BOOK PROBLEMS C 1
2 a. 1 cm2 [1 in/2.54 cm]2 = 1 cm2(1 in2/6.452cm2) = 0.155 in2
b. 1 cm3 [1 in/2.54 cm]3 = 1 cm3(1 in3/16.387 cm3) = 0.061 in3
c. 1 L (1000 mL/L) = 1000 mL
d. 1 L (1000 mL/L)(1 cm3/mL) = 1000 cm3
e. 1 L (1000 mL/L)(1 cm3/mL)[1 in/2.54 cm]3 = 1 L (1000 mL/L)(1 cm3/mL) (1 in3/16.387 cm3)
= 61.02 in3
f. 1 L (1000 mL/L)(1 cm3/mL)[(1 in/2.54 cm)(1 ft/12 in)]3 =
1 L (1000 mL/L)(1 cm3/mL)[1 ft/30.48 cm]3 =
1 L (1000 mL/L)(1 cm3/mL)[1 ft3/2.83 x 104 cm3 ) = 0.0353 ft3
4. a. Force = (mass)(acceleration) = ma
Mass units = grams (g)
Acceleration units = distance/time2 = cm/sec2
1 dyne = 1 g cm/sec2
b. 1 Newton (N) = F = ma = 1 kg m/sec2
1N[(1 kg m/sec2)/N] (1000 g/kg)(100 cm/m)[1 dyne/[1 g cm/sec2)] =
105 dyne/N
5. a. Work = energy = (force)(distance) = dynes(distance) = (mass)(acceleration)(distance) =
= ( g )(cm/sec2)(cm) = g cm2/sec2 = ergs
b. 1 joule = Work = energy = (force)(distance) = Newtons(distance) = (mass)(acceleration)(distance) =
= ( kg )(m/sec2)(m) = 1 Kg m2/sec2 = joules
1 joule = 1 kg m2/sec2 (1000 g/kg)[100 cm/m]2 = 1 x 107 ergs
A Story Moment: Newton in his own words:
Axiomata sive Leges Motus
From Philosophiae Naturalis Principia Mathematica
Lex I
Corpus omne perseverare in statu suo quiscendi vel movendi uniformiter in
directum, nisi quatenus a viribus impressis cogitur statum illum mutare.
Lex II
Mutationem motus proportionalem esse vi motrici impressæ, & fieri
secundum lineam rectam qua vis illa imprimitur.
Lex III
Actioni contrariam semper & æqualem esse reactionem: sive corporum
duorum actiones in se mutuo semper esse æquales & in partes contrarias
dirigi.
The story that Newton's hypothesis of universal gravitation was
prompted by the fall of an apple, is probably based on fact. There are several reliable accounts from
the last few years of his life which record him describing such an event. (There is no indication an
apple fell on his head though...)
The first was recorded by William Stukeley, who said that after dining with Newton in Kensington on
15 April 1726:
The weather being warm, we went into the garden and drank tea, under shade of some
apple-trees, only he and myself. Amidst other discourses, he told me, he was just in the
same situation, as when formerly, the notion of gravitation came into his mind. It was
occasion'd by the fall of an apple, as he sat in contemplative mood. Why should that
apple always descend perpendicularly to the ground, thought he to himself. Why should
it not go sideways or upwards, but constantly to the earth's centre.
From Memoirs of Sir Isaac Newton's Life pp19-20.
Take a longer look at Newton: http://www.newton.org.uk/
8. x = position. Units are something that measures distance (be it Å, lightyears or Smoots). In the
MKS system, it’s meters (m)
t= time. Units are something that measures time (be it femtoseconds hours or years). In the MKS
system, it’s seconds (sec).
.v = velocity. Units are distance covered per unit time (distance/time). In the MKS system, it’s
meters/second (m/sec).
a. x0 + v + t. = x(t)?
m + m/sec + sec. ≠ m (you can’t add apples, oranges & pears & come up w/ anything but apples,
oranges & pears)
b. x0t + v = x(t)?
msec + m/sec ≠ m
c. x0 + vt = x(t)?
m + (m/sec)(sec) = m + m = m This works.
d. x0vt = x(t)?
meters(meters/sec)(sec) = meters2 ≠ meters
9. Distance traveled = xt – x0 = 500 mi – 100 mi = 400 mi.
400 mi(1 hr/60 mi) = 6.67 hr. Should pass the maker at ≈ 6.40 AM
10.a 9.81m/sec2 : This is a number arrived at by experiment, and measurement. It is subject to
random error of measurement. 3 sig. fig.
2 : This is an EXACT number, not found by experiment but be definition. It has as many sig fig as
you need.
2.0165 sec: This is a number arrived at by experiment, and measurement. It is subject to random
error of measurement. 5 sig. fig.
THE KEY TO SIGNIFICANT FIGURES…ALL MEASUREMENTS HAVE ERROR (Unless you are
God) YOUR ANSWER CAN ONLY BE AS GOOD AS YOUR WORST (least precise)
MEASUREMENT. Consider this: You make 2 measurements: 1) You walk from your dorm to the
front door of the Blue Line Diner. It takes you 25 minutes. Thus, you figure it’s about 2 kilometers to
the Blue Line. (What you are saying in this measurement is that it’s not 1 km to the Blue Line, you
know that wouldn’t take you 25 minutes, but 3 km seems too far). 2) At the Blue Line, you get out a
meter stick and find that it is 4.234 meters from the front door to your booth. That is, you are
confident that the distance to the booth is 4 meters, 234 mmeters, not 4 meters, 235 mm. Now you
have 2 measurements. How far is it from your dorm to the booth in the Blue Line? Can you really
say it’s 2.004234 km? Naw. It’s 2 km. YOUR ANSWER CAN ONLY BE AS GOOD AS YOUR
WORST (least precise) MEASUREMENT. The number of digits in a number tells you how far
you can be confident of the measurement.
b. YOUR ANSWER CAN ONLY BE AS GOOD AS YOUR WORST (least precise)
MEASUREMENT.
Your least precise measurement is 9.81 m/sec2 (3 sig. Fig.) Thus, your answer can only have 3 sig .
fig.
c. d = 9.81 m/sec2(4.0663 sec2)/2 = 19.9 m
14. a. g = 9.81m/sec2 (1 in/2.54 cm)(100 cm/m)(1 ft/12 in) = 32.2 ft/sec2 ( You get 3 sig fig. In your
answer, because you have one measured quantity, 9.81m/sec2, with 3 sig. fig. 2.54 cm/in, 100 cm/m
and 12 in/ft are all exact numbers. They have all the sig. fig. you want.)
b. 1 lb = 1 slug(1 ft/sec2)
1 lb = 1 slug(1 ft/sec2)(14.594 Kg/slug) (12 in/1 ft)(2.54 cm/in)(1 m/100 cm) = 4.45 Kg m/sec2 = 4.55
N
4.45 N/lb or 1 lb/4.45 N = 0.25lb/N or 0.225 lb = 1 N
21. a. Mass = 50.0 g REM: Convert to kg for MKS units.
Velocity = 100.0 m/sec.
KE = 1/2 mv2 = ½(50.0 g)(100.0 m/sec)2 = 2.50 x 105 gm2/sec2.
KE is typically expressed in units of kg m2/sec2, or Joules (J), so:
2.50 x 105 gm2/sec2(1 kg/1000g) = 250. J. And we are entitled to 3 sig. fig.
Another Story Moment:
The Age of Romanticism and the Industrial Revolution were in full
swing when James Prescott Joule was born in Manchester, England
to a prosperous brewery owner.
Educated at home, he was sent to Cambridge at the age of sixteen to
study with the eminent British chemist John Dalton.
Having completed his education, he returned home. His independent
means allowed him to begin independent researches in a laboratory
built in his father's home. Hoping to replace steam engines by electric
motors, his first research sought to improve electric motor efficiency.
His investigations concerned themselves with the production of heat. His first important paper
presented to the Royal Society showed that the heat generated by an electrical current was
proportional to the square of the current. This was one of the first of many reports establishing the
linkage between heat and other forms of energy.
In 1850, he established the equivalence between amounts of heat and mechanical work with his
famous "paddle-wheel experiment". He also established the cooling effect that occurs when a gas is
rapidly expanded, an effect that is used in refrigeration systems.
As is characteristic of many researchers, Joule's accomplishments were not limited to those things we
today associate with the word "research". Much of what comes from experimental investigations
results in practical devices and new technologies. In this respect one can note that James Joule was
also an inventor. Amongst his many inventions are "arc" or electrical welding and the displacement
pump.
Most of his researches were funded from his own pockets. Unfortunately, this money ran out in 1875
and the following years were ones of continuous illness until he died in 1889.
The unit of energy, the Joule, is named in his honor.
22. b. .Mass = 100.0 g REM: Convert to kg for MKS units.
Velocity = 25.0 m/sec.
KE = 1/2 mv2 = ½(100.0 g)(25.0 m/sec)2 = 3.13 x 105 gm2/sec2.
KE is typically expressed in units of kg m2/sec2, or Joules (J), so:
3.13 x 105 gm2/sec2(1 kg/1000g) = 31.3 J. And we are entitled to 3 sig. fig.
24. Mass = 4000 lb REM: Convert to kg for MKS units.
0.453592kg = 1 lb
Velocity = 60 mi/hr REM: Convert to m/sec
1 m = 0.000621miles
KE = 1/2mv2 = ½(4000 lb)(0.453592kg/lb)[(60 mi/hr)(1 m/0.000621miles)(1 hr/60 min)(1 min/60
sec)]2 =
½ (1814 kg)[2.684 x 101 m/sec]2 = 6.5 x 105 kgm2/sec2 = 7 x 105 J. And we are entitled to 1 sig. fig.
28. Massearth = 5.98 x 1024 kg.
Massmoon = 7.35 x 1022 kg
r = 3.84 x 105 km REM: Convert to meters
G = 6.67 X 10-11 N m2/kg2 REM: Convert N to kg m/sec2
1 N = 1 kg m/sec2
G = 6.67 x 10-11 N m2/kg2(1 kg m/sec2 N) = 6.67 x 10-11 kg m3/sec2 kg2 = 6.67 x 10-11 m3/sec2 kg
F = - Gmearthmmoon/r2 =
- [6.67 x 10-11 N m2/kg2(1 kg m/sec2 / N)( 5.98 x 1024 kg)( 7.35 x 1022 kg)]/[( 3.84 x 105 km)( 1000
m/km)]2 =
-1.98 x 1020 kg m/sec2 = - 1.98 x 1020 N . And we get 3 sig. Fig.
29. Let’s see what we KNOW: The Coulomb electric force is a function of charge/distance2:
F = (1/4πεo) [q2/r2 ] , where:
Π = 3.141…
εo = the permissivity of vacuum (a constant) = 8.8542 x 10-12 C2/N m2
q is the charge on a particle, in Coulombs (C)
r is the distance the charged particles are separated by, in m.
We have 1.00 g of H, that is:
We are told that the particles we are dealing with are 6.02 x 1023 protons:
Charge/proton = +1.602 x 10-19 C
Total + charge = (+1.602 x 10-19 C/proton)(6.022 x 1023 protons) = + 9.647 x 104 C
And 6.02 x 1023 electrons:
Charge/electron = -1.602 x 10-19 C
Total - charge = (-1.602 x 10-19 C/electron)(6.022 x 1023 electrons) = -9.647 x 104 Coulombs.
We are told that the force of Coulombic attraction between them is the answer to the last problem,
-1.98 x 1020 N . (the – sign says it’s an attractive force) We want to know how far apart the protons
& electrons must be to produce such a force. SO:
Rearranging the Coulombic force equation:
r2 = = (1/4πεo) [q2/F] =
(1/4(3.141)( 8.8542 x 10-12 C2/N m2)[ (+ 9.647 x 104 C)(- 9.647 x 104 C)/-1.98 x 1020 N =
(8.977 x 109 N m2/C2)(-9.3065 x 109 C2 )/-1.98 x 1020 N ) = 4.2196 x 10-1 m2
r = √(4.2196 x 10-1 m2) = 6.4958 x 10-1 m and we get 3 sig fig. (From 1.00 g). Final answer:
0.650 m.
We are comparing the force holding together this 1.00 g mass to that holding together the earth & the
moon. Clearly, at any distance, Coulombic force is >>> gravitational force.
Charles Augustin Coulomb
1736-1806
About the same time that the American colonists and the British were shooting at each other during
the Revolutionary War, Charles Augustin Coulomb was on the other side of the Atlantic studying
magnetism and designing his torsion balance. Soon he was to use this balance to clearly establish
the inverse-square law for electrostatic forces.
Coulomb was born in 1736 to an upper middle class family in Montpellier, France. Charles' mother
guided his education carefully and planned that he would become a medical doctor. However,
Coulomb announced early in his adolescence that he would become a mathematician. He continued
to deny his mother's wishes and was temporarily disowned. Nevertheless, he studied at various
schools and presented several mathematical papers called memoirs to the local Societe des
Sciences. At age 21, he was elected to the society as Adjunct for Mathematics.
The Societe des Sciences was pleasant and stimulating, but Coulomb needed to earn a living.
Consequently, he joined the engineering corps and began studying at Mezieres in 1760. He learned
carpentry, stonecutting, drafting, some physics, and battle strategy. When he graduated, one
cantankerous officer wrote that Coulomb's work was adequate except for his battle strategy work
which included a "rather bad siege memoir." Actually Coulomb graduated with honors and received
an award of 150 livres to "arouse his interest." His first assignment found him in charge of building a
huge fortification in Martinique. For eight years he learned the practical side of construction and
gained a reputation for being an excellent engineer.
Several times in Coulomb's career he was taken advantage of by those involved in political intrigue.
In Martinique he was ordered to remain in charge, although gravely ill, while another more
unscrupulous and healthy officer returned to Paris. Later Charles became the scapegoat for a minor
political scandal and consequently spent a week in prison.
When his duty was over, Coulomb returned to Paris and read his first engineering memoir to the
prestigious Academy of Science. Several more of his engineering memoirs followed, including one on
underwater construction in which Coulomb outlined the design of a submarine-like chamber. In 1775
he won an Academy prize for his memoir on constructing magnetic needles, and in 1781 he won the
Academy contest for the best memoir on friction. Finally he was elected to the Academy as Adjunct in
Mechanics.
By 1784 Coulomb had completed his study of torsion in fine threads and had designed a sensitive
torsion balance. Then came his seven memoirs on electricity and magnetism. He wasn't the first to
state the inverse-square law; there were at least ten statements of the law prior to Coulomb's
memoirs. Yet it was Coulomb who presented the convincing experimental proof. He soundly refuted
the existence of "electric atmospheres" or "magnetic vortices" and showed that electrostatic forces
acted at a distance.
Coulomb spent the rest of his life as a public servant. He was the King's water inspector, then director
of hospital reform, and consultant for weights and measures. He left Paris with his wife and two sons
during the Reign of Terror, and returned later when appointed by Napoleon as inspector of education.
Shortly after, he contracted a fever and died in 1806. http://physics.uwstout.edu/SOTW/coulomb.htm
31. Volts are units of electrical potential energy. By definition, 1 volt of PE exists if 1 coulomb of
charge gains or loses 1 Joule (1 kg m2/sec2) of energy in moving from point A to point B:
1 V = 1 J/C
REM: Joules are units of energy, or work:
An electron has a charge of 1.602 x 10-19 C.
It is in a field of 1 x 103 volts.
Energy imparted to the electron = 1 x 103 volt(1.602 x 10-19 C) = 1.602 x 10-16 J =
1.602 x 10-16 kg m2/sec2
The potential energy the electron loses in dropping down the voltage field is transformed into kinetic
energy. (Analogy: If you are in an airplane flying over SMC , you have high potential energy in the
earth’s gravitational field. If you jump out, you lose potential energy as you near the earth’s surface,
and you gain kinetic energy, as you fall faster and faster. Energy cannot be created or destroyed,
only interconverted. Don’t try this experiment at home without a parachute & a qualified
jumpmaster.)
PE lost = KE gained
PE lost = 1.602 x 10-16 J = KE gained = ½ mv2
Mass of an electron is: 9.109 x 10-31 kg.
Rearranging: v2 = 2KE/m
v2 = 2(1.602 x 10-16 kg m2/sec2)/9.109 x 10-31 kg = 3.5174 x 1014 m2/sec2
v = √3.5174x 1014 m2/sec2 = 1.8755 x 107 m/sec. And we get 3 sig fig (1.00 x 103 V) so:
1.88 x 107 m/sec.
Of course you realize that this is getting close to the speed of light, 2.99 x 108 m/sec.
At high velocities, Einstein, Lorenz and others showed that:
KE = (γ-1)(mc2), where γ = 1/[1-(v2/c2)]1/2, where c = speed of light.
γ = 1/[1-(v2/c2)]1/2 = 1/[1-(1.88 x 107 m/sec)2/(2.99 x 108 m/sec)2]1/2 = 1/(1-0.0039534)1/2 = 1/(0.99605)1/2 =
1/0.99802 = 1.002.
v2 = 2KE/m = 2(1.002 –1)( 1.602 x 10-16 kg m2/sec2)/9.109 x 10-31 kg= 7.0348 x 1011 m2/sec2
v = 8.39 x 105 m/sec.
Visit the Molecule of the Month at Oxford University:
http://www.chem.ox.ac.uk/mom/taxol/taxol.html