves a 2.00 × 102 kg
iceboat 0.90 km. Frictional force is negligible, and the
132. The Queen Mary had a mass of 7.5 × 107 kg and a top boat starts at rest. Find the boat’s final speed.
cruising speed of 57 km/h. What was the kinetic energy
of the ship at that speed? 148. A 50.0 kg circus clown jumps from a platform into a net
1.00 m above the ground. The net is stretched 0.65 m
1 33. How fast is a 55.0 kg sky diver falling when her kinetic and has a force constant of 3.4 × 104 N/m. What is the
energy is 7.81 × 104 J? height of the platform?

134. A hockey puck with an initial speed of 8.0 m/s coasts Momentum and Collisions
45 m to a stop. If the force of friction on the puck is
0.12 N, what is the puck’s mass? 1 49. If a 50.0 kg cheetah, initially at rest, runs 274 m north in
8.65 s, what is its momentum?
135. How far does a 1.30 × 104 kg jet travel if it is slowed
from 2.40 × 102 km/h to 0 km/h by an acceleration 1 50. If a 1.46 × 105 kg whale has a momentum of 9.73 × 105
of −30.8 m/s2? kg•m/s to the south, what is its velocity?

1 36. An automobile is raised 7.0 m, resulting in an increase 1 51. A star has a momentum of 8.62 × 1036 kg•m/s and a
in gravitational potential energy of 6.6 × 104 J. What is speed of 255 km/s. What is its mass?
the automobile’s mass?
1 52. A 5.00 g projectile has a velocity of 255 m/s right. Find
137. A spring in a pogo stick has a force constant of the force to stop this projectile in 1.45 s.
1.5 × 104 N/m. How far is the spring compressed when
its elastic potential energy is 120 J? 1 53. How long does it take a 0.17 kg hockey puck to decrease
its speed by 9.0 m/s if the coefficient of kinetic friction
138. A 100.0 g arrow is pulled back 30.0 cm against a is 0.050?
bowstring. The bowstring’s force constant is 1250 N/m.
At what speed will the arrow leave the bow?

R56 Appendix I

1 54. A 705 kg racecar driven by a 65 kg driver moves with a 1 69. A 1.1 × 103 kg walrus starts swimming east from rest
velocity of 382 km/h right. Find the force to bring the and reaches a velocity of 9.7 m/s in 19 s. What is the net
car and driver to a stop in 12.0 s. force acting on the walrus?

1 55. Find the stopping distance in problem 154. 1 70. A 12.0 kg wagon at rest is pulled by a 15.0 N force at an
angle of 20.0° above the horizontal. If an 11.0 N
156. A 50.0 g shell fired from a 3.00 kg rifle has a speed of frictional force resists the forward force, how long will
400.0 m/s. With what velocity does the rifle recoil in the the wagon take to reach a speed of 4.50 m/s?
opposite direction?
171. A 42 g meteoroid moving forward at 7.82 × 103 m/s
1 57. A twig at rest in a pond moves with a speed of 0.40 cm/s collides with a spacecraft. What force is needed to stop
opposite a 2.5 g snail, which has a speed of 1.2 cm/s. the meteoroid in 1.0 × 10−6 s?
What is the mass of the twig?
1 72. A 455 kg polar bear slides for 12.2 s across the ice. If the
158. A 25.0 kg sled holding a 42.0 kg child has a speed of 3.50 coefficient of kinetic friction between the bear and the
m/s. They collide with and pick up a snowman, initially ice is 0.071, what is the change in the bear’s momentum
at rest. The resulting speed of the snowman, sled, and as it comes to a stop?
child is 2.90 m/s. What is the snowman’s mass?
1 73. How far does the bear in problem 172 slide?
1 59. An 8500 kg railway car moves right at 4.5 m/s, and a
9800 kg railway car moves left at 3.9 m/s. The cars 1 74. How long will it take a −1.26 × 104 N force to stop a
collide and stick together. What is the final velocity of 2.30 × 103 kg truck moving at a speed of 22.2 m/s?
the system?
1 75. A 63 kg skater at rest catches a sandbag moving north at
1 60. What is the change in kinetic energy for the two railway 5.4 m/s. The skater and bag then move north at 1.5 m/s.
cars in problem 159? Find the sandbag’s mass.

161. A 55 g clay ball moving at 1.5 m/s collides with a 55 g 1 76. A 1.36 × 104 kg barge is loaded with 8.4 × 103 kg of coal.
clay ball at rest. By what percentage does the kinetic What was the unloaded barge’s speed if the loaded
energy change after the inelastic collision? barge has a speed of 1.3 m/s?

162. A 45 g golf ball collides elastically with an identical ball 1 77. A 1292 kg automobile moves east at 88.0 km/h. If all
at rest and stops. If the second ball’s final speed is 3.0 forces remain constant, what is the car’s velocity if its
m/s, what was the first ball’s initial speed? mass is reduced to 1255 kg?

1 63. A 5.00 × 102 kg racehorse gallops with a momentum of 1 78. A 68 kg student steps into a 68 kg boat at rest, causing
8.22 × 103 kg•m/s to the west. What is the horse’s both to move west at a speed of 0.85 m/s. What was the
velocity? student’s initial velocity?

164. A 3.0 × 107 kg ship collides elastically with a 2.5 × 107 kg 179. A 1400 kg automobile, heading north at 45 km/h,
ship moving north at 4.0 km/h. After the collision, the collides inelastically with a 2500 kg truck traveling east
first ship moves north at 3.1 km/h and the second ship at 33 km/h. What is the vehicles’ final velocity?
moves south at 6.9 km/h. Find the unknown velocity.
1 80. An artist throws 1.3 kg of paint onto a 4.5 kg canvas at
1 65. A high-speed train has a mass of 7.10 × 105 kg and rest. The paint-covered canvas slides backward at
moves at a speed of 270.0 km/h. What is the magnitude 0.83 m/s. What is the change in the kinetic energy of the
of the train’s momentum? paint and canvas?

166. A bird with a speed of 50.0 km/h has a momentum of 181. Find the change in kinetic energy if a 0.650 kg fish
magnitude of 0.278 kg•m/s. What is the bird’s mass? leaping to the right at 15.0 m/s collides inelastically
with a 0.950 kg fish leaping to the left at 13.5 m/s.
1 67. A 75 N force pulls a child and sled initially at rest down
a snowy hill. If the combined mass of the sled and child 1 82. A 10.0 kg cart moving at 6.0 m/s hits a 2.5 kg cart
is 55 kg, what is their speed after 7.5 s? moving at 3.0 m/s in the opposite direction. Find the
carts’ final speed after an inelastic collision.
1 68. A student exerts a net force of −1.5 N over a period of
0.25 s to bring a falling 60.0 g egg to a stop. What is the 1 83. A ball, thrown right 6.00 m/s, hits a 1.25 kg panel at rest,
egg’s initial speed? then bounces back at 4.90 m/s. The panel moves right
at 1.09 m/s. Find the ball’s mass.

Additional Problems R57

1 84. A 2150 kg car, moving east at 10.0 m/s, collides and 1 97. A 2.05 × 108 kg asteroid has an orbit with a 7378 km
joins with a 3250 kg car. The cars move east together at radius. The centripetal force on the asteroid is
5.22 m/s. What is the 3250 kg car’s initial velocity? 3.00 × 109 N. Find the asteroid’s tangential speed.

185. Find the change in kinetic energy in problem 184. 198. Find the gravitational force between a 0.500 kg mass
and a 2.50 × 1012 kg mountain that is 10.0 km away.
1 86. A 15.0 g toy car moving to the right at 20.0 cm/s collides
elastically with a 20.0 g toy car moving left at 30.0 cm/s. 1 99. The gravitational force between Ganymede and Jupiter
The 15.0 g car then moves left at 37.1 cm/s. Find the is 1.636 × 1022 N. Jupiter’s mass is 1.90 × 1027 kg, and
20.0 g car’s final velocity. the distance between the two bodies is 1.071 × 106 km.
What is Ganymede’s mass?
187. A remora swimming right at 5.0 m/s attaches to a 150.0
kg shark moving left at 7.00 m/s. Both move left at 6.25 2 00. At the sun’s surface, the gravitational force on 1.00 kg is
m/s. Find the remora’s mass. 274 N. The sun’s mass is 1.99 × 1030 kg. If the sun is
assumed spherical, what is the sun’s radius?
1 88. A 6.5 × 1012 kg comet, moving at 420 m/s, catches up to
and collides inelastically with a 1.50 × 1013 kg comet 2 01. At the surface of a red giant star, the gravitat­ ional force
moving at 250 m/s. Find the change in the comets’ on 1.00 kg is only 2.19 × 10−3 N. If its mass equals
kinetic energy. 3.98 × 1031 kg, what is the star’s radius?

189. A 7.00 kg ball moves east at 2.00 m/s, collides with a 202. Uranus has a mass of 8.6 × 1025 kg. The mean distance
7.00 kg ball at rest, and then moves 30.0° north of east at between the centers of the planet and its moon
1.73 m/s. What is the second ball’s final velocity? Miranda is 1.3 × 105 km. If the orbit is circular, what is
Miranda’s period in hours?
190. A 2.0 kg block moving at 8.0 m/s on a frictionless
surface collides elastically with a block at rest. The first 203. What is the tangential speed in problem 202?
block moves in the same direction at 2.0 m/s. What is
the second block’s mass? 204. The rod connected halfway along the 0.660 m radius of
a wheel exerts a 2.27 × 105 N force. How large is the
Circular Motion and Gravitation maximum torque?

1 91. A pebble that is 3.81 m from the eye of a torn­ ado has a 2 05. A golfer exerts a torque of 0.46 N•m on a golf club. If the
tangential speed of 124 m/s. What is the magnitude of club exerts a force of 0.53 N on a stationary golf ball,
the pebble’s centripetal acceleration? what is the length of the club?

192. A racecar speeds along a curve with a tangential speed 206. What is the orbital radius of the Martian moon Deimos
of 75.0 m/s. The centripetal acceleration on the car is if it orbits 6.42 × 1023 kg Mars in 30.3 h?
22.0 m/s2. Find the radius of the curve.
207. A 4.00 × 102 N•m torque is produced applying a force
1 93. A subject in a large centrifuge has a radius of 8.9 m and 1.60 m from the fulcrum and at an angle of 80.0° to the
a centripetal acceleration of 20g (g = 9.81 m/s2). What lever. How large is the force?
is the tangential speed of the subject?
2 08. A customer 11 m from the center of a revolving
1 94. A 1250 kg automobile with a tangential speed of restaurant has a speed of 1.92 × 10−2 m/s. How large a
48.0 km/h follows a circular road that has a radius of centripetal acceleration acts on the customer?
35.0 m. How large is the centripetal force?
209. A toy train on a circular track has a tangential speed of
195. A rock in a sling is 0.40 m from the axis of rotation and 0.35 m/s and a centripetal acceleration of 0.29 m/s2.
has a tangential speed of 6.0 m/s. What is the rock’s What is the radius of the track?
mass if the centripetal force is 8.00 × 102 N?
210. A person against the inner wall of a hollow cylinder
1 96. A 7.55 × 1013 kg comet orbits the sun with a speed of with a 150 m radius feels a centripetal acceleration of
0.173 km/s. If the centripetal force on the comet is 9.81 m/s2. Find the cylinder’s tangential speed.
505 N, how far is it from the sun?
2 11. The tangential speed of 0.20 kg toy carts is 5.6 m/s when
they are 0.25 m from a turning shaft. How large is the
centripetal force on the carts?

R58 Appendix I

212. A 1250 kg car on a curve with a 35.0 m radius has a 2 28. A block of ebony with a volume of 2.5 × 10−3 m3 is
centripetal force from friction and gravity of placed in fresh water. If the apparent weight of the
8.07 × 103 N. What is the car’s tangential speed? block is 7.4 N, what is the density of ebony?

2 13. Two wrestlers, 2.50 × 10−2 m apart, exert a 229. One piston of a hydraulic lift holds 1.40 × 103 kg. The
2.77 × 10−3 N gravitational force on each other. One other holds an ice block (ρ = 917 kg/m3) that is 0.076 m
has a mass of 157 kg. What is the other’s mass? thick. Find the first piston’s area.

214. A 1.81 × 105 kg blue whale is 1.5 m from a 2.04 × 104 kg 2 30. A hydraulic-lift piston raises a 4.45 × 104 N weight by
whale shark. What is the gravitational force between 448 m. How large is the force on the other piston if it is
them? pushed 8.00 m downward?

2 15. Triton’s orbit around Neptune has a radius of 3.56 × 105 2 31. A platinum flute with a density of 21.5 g/cm3 is
km. Neptune’s mass is 1.03 × 1026 kg. What is Triton’s submerged in fresh water. If its apparent weight is
period? 40.2 N, what is the flute’s mass?

216. Find the tangential speed in problem 215. Heat

2 17. A moon orbits a 1.0 × 1026 kg planet in 365 days. What 2 32. Surface temperature on Mercury ranges from 463 K
is the radius of the moon’s orbit? during the day to 93 K at night. Express this tempera-
ture range in degrees Celsius.
218. What force is required to produce a 1.4 N•m torque
when applied to a door at a 60.0° angle and 0.40 m from 2 33. Solve problem 233 for degrees Fahrenheit.
the hinge?
2 34. The temperature in Fort Assiniboine, Montana, went
2 19. What is the maximum torque that the force in prob- from −5°F to +37°F on January 19, 1892. Calculate this
lem 218 can exert? change in temperature in kelvins.

220. A worker hanging 65.0° from the vane of a windmill 2 35. An acorn falls 9.5 m, absorbing 0.85 of its initial
exerts an 8.25 × 103 N•m torque. If the worker weighs potential energy. If 1200 J/kg will raise the acorn’s
587 N, what is the vane’s length? temperature 1.0°C, what is its temperature increase?

Fluid Mechanics 236. A bicyclist on level ground brakes from 13.4 m/s to
0 m/s. What is the cyclist’s and bicycle’s mass if the
221. A cube of volume 1.00 m3 floats in gasoline, which has a increase in internal energy is 5836 J?
density of 675 kg/m3. How large a buoyant force acts on
the cube? 237. A 61.4 kg roller skater on level ground brakes from
20.5 m/s to 0 m/s. What is the total change in the
222. A cube 10.0 cm on each side has a density of internal energy of the system?
2.053 × 104 kg/m3. Its apparent weight in fresh water is
192 N. Find the buoyant force. 238. A 0.225 kg tin can (cp = 2.2 × 103 J/kg•°C) is cooled in
water, to which it transfers 3.9 × 104 J of energy. By how
223. A 1.47 × 106 kg steel hull has a base that is 2.50 × 103 much does the can’s temp­ erature change?
m2 in area. If it is placed in sea water (ρ = 1.025 × 103
kg/m3), how deep does the hull sink? 2 39. What mass of bismuth (cp = 121 J/kg•°C) increases
temperature by 5.0°C when 25 J are added by heat?
2 24. What size force will open a door of area 1.54 m2 if the
net pressure on the door is 1.013 × 103 Pa? 2 40. Placing a 0.250 kg pot in 1.00 kg of water raises the
water’s temperature 1.00°C. The pot’s temperature
225. Gas at a pressure of 1.50 × 106 Pa exerts a force of drops 17.5°C. Find the pot’s specific heat capacity.
1.22 × 104 N on the upper surface of a piston. What is
the piston’s upper surface area? 241. Lavas at Kilauea in Hawaii have temperatures of 2192°F.
Express this quantity in degrees Celsius.
226. In a barometer, the mercury column’s weight equals the
force from air pressure on the mercury’s surface. 242. The present temperature of the background radiation
Mercury’s density is 13.6 × 103 kg/m3. What is the air’s in the universe is 2.7 K. What is this temperature in
pressure if the column is 760 mm high? degrees Celsius?

2 27. A cube of osmium with a volume of 166 cm3 is placed in
fresh water. The cube’s apparent weight is 35.0 N. What
is the density of osmium?

Additional Problems R59

243. The human body cannot survive at a temperature of 2 58. Find the efficiency of an engine that receives 571 J as
42°C for very long. Express this quantity in kelvins. heat and loses 463 J as heat per cycle.

244. Two sticks rubbed together gain 2.15 × 104 J from 2 59. A 5.4 × 10−4 m3 increase in steam’s volume does 1.3 J of
kinetic energy and lose 33 percent of it to the air. How work on a piston. What is the pressure?
much does the sticks’ internal energy change?
2 60. A pressure of 655 kPa does 393 J of work inflating a bike
245. A stone falls 561.7 m. When the stone lands, the internal tire. Find the change in volume.
energy of the ground and the stone increases by 105 J.
What is the stone’s mass? 261. An engine’s internal energy changes from 8093 J to
2.0920 × 104 J. If 6932 J are added as heat, how much
2 46. A 2.5 kg block of ice at 0.0°C slows on a level floor from work is done on or by the system?
5.7 m/s to 0 m/s. If 3.3 × 105 J cause 1.0 kg of ice to melt,
how much of the ice melts? 262. Steam expands from a geyser to do 192 kJ of work. If the
system’s internal energy increases by 786 kJ, how much
247. Placing a 3.0 kg skillet in 5.0 kg of water raises the energy is transferred as heat?
water’s temperature 2.25°C and lowers the skillet’s
temperature 29.6°C. Find the skillet’s specific heat. 263. If 632 kJ are added to a boiler and 102 kJ of work are
done as steam escapes from a safety valve, what is the
248. Air has a specific heat of 1.0 × 103 J/kg•°C. If air’s net change in the system’s internal energy?
temperature increases 55°C when 45 × 106 J are added
to it by heat, what is the air’s mass? 2 64. A power plant with an efficiency of 0.35 percent
requires 7.37 × 108 J of energy as heat. How much work
249. A 0.23 kg tantalum part has a specific heat capacity of is done by the power plant?
140 J/kg•°C. By how much does the part’s temperature
change if it gives up 3.0 × 104 J as heat? 265. An engine with an efficiency of 0.11 does 1150 J of work.
How much energy is taken in as heat?
Thermodynamics
2 66. A test engine performs 128 J of work and receives 581 J
250. A volume of air increases 0.227 m3 at a net pressure of of energy as heat. What is the engine’s efficiency?
2.07 × 107 Pa. How much work is done on the air?
Vibrations and Waves
2 51. The air in a hot-air balloon does 3.29 × 106 J of work,
increasing the balloon’s volume by 2190 m3. What is the 267. A scale with a spring constant of 420 N/m is
net pressure in the balloon? compressed 4.3 cm. What is the spring force?

252. Filling a fire extinguisher with nitrogen gas at a net 268. A 669 N weight attached to a giant spring stretches it
pressure of 25.0 kPa requires 472.5 J of work on the gas. 6.5 cm. What is the spring constant?
Find the change in the gas’s volume.
269. An archer applies a force of 52 N on a bowstring with a
253. The internal energy of air in a closed car rises 873 J. spring constant of 490 N/m. What is the bowstring’s
How much heat energy is transferred to the air? displacement?

2 54. A system’s initial internal energy increases from 39 J to 270. On Mercury, a pendulum 1.14 m long would have a
163 J. If 114 J of heat are added to the system, how much 3.55 s period. Calculate ag for Mercury.
work is done on the system?
2 71. Find the length of a pendulum that oscillates with a
2 55. A gas does 623 J of work on its surroundings when 867 J frequency of 2.5 Hz.
are added to the gas as heat. What is the change in the
internal energy of the gas? 2 72. Calculate the period of a 6.200 m long pendulum in
Oslo, Norway, where ag = 9.819 m/s2.
2 56. An engine with an efficiency of 0.29 takes in 693 J as
heat. How much work does the engine do? 273. Find the pendulum’s frequency in problem 272.

2 57. An engine with an efficiency of 0.19 does 998 J of work. 2 74. A 24 kg child jumps on a trampoline with a spring
How much energy is taken in by heat? constant of 364 N/m. What is the oscillation period?

2 75. A 32 N weight oscillates with a 0.42 s period when on a
spring scale. Find the spring constant.

R60 Appendix I

2 76. Find the mass of a ball that oscillates at a period of 295. A 1.53 m long pipe that is closed on one end has a
0.079 s on a spring with a constant of 63 N/m. seventh harmonic frequency of 466.2 Hz. What is the
speed of the waves in the pipe?
2 77. A dolphin hears a 280 kHz sound with a wavelength of
0.51 cm. What is the wave’s speed? 2 96. A pipe open at both ends has a fundamental frequency
of 125 Hz. If the pipe is 1.32 m long, what is the speed of
2 78. If a sound wave with a frequency of 20.0 Hz has a speed the waves in the pipe?
of 331 m/s, what is its wavelength?
297. Traffic has a power output of 1.57 × 10−3 W. At what
279. A sound wave has a speed of 2.42 × 104 m/s and a distance is the intensity 5.20 × 10−3 W/m2?
wavelength of 1.1 m. Find the wave’s frequency.
2 98. If a mosquito’s buzzing has an intensity of 9.3 × 10−8
280. An elastic string with a spring constant of 65 N/m is W/m2 at a distance of 0.21 m, how much sound power
stretched 15 cm and released. What is the spring force does the mosquito generate?
exerted by the string?
2 99. A note from a flute (a pipe with a closed end) has a first
281. The spring in a seat compresses 7.2 cm under a 620 N harmonic of 392.0 Hz. How long is the flute if the
weight. What is the spring constant? sound’s speed is 331 m/s?

2 82. A 3.0 kg mass is hung from a spring with a spring 300. An organ pipe open at both ends has a first harmonic of
constant of 36 N/m. Find the displacement. 370.0 Hz when the speed of sound is 331 m/s. What is
the length of this pipe?
2 83. Calculate the period of a 2.500 m long pendulum in
Quito, Ecuador, where ag = 9.780 m/s2. Light and Reflection

2 84. How long is a pendulum with a frequency of 0.50 Hz? 3 01. A 7.6270 × 108 Hz radio wave has a wavelength of
39.296 cm. What is this wave’s speed?
285. A tractor seat supported by a spring with a spring
constant of 2.03 × 103 N/m oscillates at a frequency of 3 02. An X ray’s wavelength is 3.2 nm. Using the speed of light
0.79 Hz. What is the mass on the spring? in a vacuum, calculate the frequency of the X ray.

286. An 87 N tree branch oscillates with a period of 0.64 s. 3 03. What is the wavelength of ultraviolet light with a
What is the branch’s spring constant? frequency of 9.5 × 1014 Hz?

2 87. What is the oscillation period for an 8.2 kg baby in a 3 04. A concave mirror has a focal length of 17 cm. Where
seat that has a spring constant of 221 N/m? must a 2.7 cm tall coin be placed for its image to appear
23 cm in front of the mirror’s surface?
288. An organ creates a sound with a speed of 331 m/s and a
wavelength of 10.6 m. Find the frequency. 305. How tall is the coin’s image in problem 304?

2 89. What is the speed of an earthquake S-wave with a 3 06. A concave mirror’s focal length is 9.50 cm. A 3.0 cm tall
2.3 × 104 m wavelength and a 0.065 Hz frequency? pin appears to be 15.5 cm in front of the mirror. How far
from the mirror is the pin?
Sound
307. How tall is the pin’s image in problem 306?
2 90. What is the distance from a sound with 5.88 × 10−5 W
power if its intensity is 3.9 × 10−6 W/m2? 308. A convex mirror’s magnification is 0.11. Suppose you
are 1.75 m tall. How tall is your image?
291. Sound waves from a stereo have a power output of
3.5 W at 0.50 m. What is the sound’s intensity? 3 09. How far in front of the mirror in problem 308 are you if
your image is 42 cm behind the mirror?
2 92. What is a vacuum cleaner’s power output if the sound’s
intensity 1.5 m away is 4.5 × 10−4 W/m2? 3 10. A mirror’s focal length is −12 cm. What is the object
distance if an image forms 9.00 cm behind the surface
293. Waves travel at 499 m/s on a 0.850 m long cello string. of the mirror?
Find the string’s fundamental frequency.
311. What is the magnification in problem 310?
2 94. A mandolin string’s first harmonic is 392 Hz. How long
is the string if the wave speed on it is 329 m/s? 312. A metal bowl is like a concave spherical mirror. You are
35 cm in front of the bowl and see an image at 42 cm.
What is the bowl’s focal length?

Additional Problems R61

3 13. For problem 312, find the bowl’s radius of curvature. 3 28. Consider the lamp and location in problem 327. If your
nose is 6.0 cm long, how long does the image appear?
3 14. A concave spherical mirror on a dressing table has a
focal length of 60.0 cm. If someone sits 35.0 cm in front 329. How fast does microwave radiation that has a frequency
of it, where is the image? of 1.173 06 × 1011 Hz and a wavelength of 2.5556 mm
travel?
3 15. What is the magnification in problem 314?
3 30. Suppose the microwaves in your microwave oven have
3 16. An image appears 5.2 cm behind the surface of a a frequency of 2.5 × 1010 Hz. What is the wavelength of
convex mirror when the object is 17 cm in front of the these microwaves?
mirror. What is the mirror’s focal length?
3 31. You place an electric heater 3.00 m in front of a concave
317. If the object in problem 316 is 3.2 cm tall, how tall is its spherical mirror that has a focal length of 30.0 cm.
image? Where would your hand feel warmest?

3 18. In order for someone to observe an object, the wave- 3 32. You see an image of your hand as you reach for a
length of the light must be smaller than the object. The doorknob with a focal length of 6.3 cm. How far from
Bohr radius of a hydrogen atom is 5.291 770 × 10−11 m. the doorknob is your hand when the image appears at
What is the lowest frequency that can be used to locate 5.1 cm behind the doorknob?
a hydrogen atom?
333. What is the magnification of the image in problem 332?
3 19. Meteorologists use Doppler radar to watch the
movement of storms. If a weather station uses electro- Refraction
magnetic waves with a frequency of 2.85 × 109 Hz, what
is the wavelength of the radiation? 3 34. A ray of light in air enters an amethyst crystal
(n = 1.553). If the angle of refraction is 35°, what is the
3 20. PCS cellular phones have antennas that use radio angle of incidence?
frequencies from 1800 to 2000 MHz. What range of
wavelengths corresponds to these frequencies? 335. Light passes from air at an angle of incidence of 59.2°
into a nephrite jade vase (n = 1.61). Determine the
3 21. Suppose you have a mirror with a focal length of angle of refraction in the jade.
32.0 cm. Where would you place your right hand so that
you appear to be shaking hands with yourself? 3 36. Light entering a pearl travels at a speed of 1.97 × 108
m/s. What is the pearl’s index of refraction?
3 22. A car’s headlamp is made of a light bulb in front of a
concave spherical mirror. If the bulb is 5.0 cm in front of 337. An object in front of a diverging lens of focal length
the mirror, what is the radius of the mirror? 13.0 cm forms an image with a magnification of +5.00.
How far from the lens is the object placed?
323. Suppose you are 19 cm in front of the bell of your
friend’s trumpet and you see your image at 14 cm. If the 3 38. An object with a height of 18 cm is placed in front of a
trumpet’s bell is a concave mirror, what would be its converging lens. The image height is −9.0 cm. What is
focal length? the magnification of the lens?

324. A soup ladle is like a spherical convex mirror with a 339. If the focal length of the lens in problem 338 is 6.0 cm,
focal length of 27 cm. If you are 43 cm in front of the how far in front of the lens is the object?
ladle, where does the image appear?
340. Where does the image appear in problem 339?
325. What is the magnification in problem 324?
3 41. The critical angle for light traveling from a green
3 26. Just after you dry a spoon, you look into the convex part tourmaline gemstone into air is 37.8°. What is tourma-
of the spoon. If the spoon has a focal length of −8.2 cm line’s index of refraction?
and you are 18 cm in front of the spoon, where does the
image appear? 342. Find the critical angle for light traveling from ruby
(n = 1.766) into air.
3 27. The base of a lamp is made of a convex spherical mirror
with a focal length of −39 cm. Where does the image 3 43. Find the critical angle for light traveling from emerald
appear when you are 16 cm from the base? (n =1.576) into air.

R62 Appendix I

344. Malachite has two indices of refraction: n1 = 1.91 and 3 60. Light passing through two slits with a separation of
n2 = 1.66. A ray of light in air enters malachite at an 8.04 × 10−6 m forms a third bright fringe 13.1° from the
incident angle of 35.2°. Calculate both of the angles of center. Find the wavelength.
refraction.
3 61. Two slits are separated by 0.0220 cm. Find the angle at
345. A ray of light in air enters a serpentine figurine which a first-order bright fringe is observed for light
(n = 1.555). If the angle of refraction is 33°, what is the with a wavelength of 527 nm.
angle of incidence?
3 62. For 546.1 nm light, the first-order maximum for a
346. The critical angle for light traveling from an aquama- diffraction grating forms at 75.76°. How many lines per
rine gemstone into air is 39.18°. What is the index of centimeter are on the grating?
refraction for aquamarine?
3 63. Infrared light passes through a diffraction grating of
3 47. A 15 cm tall object is placed 44 cm in front of a diverg- 3600 lines/cm. The angle of the third-order maximum
ing lens. A virtual image appears 14 cm in front of the is 76.54°. What is the wavelength?
lens. What is the lens’s focal length?
364. A diffraction grating with 1950 lines/cm is used to
3 48. What is the image height in problem 347? examine light with a wavelength of 497.3 nm. Find the
angle of the first-order maximum.
349. A lighthouse converging lens has a focal length of 4 m.
What is the image distance for an object placed 4 m in 3 65. At what angle does the second-order maximum in
front of the lens? problem 364 appear?

350. What is the magnification in problem 349? 366. Light passes through two slits separated by
3.92 × 10−6 m to form a second-order bright fringe at
351. Light moves from olivine (n = 1.670) into onyx. If the an angle of 13.1°. What is the light’s wavelength?
critical angle for olivine is 62.85°, what is the index of
refraction for onyx? 367. Light with a wavelength of 430.8 nm shines on two slits
that are 0.163 mm apart. What is the angle at which a
352. When light in air enters an opal mounted on a ring, the second dark fringe is observed?
light travels at a speed of 2.07 × 108 m/s. What is opal’s
index of refraction? 368. Light of wavelength 656.3 nm passes through two slits.
The fourth-order dark fringe is 0.548° from the central
353. When light in air enters albite, it travels at a velocity of maximum. Find the slit separation.
1.95 × 108 m/s. What is albite’s index of refraction?
369. The first-order maximum for light with a wavelength of
3 54. A searchlight is constructed by placing a 500 W bulb 447.1 nm is found at 40.25°. How many lines per
0.5 m in front of a converging lens. The focal length of centimeter does the grating have?
the lens is 0.5 m. What is the image distance?
3 70. Light through a diffraction grating of 9550 lines/cm
3 55. A microscope slide is placed in front of a converging forms a second-order maximum at 54.58°. What is the
lens with a focal length of 3.6 cm. The lens forms a real wavelength of the light?
image of the slide 15.2 cm behind the lens. How far is
the lens from the slide? Electric Forces and Fields

356. Where must an object be placed to form an image 371. Charges of −5.3 μC and +5.3 μC are separated by
12 cm in front of a diverging lens with a focal length 4.2 cm. Find the electric force between them.
of 44 cm?
372. A dog’s fur is combed, and the comb gains a charge of
357. The critical angle for light traveling from almandine 8.0 nC. Find the electric force between the fur and
garnet into air ranges from 33.1° to 35.3°. Calculate the comb when they are 2.0 cm apart.
range of almandine garnet’s index of refraction.
373. Two equal charges are separated by 6.5 × 10−11 m. If
358. Light moves from a clear andalusite (n = 1.64) crystal the magnitude of the electric force between the charges
into ivory. If the critical angle for andalusite is 69.9°, is 9.92 × 10−4 N, what is the value of q?
what is the index of refraction for ivory?

Interference and Diffraction

359. Light with a 587.5 nm wavelength passes through two
slits. A second-order bright fringe forms 0.130° from the
center. Find the slit separation.

Additional Problems R63

3 74. Two point charges of −13.0 μC and −16.0 μC exert 3 89. Find the electric force vector on a 5.0 nC charge in a
repulsive forces on each other of 12.5 N. What is the 1500 N/C electric field directed along the y-axis.
distance between the two charges?
390. What electric charge experiences an 8.42 × 10−9 N
3 75. Three equal point charges of 4.00 nC lie 4.00 m apart on electric force in an electric field of 1663 N/C?
a line. Calculate the magnitude and direction of the net
force on the middle charge. 391. Two 3.00 μC charges lie 2.00 m apart on the x-axis. Find
the resultant electric field vector at a point 0.250 m on
3 76. A proton is at each corner of a square with sides the y-axis, above the charge on the left.
1.52 × 10−9 m long. Calculate the resultant force vector
on the proton at the upper right corner. 392. Two electrons are 2.00 × 10−10 m and 3.00 × 10−10 m,
respectively, from a point. Where with respect to that
377. Three 2.0 nC charges are located at coordinates (0 m, point must a proton be placed so that the resultant
0 m), (1.0 m, 0 m), and (1.0 m, 2.0 m). Find the resultant electric field strength is zero?
force on the first charge.
3 93. A −7.0 C charge is in equilibrium with a 49 C charge
3 78. Charges of 7.2 nC and 6.7 nC are 32 cm apart. Find the 18 m to the right and an unknown charge 25 m to the
equilibrium position for a −3.0 nC charge. right. What is the unknown charge?

379. A −12.0 μC charge is between two 6.0 μC charges, 394. Suppose two pions are separated by 8.3 × 10−10 m. If
5.0 cm away from each. What electric force keeps the the magnitude of the electric force between the charges
central charge in equilibrium? is 3.34 × 10−10 N, what is the value of q?

380. A 9.0 N/C electric field is directed along the x-axis. Find 395. Suppose two muons having equal but opposite charge
the electric force vector on a −6.0 C charge. are separated by 6.4 × 10−8 m. If the magnitude of the
electric force between the charges is 5.62 × 10−14 N,
381. What charge experiences an electric force of what is the value of q?
6.43 × 10−9 N in an electric field of 4.0 × 103 N/C?
396. Consider four electrons at the corners of a square. Each
382. A 5.00 μC charge is 0.500 m above a 15.0 μC charge. side of the square is 3.02 × 10−5 m. Find the magnitude
Calculate the electric field at a point 1.00 m above the and direction of the resultant force on q3 if it is at the
15.0 mC charge. origin.

3 83. Two static point charges of 99.9 μC and 33.3 μC exert 3 97. A charge of 5.5 nC and a charge of 11 nC are separated
repulsive forces on each other of 87.3 N. What is the by 88 cm. Find the equilibrium position for a −22 nC
distance between the two charges? charge.

3 84. Two particles are separated by 9.30 × 10−11 m. If the 398. Three charges are on the y-axis. At the origin is a
magnitude of the electric force between the charges charge, q1 = 72 C; an unknown charge, q2, is at
is 2.66 × 10−8 N, what is the value of q? y = 15 mm. A third charge, q3 = −8.0 C, is placed at
y = −9.0 mm so that it is in electrostatic equilibrium
385. A −23.4 nC charge is 0.500 m below a 4.65 nC charge with q1 and q2. What is the charge on q2?
and 1.00 m below a 0.299 nC charge. Find the resultant
force vector on the −23.4 nC charge. Electrical Energy and Current

386. Three point charges are on the corners of a triangle: 3 99. A helium-filled balloon with a 14.5 nC charge rises
q1 = −9.00 nC is at the origin; q2 = −8.00 nC is at 290 m above Earth’s surface. By how much does the
x = 2.00 m; and q3 = 7.00 nC is at y = 3.00 m. Find the electrical potential energy change if Earth’s electric field
magnitude and direction of the resultant force on q1. is −105 N/C?

387. Charges of −2.50 nC and −7.50 nC are 20.0 cm apart. 4 00. A charged airplane rises 7.3 km in a 3.4 × 105 N/C
Find a 5.0 nC charge’s equilibrium position. electric field. The electrical potential energy changes
by −1.39 × 1011 J. What is the charge on the plane?
3 88. A −4.6 C charge is in equilibrium with a −2.3 C charge
2.0 m to the right, and an unknown charge 4.0 m to the
right. What is the unknown charge?

R64 Appendix I

4 01. Earth’s radius is 6.4 × 106 m. What is Earth’s capacitance 4 20. The potential difference across an electric eel is 650 V.
if it is regarded as a conducting sphere? How much current would an electric eel deliver to a
body with a resistance of 1.0 × 102 Ω?
402. A 0.50 pF capacitor is connected across a 1.5 V battery.
How much charge can this capacitor store? 4 21. If a garbage-disposal motor has a resistance of 25.0 Ω
and carries a current of 4.66 A, what is the potential
4 03. A 76 C charge passes through a wire’s cross-sectional difference across the motor’s terminals?
area in 19 s. Find the current in the wire.
4 22. A medium-sized oscillating fan draws 545 mA of
4 04. The current in a telephone is 1.4 A. How long does 98 C current when the potential difference across its motor
of charge take to pass a point in the wire? is 120 V. How large is the fan’s resistance?

4 05. What is a television’s total resistance if it is plugged into 423. A generator produces a 2.5 × 104 V potential difference
a 120 V outlet and carries 0.75 A of current? across power lines that carry 20.0 A of current. How
much power is generated?
406. A motor with a resistance of 12.2 Ω is plugged into a
120.0 V outlet. What is the current in the motor? 424. A computer with a resistance of 91.0 Ω uses 230.0 W of
power. Find the current in the computer.
407. The potential difference across a motor with a 0.30 Ω
resistance is 720 V. How much power is used? 4 25. A laser uses 6.0 × 1013 W of power. What is the potential
difference across the laser’s circuit if the current in the
408. What is a microwave oven’s resistance if it uses 1750 W circuit is 8.0 × 106 A?
of power at a voltage of 120.0 V?
4 26. A blender with a 75 Ω resistance uses 350 W of power.
409. A 64 nC charge moves 0.95 m with an electrical What is the current in the blender’s circuit?
potential energy change of −3.88 × 10−5 J. What is the
electric field strength? Circuits and Circuit Elements

410. A −14 nC charge travels through a 156 N/C electric field 427. A theater has 25 surround-sound speakers wired in
with a change of 2.1 × 10−6 J in the electrical potential series. Each speaker has a resistance of 12.0 Ω. What
energy. How far does the charge travel? is the equivalent resistance?

411. A 5.0 × 10−5 F polyester capacitor stores 6.0 × 10−4 C. 428. In case of an emergency, a corridor on an airplane has
Find the potential difference across the capacitor. 57 lights wired in series. Each light bulb has a resistance
of 2.00 Ω. Find the equival­ent resistance.
4 12. Some ceramic capacitors can store 3 × 10−2 C with a
potential difference of 30 kV across them. What is the 4 29. Four resistors with resistances of 39 Ω, 82 Ω, 12 Ω, and
capacitance of such a capacitor? 42 Ω are connected in parallel across a 3.0 V potential
difference. Find the equivalent resistance.
4 13. The area of the plates in a 4550 pF parallel-plate
capacitor is 6.4 × 10−3 m2. Find the plate separation. 4 30. Four resistors with resistances of 33 Ω, 39 Ω, 47 Ω, and
68 Ω are connected in parallel across a 1.5 V potential
414. A television receiver contains a 14 μF capacitor charged difference. Find the equivalent resistance.
across a potential difference of 1.5 × 104 V. How much
charge does this capacitor store? 431. A 16 Ω resistor is connected in series with another
resistor across a 12 V battery. The current in the circuit
4 15. A photocopier uses 9.3 A in 15 s. How much charge is 0.42 A. Find the unknown resistance.
passes a point in the copier’s circuit in this time?
432. A 24 Ω resistor is connected in series with another
4 16. A 114 μC charge passes through a gold wire’s cross- resistor across a 3.0 V battery. The current in the circuit
sectional area in 0.36 s. What is the current? is 62 mA. Find the unknown resistance.

4 17. If the current in a blender is 7.8 A, how long do 56 C 4 33. A 3.3 Ω resistor and another resistor are connected in
of charge take to pass a point in the circuit? parallel across a 3.0 V battery. The current in the circuit
is 1.41 A. Find the unknown resistance.
4 18. A computer uses 3.0 A in 2.0 min. How much charge
passes a point in the circuit in this time?

419. A battery-powered lantern has a resistance of 6.4 Ω.
What potential difference is provided by the battery
if the total current is 0.75 A?

Additional Problems R65

4 34. A 56 Ω resistor and another resistor are connected in 4 45. For the figure above, what is the current in the 3.0 Ω
parallel across a 12 V battery. The current in the circuit resistors?
is 3.21 A. Find the unknown resistance.
8.0 Ω 2.0 Ω

4 35. Three bulbs with resistances of 56 Ω, 82 Ω, and 24 Ω are 4.0 Ω 3.0 Ω 3.0 Ω
wired in series. If the voltage across the circuit is 9.0 V,
what is the current in the circuit? 24.0 V

4 36. Three bulbs with resistances of 96 Ω, 48 Ω, and 29 Ω are 5.0Ω 2.0Ω 4.0 Ω
wired in series. What is the current through the bulbs if 8.0 Ω
the voltage across them is 115 V?

437. A refrigerator (R1 = 75 Ω) wired in parallel with an oven 4 46. For the figure above, calculate the equivalent resistance
(R2 = 91 Ω) is plugged into a 120 V outlet. What is the of the circuit.
current in the circuit of each appliance?
447. For the figure above, what is the total current in the
438. A computer (R1 = 82 Ω) and printer (R2 = 24 Ω) are circuit?
wired in parallel across a 120 V potential difference.
Find the current in each machine’s circuit. 448. For the figure above, what is the current in either of the
8.0 Ω resistors?
5.0 Ω 5.0 Ω
2.0 Ω 4.0 Ω 6.0 Ω

3.0Ω 1.5Ω Magnetism

12.0 V 449. A proton moves at right angles to a magnetic field of
0.8 T. If the proton’s speed is 3.0 × 107 m/s, how large
4 39. For the figure above, what is the equivalent resistance is the magnetic force exerted on the proton?
of the circuit?
450. A weak magnetic field exerts a 1.9 × 10−22 N force on
4 40. For the figure above, find the current in the circuit. an electron moving 3.9 × 106 m/s perpendicular to the
field. What is the magnetic field strength?
4 41. For the figure above, what is the potential difference
across the 6.0 Ω resistor? 4 51. A 5.0 × 10−5 T magnetic field exerts a 6.1 × 10−17 N
force on a 1.60 × 10−19 C charge, which moves at a
4 42. For the figure above, what is the current through the right angle to the field. What is the charge’s speed?
6.0 Ω resistor?
452. A 14 A current passes through a 2 m wire. A
5.0 Ω 3.6 × 10−4 T magnetic field is at right angles to the
wire. What is the magnetic force on the wire?
5.0 Ω 3.0 Ω
5.0 Ω 4 53. A 1.0 m printer cable is perpendicular to a 1.3 × 10−4 T
3.0 Ω 5.0 Ω 15.0 V magnetic field. What current must the cable carry to
5.0 Ω experience a 9.1 × 10−5 N magnetic force?
5.0 Ω
4 54. A wire perpendicular to a 4.6 × 10−4 T magnetic field
443. For the figure above, calculate the equivalent resistance experiences a 2.9 × 10−3 N magnetic force. How long is
of the circuit. the wire if it carries a 10.0 A current?

4 44. For the figure above, what is the total current in the 4 55. A 12 m wire carries a 12 A current. What magnetic field
circuit? causes a 7.3 × 10−2 N magnetic force to act on the wire
when it is perpendicular to the field?

4 56. A magnetic force of 3.7 × 10−13 N is exerted on an
electron moving at 7.8 × 106 m/s perpend­ icular to a
sunspot. How large is the sunspot’s magnetic field?

R66 Appendix I