Chapter 1
Systems of Measurement
Conceptual Problems
*1 •
Determine the Concept The fundamental physical quantities in the SI system include
mass, length, and time. Force, being the product of mass and acceleration, is not a
fundamental quantity. (c) is correct.
2 •
Picture the Problem We can express and simplify the ratio of m/s to m/s2 to determine
the final units.
Express and simplify the ratio of m
s = m ⋅ s = s and (d ) is correct.
2
m/s to m/s2:
m m ⋅s
2
s
3 •
Determine the Concept Consulting Table 1-1 we note that the prefix giga
means 109. (c ) is correct.
4 •
Determine the Concept Consulting Table 1-1 we note that the prefix mega
means 106. (d ) is correct.
*5 •
Determine the Concept Consulting Table 1-1 we note that the prefix pico
means 10−12. (a ) is correct.
6 •
Determine the Concept Counting from left to right and ignoring zeros to the left
of the first nonzero digit, the last significant figure is the first digit that is in doubt.
Applying this criterion, the three zeros after the decimal point are not significant figures,
but the last zero is significant. Hence, there are four significant figures in this number.
(c) is correct.
1
,2 Chapter 1
7 •
Determine the Concept Counting from left to right, the last significant figure is
the first digit that is in doubt. Applying this criterion, there are six significant
figures in this number. (e) is correct.
8 •
Determine the Concept The advantage is that the length measure is always with you. The
disadvantage is that arm lengths are not uniform; if you wish to purchase a board of ″two
arm lengths″ it may be longer or shorter than you wish, or else you may have to physically
go to the lumberyard to use your own arm as a measure of length.
9 •
(a) True. You cannot add ″apples to oranges″ or a length (distance traveled) to a volume
(liters of milk).
(b) False. The distance traveled is the product of speed (length/time) multiplied by the
time of travel (time).
(c) True. Multiplying by any conversion factor is equivalent to multiplying by 1.
Doing so does not change the value of a quantity; it changes its units.
Estimation and Approximation
*10 ••
Picture the Problem Because θ is small, we can approximate it by θ ≈ D/rm
provided that it is in radian measure. We can solve this relationship for the diameter of
the moon.
Express the moon’s diameter D in D = θ rm
terms of the angle it subtends at the
earth θ and the earth-moon distance
rm:
Find θ in radians: 2π rad
θ = 0.524° × = 0.00915 rad
360°
Substitute and evaluate D: D = (0.00915 rad )(384 Mm )
= 3.51 × 106 m
, Systems of Measurement 3
*11 ••
Picture the Problem We’ll assume that the sun is made up entirely of hydrogen. Then we
can relate the mass of the sun to the number of hydrogen atoms and the mass of each.
Express the mass of the sun MS as M S = NHM H
the product of the number of
hydrogen atoms NH and the mass of
each atom MH:
Solve for NH: MS
NH =
MH
Substitute numerical values and 1.99 × 1030 kg
NH = = 1.19 × 1057
evaluate NH: 1.67 × 10 −27 kg
12 ••
Picture the Problem Let P represent the population of the United States, r the rate of
consumption and N the number of aluminum cans used annually. The population of the
United States is roughly 3×108 people. Let’s assume that, on average, each person drinks
one can of soft drink every day. The mass of a soft-drink can is approximately
1.8 ×10−2 kg.
(a) Express the number of cans N N = rP∆t
used annually in terms of the daily
rate of consumption of soft drinks r
and the population P:
⎛ 1can ⎞
Substitute numerical values and
approximate N:
N = ⎜⎜
⋅
(
⎟⎟ 3 × 108 people )
⎝ person d ⎠
⎛ d⎞
× (1 y )⎜⎜ 365.24 ⎟⎟
⎝ y⎠
≈ 1011 cans
(b) Express the total mass of M = Nm
aluminum used per year for soft
drink cans M as a function of the
number of cans consumed and the
mass m per can:
, 4 Chapter 1
Substitute numerical values and ( )(
M = 1011 cans/y 1.8 × 10−2 kg/can )
evaluate M:
≈ 2 × 109 kg/y
(c) Express the value of the Value = ($1 / kg )M
aluminum as the product of M and (
= ($1 / kg ) 2 × 109 kg/y )
the value at recycling centers:
= $2 × 10 / y 9
= 2 billion dollars/y
13 ••
Picture the Problem We can estimate the number of words in Encyclopedia Britannica
by counting the number of volumes, estimating the average number of pages per volume,
estimating the number of words per page, and finding the product of these measurements
and estimates. Doing so in Encyclopedia Britannica leads to an estimate of
approximately 200 million for the number of words. If we assume an average word
length of five letters, then our estimate of the number of letters in Encyclopedia
Britannica becomes 109.
(a) Relate the area available for one π
Ns 2 = d 2 where d is the diameter of the
letter s2 and the number of letters N 4
to be written on the pinhead to the pinhead.
area of the pinhead:
Solve for s to obtain: πd 2
s=
4N
Substitute numerical values and 2
⎡ ⎛ cm ⎞⎤
π ⎢(161 in )⎜ 2.54 ⎟
in ⎠⎥⎦
evaluate s:
⎣ ⎝
s= ≈ 10−8 m
( )
4 10 9
(b) Express the number of atoms per s
n=
letter n in terms of s and the atomic d atomic
spacing in a metal datomic:
Substitute numerical values and 10 −8 m
n= ≈ 20 atoms
evaluate n: 5 × 10 −10 atoms/m
*14 ••
Picture the Problem The population of the United States is roughly 3 × 108 people.
Assuming that the average family has four people, with an average of two cars per
Systems of Measurement
Conceptual Problems
*1 •
Determine the Concept The fundamental physical quantities in the SI system include
mass, length, and time. Force, being the product of mass and acceleration, is not a
fundamental quantity. (c) is correct.
2 •
Picture the Problem We can express and simplify the ratio of m/s to m/s2 to determine
the final units.
Express and simplify the ratio of m
s = m ⋅ s = s and (d ) is correct.
2
m/s to m/s2:
m m ⋅s
2
s
3 •
Determine the Concept Consulting Table 1-1 we note that the prefix giga
means 109. (c ) is correct.
4 •
Determine the Concept Consulting Table 1-1 we note that the prefix mega
means 106. (d ) is correct.
*5 •
Determine the Concept Consulting Table 1-1 we note that the prefix pico
means 10−12. (a ) is correct.
6 •
Determine the Concept Counting from left to right and ignoring zeros to the left
of the first nonzero digit, the last significant figure is the first digit that is in doubt.
Applying this criterion, the three zeros after the decimal point are not significant figures,
but the last zero is significant. Hence, there are four significant figures in this number.
(c) is correct.
1
,2 Chapter 1
7 •
Determine the Concept Counting from left to right, the last significant figure is
the first digit that is in doubt. Applying this criterion, there are six significant
figures in this number. (e) is correct.
8 •
Determine the Concept The advantage is that the length measure is always with you. The
disadvantage is that arm lengths are not uniform; if you wish to purchase a board of ″two
arm lengths″ it may be longer or shorter than you wish, or else you may have to physically
go to the lumberyard to use your own arm as a measure of length.
9 •
(a) True. You cannot add ″apples to oranges″ or a length (distance traveled) to a volume
(liters of milk).
(b) False. The distance traveled is the product of speed (length/time) multiplied by the
time of travel (time).
(c) True. Multiplying by any conversion factor is equivalent to multiplying by 1.
Doing so does not change the value of a quantity; it changes its units.
Estimation and Approximation
*10 ••
Picture the Problem Because θ is small, we can approximate it by θ ≈ D/rm
provided that it is in radian measure. We can solve this relationship for the diameter of
the moon.
Express the moon’s diameter D in D = θ rm
terms of the angle it subtends at the
earth θ and the earth-moon distance
rm:
Find θ in radians: 2π rad
θ = 0.524° × = 0.00915 rad
360°
Substitute and evaluate D: D = (0.00915 rad )(384 Mm )
= 3.51 × 106 m
, Systems of Measurement 3
*11 ••
Picture the Problem We’ll assume that the sun is made up entirely of hydrogen. Then we
can relate the mass of the sun to the number of hydrogen atoms and the mass of each.
Express the mass of the sun MS as M S = NHM H
the product of the number of
hydrogen atoms NH and the mass of
each atom MH:
Solve for NH: MS
NH =
MH
Substitute numerical values and 1.99 × 1030 kg
NH = = 1.19 × 1057
evaluate NH: 1.67 × 10 −27 kg
12 ••
Picture the Problem Let P represent the population of the United States, r the rate of
consumption and N the number of aluminum cans used annually. The population of the
United States is roughly 3×108 people. Let’s assume that, on average, each person drinks
one can of soft drink every day. The mass of a soft-drink can is approximately
1.8 ×10−2 kg.
(a) Express the number of cans N N = rP∆t
used annually in terms of the daily
rate of consumption of soft drinks r
and the population P:
⎛ 1can ⎞
Substitute numerical values and
approximate N:
N = ⎜⎜
⋅
(
⎟⎟ 3 × 108 people )
⎝ person d ⎠
⎛ d⎞
× (1 y )⎜⎜ 365.24 ⎟⎟
⎝ y⎠
≈ 1011 cans
(b) Express the total mass of M = Nm
aluminum used per year for soft
drink cans M as a function of the
number of cans consumed and the
mass m per can:
, 4 Chapter 1
Substitute numerical values and ( )(
M = 1011 cans/y 1.8 × 10−2 kg/can )
evaluate M:
≈ 2 × 109 kg/y
(c) Express the value of the Value = ($1 / kg )M
aluminum as the product of M and (
= ($1 / kg ) 2 × 109 kg/y )
the value at recycling centers:
= $2 × 10 / y 9
= 2 billion dollars/y
13 ••
Picture the Problem We can estimate the number of words in Encyclopedia Britannica
by counting the number of volumes, estimating the average number of pages per volume,
estimating the number of words per page, and finding the product of these measurements
and estimates. Doing so in Encyclopedia Britannica leads to an estimate of
approximately 200 million for the number of words. If we assume an average word
length of five letters, then our estimate of the number of letters in Encyclopedia
Britannica becomes 109.
(a) Relate the area available for one π
Ns 2 = d 2 where d is the diameter of the
letter s2 and the number of letters N 4
to be written on the pinhead to the pinhead.
area of the pinhead:
Solve for s to obtain: πd 2
s=
4N
Substitute numerical values and 2
⎡ ⎛ cm ⎞⎤
π ⎢(161 in )⎜ 2.54 ⎟
in ⎠⎥⎦
evaluate s:
⎣ ⎝
s= ≈ 10−8 m
( )
4 10 9
(b) Express the number of atoms per s
n=
letter n in terms of s and the atomic d atomic
spacing in a metal datomic:
Substitute numerical values and 10 −8 m
n= ≈ 20 atoms
evaluate n: 5 × 10 −10 atoms/m
*14 ••
Picture the Problem The population of the United States is roughly 3 × 108 people.
Assuming that the average family has four people, with an average of two cars per
