General Physics I:
Classical Mechanics
D.G. Simpson, Ph.D.
Department of Physical Sciences and Engineering
Prince George’s C ommunity College
Largo, Maryland
Fall 2013
Last updated: December 16, 2013
Contents
Acknowledgments 11
1WhatisPhysics? 12
2Units 14
2.1 SystemsofUnits 14
2.2 SIUnits 15
2.3 CGSSystemsofUnits 18
2.4 British Engineering Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 UnitsasanError-CheckingTechnique 18
2.6 UnitConversions 19
2.7 CurrencyUnits 20
2.8 OddsandEnds 21
3 Problem-Solving Strategies 22
4Density 24
4.1 SpecificGravity 25
4.2 DensityTrivia 25
5 Kinematics in One Dimension 27
5.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Velocity 27
5.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.4 HigherDerivatives 29
5.5 DotNotation 29
5.6 InverseRelations 29

9.5 Hitting a Target on a Hill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.6 OtherConsiderations 57
9.7 The Monkey and the Hunter Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.8 Summary 59
10 Newton’s Method 60
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
10.2 TheMethod 60
10.3 Example:SquareRoots 60
10.4 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
11 Mass 63
12 Force 64
12.1 TheFourForcesofNature 64
12.2 Hooke’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.3 Weight 65
12.4 NormalForce 65
12.5 Tension 65
13 Newton’s Laws of Motion 66
13.1 FirstLawofMotion 67
13.2 SecondLawofMotion 67
13.3 ThirdLawofMotion 67
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Prince George’s Community College General Physics I D.G. Simpson
14 The Inclined Plane 68
15 Atwood’s Machine 69
16 Statics 73
16.1 MassSuspendedbyTwoRopes 73
16.2 ThePulley 76
16.3 TheElevator 76
17 Friction 78
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

21.3 PotentialEnergy 94
21.4 OtherFormsofEnergy 97
21.5 ConservationofEnergy 97
21.6 TheWork-EnergyTheorem 98
21.7 TheVirialTheorem 98
22 Conservative Forces 100
23 Power 101
23.1 Energy Conversion of a Falling Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
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Prince George’s Community College General Physics I D.G. Simpson
23.2 RateofChangeofPower 102
23.3 VectorEquation 103
24 Linear Momentum 104
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
24.2 ConservationofMomentum 104
24.3 Newton’sSecondLawofMotion 104
25 Impulse 106
26 Collisions 108
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
26.2 The Coefficient of Restitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
26.3 Perfectly Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
26.4 Perfectly Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
26.5 Newton’sCradle 111
26.6 Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
26.7 Collisions in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
27 The Ballistic Pendulum 114
28 Rockets 116
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
28.2 TheRocketEquation 116
28.3 MassFraction 117

34.3 The Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
34.4 The Conical Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
34.5 The Torsional Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
34.6 The Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
34.7 Other Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
35 Simple Harm onic Motion 148
35.1 Energy 150
35.2 FrequencyandPeriod 152
35.3 MassonaSpring 152
36 Rolling Bodies 154
36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
36.2 Velocity 154
36.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
36.4 KineticEnergy 156
36.5 TheWheel 157
36.6 Ball Rolling in a Bowl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
37 Galileo’s Law 160
37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
37.2 ModernTreatment 160
38 The Coriolis Force 162
38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
38.2 Examples 163
39 Angular Momentum 164
39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
39.2 Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
40 Conservation Laws 166
41 The Gyroscope 167
41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
41.2 Precession 167
41.3 Nutation 168

45 Hydraulics 188
45.1 TheHydraulicPress 188
46 Pneumatics 190
47 Gravity 191
47.1 Newton’sLawofGravity 191
47.2 GravitationalPotential 191
47.3 TheCavendishExperiment 192
47.4 Helmert’sEquation 192
47.5 EscapeVelocity 193
47.6 Gauss’sFormulation 193
47.7 GeneralRelativity 197
47.8 BlackHoles 198
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Prince George’s Community College General Physics I D.G. Simpson
48 Earth Rotation 199
48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
48.2 Precession 199
48.3 Nutation 199
48.4 PolarMotion 201
48.5 RotationRate 201
49 Geodesy 203
49.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
49.2 TheCosineFormula 203
49.3 Vincenty’s Formulæ: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
49.4 Vincenty’sFormulæ:DirectProblem 204
49.5 Vincenty’sFormulæ:InverseProblem 206
50 Celestial Mechanics 209
50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
50.2 Kepler’sLaws 209
50.3 Time 210

55 Special Relativity 241
55.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
55.2 Postulates 241
55.3 TimeDilation 241
55.4 LengthContraction 242
55.5 AnExample 242
55.6 Momentum 242
55.7 Addition of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
55.8 Energy 243
56 Quantum Mechanics 245
56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
56.2 ReviewofNewtonianMechanics 245
56.3 QuantumMechanics 245
56.4 Example: Simple Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
56.5 TheHeisenbergUncertaintyPrinciple 248
AFurtherReading 250
B Greek Alphabet 254
C Trigonometry 255
D Useful Series 258
ESIUnits 259
F Gaussian Units 262
G Units of Physical Quantities 264
H Physical Constants 267
IAstronomicalData 268
J Unit Conversion Tables 269
K Angular Measure 272
K.1 PlaneAngle 272
K.2 SolidAngle 272
L Vector Arithmetic 274
M Matrix Properties 277

T.3 HyperbolicKepler’sEquation 304
T.4 Barker’sEquation 306
T.5 ReductionofanAngle 307
T.6 Helmert’sEquation 308
T.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
T.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
U HP 48 / HP 50g Calculator Programs 313
U.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
U.2 Kepler’sEquation 314
U.3 HyperbolicKepler’sEquation 314
U.4 Barker’sEquation 315
U.5 ReductionofanAngle 315
U.6 Helmert’sEquation 316
U.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
U.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
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Prince George’s Community College General Physics I D.G. Simpson
V HP Prime Calculator Programs 318
V.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
V.2 Kepler’sEquation 319
V.3 HyperbolicKepler’sEquation 319
V.4 Barker’sEquation 320
V.5 ReductionofanAngle 321
V.6 Helmert’sEquation 321
V.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
V.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
W Round-Number Handbook of Physics 324
X Fundamental Physical Constants — Extensive Listing 326
Y Periodic Table of the Elements 333
Refer ences 333

• Relativity includes Albert E instein’s theories of special and general relativity. Special relativity de-
scribes the motion of bodies movi ng at very high speeds (near t he speed of l i ght), while general rela-
tivity is Einstein’s theory of gravity.
The fields of cross-disciplinary physics combine physics with other sciences. These include astrophysics
(physics of astronomy), geophysics (physics of geology), biophysics (physics of biology), chemical physics
(physics of chemistry), and mathematical physics (mathematical theories related to physics).
12
Prince George’s Community College General Physics I D.G. Simpson
Besides acquiring a knowledge of physics for its own sake, the st udy of physics will give you a broad tech-
nical background and set of problem-solving skills that you can apply to wide variety of other fields. Some
students of physics go on to study more advanced physics, while others find ways to apply their knowledge
of physics to such diverse subjects as mathematics, engineering, biology, medicine, and finance.
13
Chapter 2
Units
The phenomena of Nature have been found to obey certain physical laws; one of the primary goals of physics
research is to discover those l aws. It has been known for several centuries that the laws of physics are
appropriately expressed in the language of mathematics, so physics and mathematics have enj oyed a close
connection for quite a long time.
In order to connect the physical world to the mathematical worl d, we need t o make measurements of the
real world. In making a measurement, we compare a physical quantity with some agreed-upon standard, and
determine how many such standard uni t s are present . For example, we have a precise definition of a unit of
length called a mile, and have determined that t here are about 92,000,000 such miles between the Earth and
the Sun.
It is important that we have very precise definitions of physical units — not onl y for scientific use, but also
for trade and commerce. In practice, we define a few base units, and derive other units from combinations of
those base units. For example, i f we define unit s for length and ti me, then we can define a unit for speed as
the length divided by time (e.g. miles/hour).
How many base units do we need to define? There is no magic number; in fact it i s possible to define
a system of units using only one base unit (and this is in fact done for so-called natural units). For most

the krypton-86 atomic spectrum. Still more stringent accuracy requirements led to the the current definition
of the meter, which was implement ed in 1983: the meter is now defined to be t he distance light in vacuum
travels in 1=299;792;458 second. Because o f this definition, the speed of l i ght is now exactly 299;792;458
m/s.
U.S. Customary units are legally defined in terms of metric equivalents. For length, t he foot (ft) is defined
to be exactly 0.3048 meter.
Mass (Kilogram)
Originally the kilogram (kg) was defined to be the mass of 1 liter (0.001 m
3
) of water. The need for more
accuracy required the kilogram to be re-defined to be the mass of a standard mass called the International
Prototype Kilogram (IPK, frequently designated by the Gothic letter K), which is kept in a vault at the Bureau
International des Poids et Mesures (BIPM) in Paris. The kilogram is the only base unit still definedinterms
of a prototype, rather than in terms of an experiment that can be dupl icated in the laboratory.
The International Prototype Kilogram i s a small cylinder of platinum-iridium alloy (90% platinum), about
the size of a golf ball. In 1884, a set of 40 duplicates of the IPK was made; each country that requested one
got one of these duplicates. The United States received two of these: the duplicate called K20 arrived here
in 1890, and has been t he standard of mass for t he U.S. ever since. The second copy, called K4, arrived later
that same year, and is used as a constancy check on K20. Finally, in 1996 the U.S. got a third standard called
K79; this is used for mass stability studies. These duplicates are kept at the National Institutes of Standards
and Technology (NIST) in Gaithersburg, Maryland. They are kept under very controlled conditions under
several layers of glass bell jars and are periodically cleaned. From time to time they are returned to the BIPM
in Paris for re-calibration. For reasons not entirely understood, very careful calibration measurements show
that the masses of the d uplicates do not stay exactly constant. Because of this, physicists are considering
re-defining the kilogram sometime in the next few years.
Another common met ri c (but non-SI) unit of mass is the metric ton, which is 1000 kg (a little over 1 short
ton).
In U.S. customary unit s, the pound-mass (lbm) is defined to be exactly 0:45359237 kg.
Mass vs. Weight
Mass is not t he same thi ng as weight, so it ’s important not to confuse the two. The mass of a body is a

23
. You could have a mole of grai ns of sand or a mole of Volkswagens, but most often the
mole is used to count atoms or molecul es. There is a reason this number is particularly useful: since each nu-
cleon (proton and neutron) in an atomic nucleus has an average mass of 1:660538921  10
24
grams (called
an atomic mass uni t, or amu), then there are 1=.1:660538921 10
24
/,or6:02214129  10
23
nucleons per
gram. In other words, one mole of nucleons has a mass of 1 gram. Therefore, i f A is the atomic weight of an
atom, t hen A moles of nucleons has a mass of A grams. But A moles of nucleons is the same as 1 mole of
atoms, so one mole of atoms has a mass (in grams) equal to t he atomic weight.Inotherwords,
moles of atoms D
grams
atomic weight
(2.2)
Similarly, when counting molecules,
moles of molecules D
grams
molecular weight
(2.3)
In short, the mole is useful when you need to convert between the mass of a material and the number of
atoms or molecules it contains.
It’s important to be clear about what exactly you’re counting (atoms or molecules) when using moles. It
doesn’t really make sense t o talk about “a mole of oxygen”, any more t han i t would be to talk about “100 of
oxygen”. It ’s either a “mole of oxygen atoms” or a “mole of oxygen molecules”.
3
Interesting fact: t here is about

radians. To con v ert between degrees and radians, then, we hav e:
degrees D radians 
180

(2.4)
and
radians D degrees 

180
(2.5)
The easy way to remember these formulæ is to think in terms of units: 180 has units of degrees and  has
units of radians, so in the first equation units of radians cancel on the right -hand side to leave degrees, and in
the second equation units of degrees cancel on t he right -hand side to leave radians.
Occasionally you w ill see a formula that inv olves a “bare” angle that is not the ar gument of a trigonometric
function like the sine, cosine, or tangent. In such cases it is understood that the angle must be in radians.For
example, the radius of a circle r, angle  , and arc length s are related by
s D rÂ; (2.6)
where it is understood that  is in radians.
See Appendix K for a furt her discussion of plane and soli d angles.
SI Prefixes
It’s often convenient to define both large and small unit s that measure the same thing. For example, in Engl i sh
units, it’s convenient to measure small lengths in inches and large lengths in miles.
In SI units, larger and smaller units are definedinasystematicwaybytheuseofprefixes to the SI base
or derived units. For example, t he base SI unit of length is the meter (m), but small lengths may also be
measured in centimeters (cm, 0.01 m), and l arge lengths may b e measured in kilometers (km, 1000 m). Table
E-3 i n Appendix E shows all the SI prefixes and the powers of 10 they represent. You should memorize the
powers of 10 for all the SI prefixes in this table.
To use the SI prefixes, simply add the prefix to the front of the name of the SI base or derived unit. The
symbol for t he prefixed unit i s the symbol for the prefix written in front of the symbol for the unit. For
example, kilometer (km) D 10

the surface of the Earth, a mass of 1 lbm has a weight of 1 lbf, so sometimes the two are loosely used
interchangeably and called the pound (lb), as we do every day when we speak of weights i n pounds.
SI prefixes are not used in the British engineering system.
2.5 Units as an E rror-Checking Technique
Checking units can be used as an important error-checking technique called dimensional analysis. If you
derive an equation and find that the units don’t work out properly, then you can be certain you made a
mistake somewhere. If the units are correct, it doesn’t necessarily mean your derivation is correct (since you
could be off by a factor of 2, for example), but i t does give you some confidence t hat you at least haven’t
made a uni t s error. So checking units doesn’t t ell you for certain whether or not you’ve made a mistake, but
it does help.
Here are some basic principles to keep in mind when working with units:
1. Units on both sides of an equation must match.
2. When adding or subtracting two quantities, they must have the same units.
3. Quantities that appear in exponents must be dimensionless.
4. The argument for functions like sin, cos, tan, sin
1
,cos
1
,tan
1
, log, and exp must be dimensionless.
5. W hen checking units, radians and steradians can be considered dimensionless.
6. W hen checking complicated units, it may be useful to break down all derived units into base units (e.g.
replace newtons with kg m s
2
).
Sometimes it’s not clear whether or not t he unit s match on bot h sides of the equation, for example when
both sides involve derived SI units. In that case, it may be useful to break all the derived units down in terms
of base SI units (m, kg, s, A, K, mol, cd). Table E-2 in Appendix E shows each of the derived SI units broken
down in terms of base SI units.

(2.8)
Now do the arithmetic:
.7 ft/ 
12 in
1ft
D 84 inches: (2.9)
More Complex Conversions
More complex conversions may involve more t han one conversion factor. You’ll need to think about what
conversion factors you know, then put together a chain of them to get to t he uni ts you want.
Example. Convert 60 miles per hour to feet per second.
Solution. First, writ e down a chain of conversion factor ratios, filling in units so that they cancel out
correctly:
60
mile
hr

ft
mile

hr
sec
(2.10)
Units cancel out to leave ft/sec. Now fill in the numbers, putting the same length in the numerator and
denominator in the first factor, and t he same time in the numerator and denominator in the second factor:
60
mile
hr

5280 ft
1 mile

mile
furlong

km
mile

m
km

fortnight
week

week
day

day
hr

hr
min

min
sec
If you check the unit s here, you’ll see t hat almost everything cancels out; the only units left are m/s, which is
what we want to convert to. Now fill in the numbers: we want to put either the same length or the same time
in bot h the numerator and denominator:
250;000
furlong
fortnight


can convert between feet and inches. The conversion factors will look like this:
2000 ft
3

Â
in
ft
Ã
3

gal
in
3
(2.13)
With these uni t s, the whol e expression reduces to units of gallons. Now fill in the same length in the numerator
and denominator of the first factor, and the same volume in the numerator and denominator of the second
factor:
2000 ft
3

Â
12 in
1 ft
Ã
3

1 gal
231 in
3
(2.14)

D $58:71 (2.16)
2.8 Odds and E nds
We’ll end this chapter with a few miscellaneous notes about SI units:
• In a few special cases, we customarily drop the ending vowel of a p refix when combining with a unit
that begins with a vowel: it’s megohm (not “megaohm”); kilohm (not “kiloohm”); and hectare (not
“hectoare”). In all other cases, keep both vowels (e.g. microohm, kiloare, etc.). There’s no particular
reason for this—it’s just customary.
• In pharmacology (on bottles of vitamins or prescription medicine, for example), it is usual to indicate
micrograms with “mcg” rather than “g”. While thi s i s t echnically i ncorrect, it i s done to avoid mis-
reading the units. Using “mc” for “micro” is not done outside pharmacology, and you should not use it
in physics. Always use  for “micro”.
• Sometimes in electronics work the SI prefix symbol may be used in place of the decimal point. For
example, 24.9 M may be written “24M9”. This saves space on electronic diagrams and when print-
ing values on electronic component s, and also avoids problems with the decimal point being nearly
invi sible when the print is t iny. This is unofficial use, and is only encountered in electronics.
• One sometimes encounters older metric units of length called the micron (, now properly called the
micrometer, 10
6
meter) and the millimicron (m, now properly called t he nanometer, 10
9
meter).
The micron and millimicron are now obsolete.
• At one time there was a metric prefix myria- (my) that meant 10
4
.Thisprefix is obsolete and is no
longer used.
• In computer work, the SI prefixes are often used with units of bytes, but may refer t o powers of 2 that
are near the SI values. For example, t he term “1 kB” may mean 1000 bytes, or it may mean 2
10
D 1024

• If you’ve derived an algebraic equation, check the units of your answer. Make sure your equation has
the correct units, and doesn’t do something like add quantities with different units.
• If you’ve derived an al gebraic equation, you can check that it has the proper behavior for extreme
val ues of the variables. For example, does the answer make sense if time t !1? If the equation
contains an angle, does it reduce to a sensible answer when the angle is 0
ı
or 90
ı
?
• Check your answer for reasonabl eness—don’t j ust write down whatever your calculator says. For
example, suppose you’re computing the sp eed of a pendulum bob in the laboratory, and find the answer
is 14;000 miles per hour. That doesn’t seem reasonable, so you should go back and check your work.
22
Prince George’s Community College General Physics I D.G. Simpson
• You can avoi d rounding errors by carrying as many significant digits as possible t hroughout your cal-
culations; don’t round off unt i l you get to the final result.
• Write down a reasonable number of significant digits in the final answer—don’t wri t e down all the
digits i n your calculator’s display. Nor should you round t oo much and use too few significant digits.
There are rules for determining the correct number of significant digits, but for most problems in this
course, 3 or 4 significant digits will be about right.
• Don’t forget to put t he correct uni t s on the final answer! You will have points deducted for forgetting
to do this.
• The best way t o get good at problem solving (and to prepare for exams for t his course) is practice—
practice working as many problems as you have time for. Working physics p roblems is a skil l much like
learning to play a sport or musical instrument. You can’t l earn by wat ching someone else do it— you
can only learn it by doing it yourself.
23
Chapter 4
Density
As an example of a quantity involving mixed units, consider the important quantity called density. Density is

charge density (electric charge per unit volume) or a number density (number of particles per unit volume).
Unless otherwise indicated, though, the word “density” usually refers to mass density.
Often the density of a mat erial is a useful clue to determining its composition. For example, suppose
you’re handed a gold-colored brick. Is the brick solid gold, or is it just a block of lead covered with gold
paint? Of course, you could just scrat ch the brick t o see if t he gold is just painted on, but suppose you don’t
want t o damage the brick? One test you mi ght do is determine the brick’s density. First, determine the volume
of the block (either by measuring the brick or by immersing it in a calibrated beaker of water). Then place
the brick on a scale to find its mass. Now divide the mass by the volume to find the density, and compare
with the densities of gold (19.3 g/cm
3
) and lead (11.3 g/cm
3
).
Densities of some common materials are shown in Table 4-1.
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