Chapter 1
Measurement and Vectors
Conceptual Problems
1 • [SSM] Which of the following is not one of the base quantities in the
SI system? (a) mass, (b) length, (c) energy, (d) time, (e) All of the above are base
quantities.
Determine the Concept The base quantities in the SI system include mass, length,
and time. Force is not a base quantity. (c) is correct.
2 • In doing a calculation, you end up with m/s in the numerator and m/s2
in the denominator. What are your final units? (a) m2/s3, (b) 1/s, (c) s3/m2, (d) s,
(e) m/s.
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 • The prefix giga means (a) 103, (b) 106, (c) 109, (d) 1012, (e) 1015.
Determine the Concept Consulting Table 1-1 we note that the prefix giga
means 109. (c) is correct.
4 • The prefix mega means (a) 10–9, (b) 10–6, (c) 10–3, (d) 106, (e) 109.
Determine the Concept Consulting Table 1-1 we note that the prefix mega
means 106. (d ) is correct.
5 • [SSM] Show that there are 30.48 cm per foot. How many centimeters
are there in one mile?
Picture the Problem We can use the facts that there are 2.540 centimeters in
1 inch and 12 inches in 1 foot to show that there are 30.48 cm per ft. We can then
use the fact that there are 5280 feet in 1 mile to find the number of centimeters in
one mile.
1
,2 Chapter 1
Multiply 2.540 cm/in by 12 in/ft to ⎛ cm ⎞ ⎛ in ⎞
⎜ 2.540 ⎟ ⎜12 ⎟ = 30.48 cm/ft
find the number of cm per ft: ⎝ in ⎠ ⎝ ft ⎠
Multiply 30.48 cm/ft by 5280 ft/mi to find the number of centimeters in one
mile:
⎛ cm ⎞ ⎛ ft ⎞
⎟ ⎜ 5280 ⎟ = 1.609 × 10 cm/mi
5
⎜ 30.48
⎝ ft ⎠ ⎝ mi ⎠
Remarks: Because there are exactly 2.54 cm in 1 in and exactly 12 inches in 1 ft, we
are justified in reporting four significant figures in these results.
6 • The number 0.000 513 0 has significant figures. (a) one, (b) three,
(c) four, (d) seven, (e) eight.
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.
7 • The number 23.0040 has significant figures. (a) two, (b) three,
(c) four, (d) five, (e) six.
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 • Force has dimensions of mass times acceleration. Acceleration has
dimensions of speed divided by time. Pressure is defined as force divided by area.
What are the dimensions of pressure? Express pressure in terms of the SI base
units kilogram, meter and second.
Determine the Concept We can use the definitions of force and pressure,
together with the dimensions of mass, acceleration, and length, to find the
dimensions of pressure. We can express pressure in terms of the SI base units by
substituting the base units for mass, acceleration, and length in the definition of
pressure.
, Measurement and Vectors 3
Use the definition of pressure and ML
[P] = [F ] = T 2 = M 2
2
the dimensions of force and area to
obtain: [A] L LT
Express pressure in terms of the SI m
kg ⋅ 2
base units to obtain: N s = kg
=
m 2
m 2
m ⋅ s2
9 • True or false: Two quantities must have the same dimensions in order
to be multiplied.
False. For example, the distance traveled by an object is the product of its speed
(length/time) multiplied by its time of travel (time).
10 • A vector has a negative x component and a positive y component. Its
angle measured counterclockwise from the positive x axis is (a) between zero and
90 degrees. (b) between 90 and 180 degrees. (c) More than 180 degrees.
Determine the Concept Because a vector with a negative x-component and a
positive y-component is in the second quadrant, its angle is between 90 and 180
degrees. (b ) is correct.
r
11 • [SSM] A vector A points in the +x direction. Show graphically at
r r r
least three choices for a vector B such that B + A points in the +y direction.
Determine the Concept The figure y
r
shows a vector A pointing in the r
positive x direction and three unlabeled Several B choices
r
possibilities for vector B. Note that the
r
choices
r for B start at the end of vector
A rather than at its initial point. Note
further that this configuration could be
r
in any quadrant of the reference system A
shown. x
r
12 • A vector A points in the +y direction. Show graphically at least three
r r r
choices for a vector B such that B − A points in the + x direction.
, 4 Chapter 1
Determine the Concept Let the +x y
direction be to the right and the +y
direction be upward. The figure shows
r
the vector − A pointing in the −y
r
direction and three unlabeled −A
r r
possibilities for vector B. Note that the Several B choices
r
choices for B start at the end of vector
r
− A rather than at its initial point.
x
13 • [SSM] Is it possible for three equal magnitude vectors to add to
zero? If so, sketch a graphical answer. If not, explain why not.
Determine the Concept In order for
the three equal magnitude vectors to
add to zero, the sum of the three vectors r r
must form a triangle. The equilateral C B
triangle shown to the right satisfies this
r r
condition for the vectors A , B , and
r
C for which it is true that A = B = C,
r r r r
whereas A + B + C = 0. A
Estimation and Approximation
14 • The angle subtended by the moon’s diameter at a point on Earth is
about 0.524º (Fig. 1-2). Use this and the fact that the moon is about 384 Mm away
to find the diameter of the moon. HINT: The angle can be determined from the
diameter of the moon and the distance to the moon.
Picture the Problem Let θ represent the angle subtended by the moon’s
diameter, D represent the diameter of the moon, and rm the distance to the moon.
Because θ is small, we can approximate it by θ ≈ D/rm where θ 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 Earth
θ and the Earth-moon distance rm:
Substitute numerical values and ⎛ 2π rad ⎞
D = ⎜ 0.524° × ⎟ (384 Mm )
evaluate D: ⎝ 360° ⎠
= 3.51 × 10 6 m
Measurement and Vectors
Conceptual Problems
1 • [SSM] Which of the following is not one of the base quantities in the
SI system? (a) mass, (b) length, (c) energy, (d) time, (e) All of the above are base
quantities.
Determine the Concept The base quantities in the SI system include mass, length,
and time. Force is not a base quantity. (c) is correct.
2 • In doing a calculation, you end up with m/s in the numerator and m/s2
in the denominator. What are your final units? (a) m2/s3, (b) 1/s, (c) s3/m2, (d) s,
(e) m/s.
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 • The prefix giga means (a) 103, (b) 106, (c) 109, (d) 1012, (e) 1015.
Determine the Concept Consulting Table 1-1 we note that the prefix giga
means 109. (c) is correct.
4 • The prefix mega means (a) 10–9, (b) 10–6, (c) 10–3, (d) 106, (e) 109.
Determine the Concept Consulting Table 1-1 we note that the prefix mega
means 106. (d ) is correct.
5 • [SSM] Show that there are 30.48 cm per foot. How many centimeters
are there in one mile?
Picture the Problem We can use the facts that there are 2.540 centimeters in
1 inch and 12 inches in 1 foot to show that there are 30.48 cm per ft. We can then
use the fact that there are 5280 feet in 1 mile to find the number of centimeters in
one mile.
1
,2 Chapter 1
Multiply 2.540 cm/in by 12 in/ft to ⎛ cm ⎞ ⎛ in ⎞
⎜ 2.540 ⎟ ⎜12 ⎟ = 30.48 cm/ft
find the number of cm per ft: ⎝ in ⎠ ⎝ ft ⎠
Multiply 30.48 cm/ft by 5280 ft/mi to find the number of centimeters in one
mile:
⎛ cm ⎞ ⎛ ft ⎞
⎟ ⎜ 5280 ⎟ = 1.609 × 10 cm/mi
5
⎜ 30.48
⎝ ft ⎠ ⎝ mi ⎠
Remarks: Because there are exactly 2.54 cm in 1 in and exactly 12 inches in 1 ft, we
are justified in reporting four significant figures in these results.
6 • The number 0.000 513 0 has significant figures. (a) one, (b) three,
(c) four, (d) seven, (e) eight.
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.
7 • The number 23.0040 has significant figures. (a) two, (b) three,
(c) four, (d) five, (e) six.
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 • Force has dimensions of mass times acceleration. Acceleration has
dimensions of speed divided by time. Pressure is defined as force divided by area.
What are the dimensions of pressure? Express pressure in terms of the SI base
units kilogram, meter and second.
Determine the Concept We can use the definitions of force and pressure,
together with the dimensions of mass, acceleration, and length, to find the
dimensions of pressure. We can express pressure in terms of the SI base units by
substituting the base units for mass, acceleration, and length in the definition of
pressure.
, Measurement and Vectors 3
Use the definition of pressure and ML
[P] = [F ] = T 2 = M 2
2
the dimensions of force and area to
obtain: [A] L LT
Express pressure in terms of the SI m
kg ⋅ 2
base units to obtain: N s = kg
=
m 2
m 2
m ⋅ s2
9 • True or false: Two quantities must have the same dimensions in order
to be multiplied.
False. For example, the distance traveled by an object is the product of its speed
(length/time) multiplied by its time of travel (time).
10 • A vector has a negative x component and a positive y component. Its
angle measured counterclockwise from the positive x axis is (a) between zero and
90 degrees. (b) between 90 and 180 degrees. (c) More than 180 degrees.
Determine the Concept Because a vector with a negative x-component and a
positive y-component is in the second quadrant, its angle is between 90 and 180
degrees. (b ) is correct.
r
11 • [SSM] A vector A points in the +x direction. Show graphically at
r r r
least three choices for a vector B such that B + A points in the +y direction.
Determine the Concept The figure y
r
shows a vector A pointing in the r
positive x direction and three unlabeled Several B choices
r
possibilities for vector B. Note that the
r
choices
r for B start at the end of vector
A rather than at its initial point. Note
further that this configuration could be
r
in any quadrant of the reference system A
shown. x
r
12 • A vector A points in the +y direction. Show graphically at least three
r r r
choices for a vector B such that B − A points in the + x direction.
, 4 Chapter 1
Determine the Concept Let the +x y
direction be to the right and the +y
direction be upward. The figure shows
r
the vector − A pointing in the −y
r
direction and three unlabeled −A
r r
possibilities for vector B. Note that the Several B choices
r
choices for B start at the end of vector
r
− A rather than at its initial point.
x
13 • [SSM] Is it possible for three equal magnitude vectors to add to
zero? If so, sketch a graphical answer. If not, explain why not.
Determine the Concept In order for
the three equal magnitude vectors to
add to zero, the sum of the three vectors r r
must form a triangle. The equilateral C B
triangle shown to the right satisfies this
r r
condition for the vectors A , B , and
r
C for which it is true that A = B = C,
r r r r
whereas A + B + C = 0. A
Estimation and Approximation
14 • The angle subtended by the moon’s diameter at a point on Earth is
about 0.524º (Fig. 1-2). Use this and the fact that the moon is about 384 Mm away
to find the diameter of the moon. HINT: The angle can be determined from the
diameter of the moon and the distance to the moon.
Picture the Problem Let θ represent the angle subtended by the moon’s
diameter, D represent the diameter of the moon, and rm the distance to the moon.
Because θ is small, we can approximate it by θ ≈ D/rm where θ 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 Earth
θ and the Earth-moon distance rm:
Substitute numerical values and ⎛ 2π rad ⎞
D = ⎜ 0.524° × ⎟ (384 Mm )
evaluate D: ⎝ 360° ⎠
= 3.51 × 10 6 m
