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Problems in Introductory Physics
B. Crowell and B. Shotwell
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Copyright 2016 B. Crowell and B. Shotwell.
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Contents
1 Measurement 7
1.1 The SI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Significant figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Proportionalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Kinematics in one dimension 15
2.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Kinematics in three dimensions 31
3.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Newton’s laws, part 1 43
4.1 Newton’s first law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Newton’s second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Newton’s third law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Newton’s laws, part 2 51
5.1 Classification of forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 Ropes, pulleys, tension, and simple machines . . . . . . . . . . . . . . . . . . . . . . 53
5.5 Analysis of forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Circular motion 69
6.1 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Rotating frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Nonuniform motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.4 Rotational kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7 Conservation of energy 79
7.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3
4 CONTENTS
7.4 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8 Conservation of momentum 97
8.1 Momentum: a conserved vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.3 The center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9 Conservation of angular momentum 105
9.1 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.2 Rigid-body dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.3 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9.4 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10 Fluids 121
10.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
11 Gravity 127
11.1 Kepler’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.2 Newton’s law of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.3 The shell theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.4 Universality of free fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
11.5 Current status of Newton’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
11.6 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12 Oscillations 141
12.1 Periodic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.2 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.3 Damped oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.4 Driven oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
13 Waves 151
13.1 Free waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
13.2 Bounded waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
14 Electrical interactions 157
14.1 Charge and Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
14.2 The electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
14.3 Conductors and insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
14.4 The electric dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
CONTENTS 5
14.5 The field of a continuous charge distribution . . . . . . . . . . . . . . . . . . . . . . . 158
14.6 Gauss’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
14.7 Gauss’s law in differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
15 The electric potential 167
15.1 The electric potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
15.2 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
15.3 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
15.4 Poisson’s equation and Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . 168
15.5 The method of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
16 Circuits 175
16.1 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
16.2 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
16.3 The loop and junction rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
17 Basics of relativity 183
17.1 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
17.2 Length contraction and time dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
18 Electromagnetism 187
18.1 Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
18.2 The magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
19 Maxwell’s equations and electromagnetic waves 195
19.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
19.2 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
19.3 Maxwell’s equations in matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
20 LRC circuits 199
20.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
20.2 LRC circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
21 Thermodynamics 205
21.1 Pressure, temperature, and heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
21.2 Kinetic theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
21.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6 CONTENTS
22 Optics 211
22.1 Geometric optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
22.2 Wave optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
23 More about relativity 225
23.1 More about relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
24 Quantum physics 227
24.1 The nucleus, half-life, and probability . . . . . . . . . . . . . . . . . . . . . . . . . . 227
24.2 Wave-particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
1 Measurement
This is not a textbook. It’s a book of problems The digits after the first few are completely
meant to be used along with a textbook. Although meaningless. Sincethecircumferencecouldhave
each chapter of this book starts with a brief sum- varied by about a centimeter in either direction,
mary of the relevant physics, that summary is the diameter is fuzzy by something like a third
not meant to be enough to allow the reader to ofacentimeter. Wesaythattheadditional,ran-
actually learn the subject from scratch. The pur- dom digits are not significant figures. If you
pose of the summary is to show what material is write down a number with a lot of gratuitous
needed in order to do the problems, and to show insignificant figures, it shows a lack of scientific
what terminology and notation are being used. literacyandimplestootherpeopleagreaterpre-
cision than you really have.
As a rule of thumb, the result of a calculation
1.1 The SI
has as many significant figures, or “sig figs,” as
the least accurate piece of data that went in. In
The Syst`eme International (SI) is a system of
the example with the soccer ball, it didn’t do us
measurementinwhichmechanicalquantitiesare
any good to know π to dozens of digits, because
expressed in terms of three basic units: the me-
the bottleneck in the precision of the result was
ter (m), the kilogram (kg), and the second (s).
the figure for the circumference, which was two
Other units can be built out of these. For ex-
sig figs. The result is 22 cm. The rule of thumb
ample, the SI unit to measure the flow of water
works best for multiplication and division.
through a pipe would be kg/s.
The numbers 13 and 13.0 mean different
Tomodifytheunitsthereisaconsistentsetof
things, because the latter implies higher preci-
prefixes. The following are common and should
sion. The number 0.0037 is two significant fig-
be memorized:
ures, not four, because the zeroes after the dec-
prefix meaning imal place are placeholders. A number like 530
nano- n 10−9 couldbeeithertwosigfigsorthree;ifwewanted
micro- µ 10−6 toremovetheambiguity,wecouldwriteitinsci-
milli- m 10−3 entific notation as 5.3×102 or 5.30×102.
kilo- k 103
mega- M 106 1.3 Proportionalities
Thesymbolµ,formicro-,isGreeklowercasemu,
Often it is more convenient to reason about the
which is equivalent to the Latin “m.” There is
also centi-, 10−2, which is only used in the cen- ratios of quantities rather than their actual val-
ues. Forexample,supposewewanttoknowwhat
timeter.
happenstotheareaofacirclewhenwetripleits
radius. We know that A = πr2, but the factor
1.2 Significant figures of π is not of interest here because it’s present
in both cases, the small circle and the large one.
The international governing body for football Throwing away the constant of proportionality,
(“soccer” in the US) says the ball should we can write A ∝ r2, where the proportionality
have a circumference of 68 to 70 cm. Tak- symbol ∝, read “is proportional to,” says that
ing the middle of this range and divid- the left-hand side doesn’t necessarily equal the
ing by π gives a diameter of approximately right-handside, butitdoesequaltheright-hand
21.96338214668155633610595934540698196 cm. side multiplied by a constant.
7
8 CHAPTER 1. MEASUREMENT
Any proportionality can be interpreted as a
statement about ratios. For example, the state-
ment A ∝ r2 is exactly equivalent to the state-
ment that A /A = (r /r )2, where the sub-
1 2 1 2
scripts 1 and 2 refer to any two circles. This in
ourexample,thegiveninformationthatr /r =
1 2
3 tells us that A /A =9.
1 2
In geometrical applications, areas are always
proportional to the square of the linear dimen-
sions, while volumes go like the cube.
1.4 Estimation
It is useful to be able to make rough estimates,
e.g., how many bags of gravel will I need to fill
my driveway? Sometimes all we need is an esti-
mate so rough that we only care about getting
theresulttoaboutthenearestfactoroften, i.e.,
to within an order of magnitude. For example,
anyone with a basic knowledge of US geography
can tell that the distance from New Haven to
NewYorkisprobablysomethinglike100km,not
10 km or 1000 km. When making estimates of
physical quantities, the following guidelines are
helpful:
1. Don’t even attempt more than one signifi-
cant figure of precision.
2. Don’t guess area, volume, or mass directly.
Guess linear dimensions and get area, vol-
ume,ormassfromthem. Massisoftenbest
found by estimating linear dimensions and
density.
3. When dealing with areas or volumes of ob-
jects with complex shapes, idealize them as
if they were some simpler shape, a cube or
a sphere, for example.
4. Check your final answer to see if it is rea-
sonable. If you estimate that a herd of
ten thousand cattle would yield 0.01 m2 of
leather,thenyouhaveprobablymadeamis-
take with conversion factors somewhere.
PROBLEMS 9
Problems superdupermean,definedas(ab)1/3. Isthisrea-
sonable?
1-a1 Convert 134 mg to units of kg, writing (cid:46) Solution, p. 233
your answer in scientific notation.
(cid:46) Solution, p. 233
1-a2 Express each of the following quantities 1-d2 (a) Based on the definitions of the sine,
in micrograms: cosine,andtangent,whatunitsmusttheyhave?
(a) 10 mg, (b) 104 g, (c) 10 kg, (d) 100×103 g, (b) A cute formula from trigonometry lets you
√
(e) 1000 ng. find any angle of a triangle if you know the
lengths of its sides. Using the notation shown
inthefigure,andlettings=(a+b+c)/2behalf
the perimeter, we have
1-a3 In the last century, the average age of
the onset of puberty for girls has decreased by (cid:115)
several years. Urban folklore has it that this (s−b)(s−c)
tanA/2= .
is because of hormones fed to beef cattle, but s(s−a)
it is more likely to be because modern girls
have more body fat on the average and possibly Showthattheunitsofthisequationmakesense.
because of estrogen-mimicking chemicals in the Inotherwords,checkthattheunitsoftheright-
environment from the breakdown of pesticides. hand side are the same as your answer to part a
A hamburger from a hormone-implanted steer of the question.
has about 0.2 ng of estrogen (about double the (cid:46) Solution, p. 233
amount of natural beef). A serving of peas
contains about 300 ng of estrogen. An adult
woman produces about 0.5 mg of estrogen per
day (note the different unit!). (a) How many
hamburgers would a girl have to eat in one
day to consume as much estrogen as an adult
woman’s daily production? (b) How many
√
servings of peas?
1-d1 The usual definition of the mean (aver-
age) of two numbers a and b is (a+b)/2. This
is called the arithmetic mean. The geometric
mean, however, is defined as (ab)1/2 (i.e., the
square root of ab). For the sake of definiteness,
let’s say both numbers have units of mass. (a)
Compute the arithmetic mean of two numbers
thathaveunitsofgrams. Thenconvertthenum-
bers to units of kilograms and recompute their
Problem 1-d2.
mean. Istheanswerconsistent? (b)Dothesame
forthegeometricmean. (c)Ifaandbbothhave
units of grams, what should we call the units 1-d3 Jae starts from the formula V = 1Ah
3
of ab? Does your answer make sense when you for the volume of a cone, where A is the area of
take the square root? (d) Suppose someone pro- its base, and h is its height. He wants to find
poses to you a third kind of mean, called the an equation that will tell him how tall a conical
10 CHAPTER 1. MEASUREMENT
tent hastobe inorderto have acertainvolume,
given its radius. His algebra goes like this:
1
V = Ah Problem 1-g2.
3
A=πr2
1
V = πr2h
1-j1 Theone-litercubeinthephotohasbeen
3
πr2 markedoffintosmallercubes,withlineardimen-
h= sionsonetenththoseofthebigone. Whatisthe
3V
volume of each of the small cubes?
Useunitstocheckwhetherthefinalresultmakes (cid:46) Solution, p. 233
sense. If it doesn’t, use units to locate the line
of algebra where the mistake happened.
(cid:46) Solution, p. 233
1-d4 The distance to the horizon is given by
√
theexpression 2rh,wherer istheradiusofthe
Earth, and h is the observer’s height above the
Earth’s surface. (This can be proved using the
Pythagorean theorem.) Show that the units of
this expression make sense. Don’t try to prove
the result, just check its units. (For an example
ofhowtodothis,seeproblem1-d3onp.9,which
has a solution given in the back of the book.)
Problem 1-j1.
1-d5 Let the function x be defined by x(t)=
Aebt, where t has units of seconds and x has
unitsofmeters. (Forb<0,thiscouldbeafairly 1-j2 How many cm2 is 1 mm2?
accurate model of the motion of a bullet shot (cid:46) Solution, p. 233
into a tank of oil.) Show that the Taylor series 1-j3 Comparethelight-gatheringpowersofa
of this function makes sense if and only if A and 3-cm-diameter telescope and a 30-cm telescope.
b have certain units. (cid:46) Solution, p. 233
1-j4 The traditional Martini glass is shaped
like a cone with the point at the bottom. Sup-
1-g1 InanarticleontheSARSepidemic, the
pose you make a Martini by pouring vermouth
May 7, 2003 New York Times discusses conflict-
intotheglasstoadepthof3cm,andthenadding
ing estimates of the disease’s incubation period
gin to bring the depth to 6 cm. What are the
(the average time that elapses from infection to
proportions of gin and vermouth?
the first symptoms). “The study estimated it to
(cid:46) Solution, p. 233
be 6.4 days. But other statistical calculations ...
1-j5 How many cubic inches are there in a
showed that the incubation period could be as
cubic foot? The answer is not 12.
long as 14.22 days.” What’s wrong here?
√
1-j6 Assume a dog’s brain is twice as great
1-g2 The photo shows the corner of a bag of
in diameter as a cat’s, but each animal’s brain
pretzels. What’s wrong here?
cells are the same size and their brains are the
same shape. In addition to being a far better