Content check — University Physics 1, Chapter 1: Units and Measurement

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 1: Units and Measurement

53 problems · 6 figures

Check Your Understanding 1.1 §1.2 source ↗

Restate

4.79 \times 10^{5} kg

using a metric prefix such that the resulting number is bigger than one but less than 1000.

Answer

4.79 \times 10^{2}

Mg or 479 Mg

Check Your Understanding 1.2 §1.3 source ↗

Light travels about 9 Pm in a year. Given that a year is about

3 \times 10^{7} s,

what is the speed of light in meters per second?

Answer

3 \times 10^{8} m/s

Check Your Understanding 1.3 §1.3 source ↗

We know from Figure 1.4 that the diameter of Earth is on the order of 10⁷ m, so the order of magnitude of its surface area is 10¹⁴ m². What is that in square kilometers (that is, km²)? (Try doing this both by converting 10⁷ m to km and then squaring it and then by converting 10¹⁴ m² directly to square kilometers. You should get the same answer both ways.)

This table of orders of magnitude of length, mass and time has three columns and thirteen rows. The first row is a header row and it labels each column, “length in meters (m),” “Masses in kilograms (kg),” and “time in seconds (s).” Under the “length in meters” column are the following entries: 10 to the minus 15 meters equals diameter of proton; 10 to the minus 14 meters equals diameter of large nucleus; 10 to the minus 10 meters equals diameter of hydrogen atom; 10 to the minus 7 meters equals diameter of typical virus; 10 to the minus 2 meters equals pinky fingernail width; 10 to the 0 meters equals height of 4 year old child, and a drawing of a child measuring himself against a meter stick is included; 10 to the 2 meters equals length of football field; 10 to the 7 meters equals diameter of earth; 10 to the 13 meters equals diameter of solar system; 10 to the 16 meters equals distance light travels in a year (one light year); 10 to the 21 meters equals milky way diameter; 10 to the 26 meters equals distance to edge of observable universe. Under the “Masses in kilograms” column are the following entries: 10 to the -30 kilograms equals mass of electron; 10 to the -27 kilograms equals mass of proton; 10 to the -15 kilograms equals mass of bacterium; 10 to the -5 kilograms equals mass of mosquito; 10 to the -2 kilograms equals mass of hummingbird; 10 to the 0 kilograms equals mass of liter of water, and a drawing of a balance scale with a liter on one side and a 1 kilogram mass on the other is shown; 10 to the 2 kilograms equals mass of person; 10 to the 19 kilograms equals mass of atmosphere; 10 to the 22 kilograms equals mass of moon; 10 to the 25 kilograms equals mass of earth; 10 to the 30 kilograms equals mass of sun; 10 to the 53 kilograms equals upper limit on mass of known universe. Under the “Time in seconds” column are the following entries: 10 to the -22 seconds equals mean lifetime of very unstable nucleus; 10 to the -17 seconds equals time for a single floating point operation in a supercomputer; 10 to the -15 seconds equals time for one oscillation of visible light; 10 to the -13 seconds equals time for one vibration of an atom in a solid; 10 to the -3 seconds equals duration of a nerve impulse; 10 to the 0 equals time for one heartbeat, and a drawing of the heart with a plot of three pulses is shown. The peak of the first pulse is labeled P. The next pulse is larger amplitude and shorter duration. The start of the second pulse is labeled Q, its peak is labeled R, and its end is labeled S. The peak of the third pulse is labeled T. The entries in the column continue as follows: 10 to the 5 seconds equals one day; 10 to the 7 seconds equals one year; 10 to the 9 seconds equals human lifetime; 10 to the 11 seconds equals recorded human history; 10 to the 17 seconds equals age of earth; 10 to the 18 seconds equals age of universe; — Figure 1.4 in the book ↗

Answer

1 0^{8} {km}^{2}

Check Your Understanding 1.4 §1.3 source ↗

Given that 1 lbf (pound-force) is 4.45 N, were the numbers being output by SM_FORCES too big or too small?

Answer

The numbers were too small, by a factor of 4.45.

Check Your Understanding 1.5 §1.4 source ↗

Suppose we want the formula for the volume of a sphere. The two expressions commonly mentioned in elementary discussions of spheres are

4 π r^{2}

and

4 π r^{3} / 3 .

One is the volume of a sphere of radius r and the other is its surface area. Which one is the volume?

Answer

4 π r^{3} / 3

Check Your Understanding 1.6 §1.4 source ↗

Is the equation v = at dimensionally consistent?

Answer

yes

Check Your Understanding 1.7 §1.5 source ↗

Figure 1.4 says the mass of the atmosphere is 10¹⁹ kg. Assuming the density of the atmosphere is 1 kg/m³, estimate the height of Earth’s atmosphere. Do you think your answer is an underestimate or an overestimate? Explain why.

Answer

3 \times 10^{4} m

or 30 km. It is probably an underestimate because the density of the atmosphere decreases with altitude. (In fact, 30 km does not even get us out of the stratosphere.)

Check Your Understanding 1.8 §1.6 source ↗

A high school track coach has just purchased a new stopwatch. The stopwatch manual states the stopwatch has an uncertainty of ±0.05 s. Runners on the track coach’s team regularly clock 100-m sprints of 11.49 s to 15.01 s. At the school’s last track meet, the first-place sprinter came in at 12.04 s and the second-place sprinter came in at 12.07 s. Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not?

Answer

No, the coach’s new stopwatch will not be helpful. The uncertainty in the stopwatch is too great to differentiate between the sprint times effectively.

1-1 §1.1 source ↗

What is physics?

Answer

Physics is the science concerned with describing the interactions of energy, matter, space, and time to uncover the fundamental mechanisms that underlie every phenomenon.

1-3 §1.1 source ↗

If two different theories describe experimental observations equally well, can one be said to be more valid than the other (assuming both use accepted rules of logic)?

Answer

No, neither of these two theories is more valid than the other. Experimentation is the ultimate decider. If experimental evidence does not suggest one theory over the other, then both are equally valid. A given physicist might prefer one theory over another on the grounds that one seems more simple, more natural, or more beautiful than the other, but that physicist would quickly acknowledge that he or she cannot say the other theory is invalid. Rather, he or she would be honest about the fact that more experimental evidence is needed to determine which theory is a better description of nature.

1-5 §1.1 source ↗

Certain criteria must be satisfied if a measurement or observation is to be believed. Will the criteria necessarily be as strict for an expected result as for an unexpected result?

Answer

Probably not. As the saying goes, “Extraordinary claims require extraordinary evidence.”

1-7 §1.2 source ↗

Identify some advantages of metric units.

Answer

Conversions between units require factors of 10 only, which simplifies calculations. Also, the same basic units can be scaled up or down using metric prefixes to sizes appropriate for the problem at hand.

1-9 §1.2 source ↗

What is the difference between a base unit and a derived unit? (b) What is the difference between a base quantity and a derived quantity? (c) What is the difference between a base quantity and a base unit?

Answer

a. Base units are defined by a particular process of measuring a base quantity whereas derived units are defined as algebraic combinations of base units. b. A base quantity is chosen by convention and practical considerations. Derived quantities are expressed as algebraic combinations of base quantities. c. A base unit is a standard for expressing the measurement of a base quantity within a particular system of units. So, a measurement of a base quantity could be expressed in terms of a base unit in any system of units using the same base quantities. For example, length is a base quantity in both SI and the English system, but the meter is a base unit in the SI system only.

1-11 §1.6 source ↗

(a) What is the relationship between the precision and the uncertainty of a measurement? (b) What is the relationship between the accuracy and the discrepancy of a measurement?

Answer

a. Uncertainty is a quantitative measure of precision. b. Discrepancy is a quantitative measure of accuracy.

1-13 §1.7 source ↗

What should you do after obtaining a numerical answer when solving a problem?

Answer

Check to make sure it makes sense and assess its significance.

1-15 §1.1 source ↗

Use the orders of magnitude you found in the previous problem to answer the following questions to within an order of magnitude. (a) How many electrons would it take to equal the mass of a proton? (b) How many Earths would it take to equal the mass of the Sun? (c) How many Earth–Moon distances would it take to cover the distance from Earth to the Sun? (d) How many Moon atmospheres would it take to equal the mass of Earth’s atmosphere? (e) How many moons would it take to equal the mass of Earth? (f) How many protons would it take to equal the mass of the Sun?

Answer

a. 10³; b. 10⁵; c. 10²; d. 10¹⁵; e. 10²; f. 10⁵⁷

1-17 §1.1 source ↗

For the remaining questions, you need to use Figure 1.4 to obtain the necessary orders of magnitude of lengths, masses, and times.

A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0?

Answer

10² generations

1-19 §1.1 source ↗

For the remaining questions, you need to use Figure 1.4 to obtain the necessary orders of magnitude of lengths, masses, and times.

Calculate the approximate number of atoms in a bacterium. Assume the average mass of an atom in the bacterium is 10 times the mass of a proton.

Answer

10¹¹ atoms

1-21 §1.1 source ↗

For the remaining questions, you need to use Figure 1.4 to obtain the necessary orders of magnitude of lengths, masses, and times.

Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

Answer

10³ nerve impulses/s

1-23 §1.1 source ↗

For the remaining questions, you need to use Figure 1.4 to obtain the necessary orders of magnitude of lengths, masses, and times.

Roughly how many floating-point operations can a supercomputer perform in a human lifetime?

Answer

10²⁶ floating-point operations per human lifetime

1-25 §1.2 source ↗

The following times are given in seconds. Use metric prefixes to rewrite them so the numerical value is greater than one but less than 1000. For example,

7.9 \times 10^{−2} s

could be written as either 7.9 cs or 79 ms. (a)

9.57 \times 10^{5} s;

(b) 0.045 s; (c)

5.5 \times 10^{−7} s;

(d)

3.16 \times 10^{7} s.

Answer

a. 957 ks; b. 4.5 cs or 45 ms; c. 550 ns; d. 31.6 Ms

1-27 §1.2 source ↗

The following lengths are given in meters. Use metric prefixes to rewrite them so the numerical value is bigger than one but less than 1000. For example,

7.9 \times 10^{−2} m

could be written either as 7.9 cm or 79 mm. (a)

7.59 \times 10^{7} m;

(b) 0.0074 m; (c)

8.8 \times 10^{−11} m;

(d)

1.63 \times 10^{13} m.

Answer

a. 75.9 Mm; b. 7.4 mm; c. 88 pm; d. 16.3 Tm

1-29 §1.2 source ↗

The following masses are given in kilograms. Use metric prefixes on the gram to rewrite them so the numerical value is bigger than one but less than 1000. For example,

7 \times 10^{−4} kg

could be written as 70 cg or 700 mg. (a)

3.8 \times 10^{−5} kg;

(b)

2.3 \times 10^{17} kg;

(c)

2.4 \times 10^{−11} kg;

(d)

8 \times 10^{15} kg;

(e)

4.2 \times 10^{−3} kg.

Answer

a. 3.8 cg or 38 mg; b. 230 Eg; c. 24 ng; d. 8 Eg e. 4.2 g

1-31 §1.3 source ↗

The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this?

Answer

a. 27.8 m/s; b. 62 mi/h

1-33 §1.3 source ↗

In SI units, speeds are measured in meters per second (m/s). But, depending on where you live, you’re probably more comfortable of thinking of speeds in terms of either kilometers per hour (km/h) or miles per hour (mi/h). In this problem, you will see that 1 m/s is roughly 4 km/h or 2 mi/h, which is handy to use when developing your physical intuition. More precisely, show that (a)

1.0 m/s =

3.6 km/h

and (b)

1.0 m/s =

2.2 mi/h .

Answer

a. 3.6 km/h; b. 2.2 mi/h

1-35 §1.3 source ↗

Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its area in square feet? (Assume that 1 m = 3.281 ft.)

Answer

1.05 \times 10^{5} {ft}^{2}

1-37 §1.3 source ↗

Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 m = 3.281 ft.)

Answer

8.847 km

1-39 §1.3 source ↗

Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate has an average speed of 4.0 cm/yr. (a) What distance does it move in 1.0 s at this speed? (b) What is its speed in kilometers per million years?

Answer

1.3 \times 10^{−9} m;

b. 40 km/My

1-41 §1.3 source ↗

The density of nuclear matter is about 10¹⁸ kg/m³. Given that 1 mL is equal in volume to cm³, what is the density of nuclear matter in megagrams per microliter (that is,

Mg/ μ L

Answer

10^{6} Mg/ μ L

1-43 §1.3 source ↗

A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm, where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?

Answer

62.4 lbm/ft³

1-45 §1.3 source ↗

It takes

2 π

radians (rad) to get around a circle, which is the same as 360°. How many radians are in 1°?

Answer

0.017 rad

1-47 §1.3 source ↗

A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.

Answer

1 light-nanosecond

1-49 §1.3 source ↗

A fluid ounce is about 30 mL. What is the volume of a 12 fl-oz can of soda pop in cubic meters?

Answer

3.6 \times 10^{−4} m^{3}

1-51 §1.4 source ↗

Consider the physical quantities s, v, a, and t with dimensions

[s] = L,

[v] = {LT}^{−1},

[a] = {LT}^{−2},

and

[t] = T .

Determine whether each of the following equations is dimensionally consistent. (a)

v^{2} = 2 a s;

(b)

s =

v t^{2} +

0.5 a t^{2};

(c)

v = s / t;

(d)

a = v / t .

Answer

a. Yes, both terms have dimension L²T^-2 b. No. c. Yes, both terms have dimension LT^-1 d. Yes, both terms have dimension LT^-2

1-53 §1.4 source ↗

Suppose quantity

s

is a length and quantity

t

is a time. Suppose the quantities

v

and

a

are defined by v = ds/dt and a = dv/dt. (a) What is the dimension of v? (b) What is the dimension of the quantity a? What are the dimensions of (c)

\int v d t,

(d)

\int a d t,

and (e) da/dt?

Answer

a. [v] = LT^–1; b. [a] = LT^–2; c.

[\int v d t] = L;

[\int a d t] =

{LT}^{–1};

[\frac{d a}{d t}] = {LT}^{–3}

1-55 §1.4 source ↗

The arc length formula says the length

s

of arc subtended by angle

Ɵ

in a circle of radius

r

is given by the equation

s = r Ɵ .

What are the dimensions of (a) s, (b) r, and (c)

Ɵ?

Answer

a. L; b. L; c. L⁰ = 1 (that is, it is dimensionless)

1-57 §1.5 source ↗

Assuming the human body is primarily made of water, estimate the number of molecules in it. (Note that water has a molecular mass of 18 g/mol and there are roughly 10²⁴ atoms in a mole.)

Answer

10²⁸ atoms

1-59 §1.5 source ↗

Estimate the number of molecules that make up Earth, assuming an average molecular mass of 30 g/mol. (Note there are on the order of 10²⁴ objects per mole.)

Answer

10⁵¹ molecules

1-61 §1.5 source ↗

Roughly how many solar systems would it take to tile the disk of the Milky Way?

Answer

10¹⁶ solar systems

1-63 §1.5 source ↗

The average density of the Sun is on the order 10³ kg/m³. (a) Estimate the diameter of the Sun. (b) Given that the Sun subtends at an angle of about half a degree in the sky, estimate its distance from Earth.

Answer

a. Volume = 10²⁷ m³, diameter is 10⁹ m.; b. 10¹¹ m

1-65 §1.5 source ↗

A floating-point operation is a single arithmetic operation such as addition, subtraction, multiplication, or division. (a) Estimate the maximum number of floating-point operations a human being could possibly perform in a lifetime. (b) How long would it take a supercomputer to perform that many floating-point operations?

Answer

a. A reasonable estimate might be one operation per second for a total of 10⁹ in a lifetime.; b. about (10⁹)(10^–17 s) = 10^–8 s, or about 10 ns

1-67 §1.6 source ↗

Suppose your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?

Answer

2 kg

1-69 §1.6 source ↗

An infant’s pulse rate is measured to be 130 ± 5 beats/min. What is the percent uncertainty in this measurement?

Answer

1-71 §1.6 source ↗

A can contains 375 mL of soda. How much is left after 308 mL is removed?

Answer

67 mL

1-73 §1.6 source ↗

(a) How many significant figures are in the numbers 99 and 100.? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers: significant figures or percent uncertainties?

Answer

a. The number 99 has 2 significant figures; 100. has 3 significant figures. b. 1.00%; c. percent uncertainties

1-75 §1.6 source ↗

(a) A person’s blood pressure is measured to be

120 \pm 2 mm Hg .

What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg?

Answer

a. 2%; b. 1 mm Hg

1-77 §1.6 source ↗

What is the area of a circle 3.102 cm in diameter?

Answer

7.557 cm²

1-79 §1.6 source ↗

Perform the following calculations and express your answer using the correct number of significant digits. (a) A woman has two bags weighing 13.5 lb and one bag with a weight of 10.2 lb. What is the total weight of the bags? (b) The force F on an object is equal to its mass m multiplied by its acceleration a. If a wagon with mass 55 kg accelerates at a rate of 0.0255 m/s², what is the force on the wagon? (The unit of force is called the newton and it is expressed with the symbol N.)

Answer

a. 37.2 lb; because the number of bags is an exact value, it is not considered in the significant figures; b. 1.4 N; because the value 55 kg has only two significant figures, the final value must also contain two significant figures

1-81 §1.AP source ↗

Consider the equation

s =

s_{0} +

v_{0} t +

a_{0} t^{2} / 2 +

j_{0} t^{3} / 6 +

S_{0} t^{4} / 24 +

c t^{5} / 120,

where s is a length and t is a time. What are the dimensions and SI units of (a)

s_{0},

(b)

v_{0},

(c)

a_{0},

(d)

j_{0},

(e)

S_{0},

and (f) c?

Answer

[s_{0}] = L

and units are meters (m); b.

[v_{0}] = {LT}^{−1}

and units are meters per second (m/s); c.

[a_{0}] = {LT}^{−2}

and units are meters per second squared (m/s²); d.

[j_{0}] = {LT}^{−3}

and units are meters per second cubed (m/s³); e.

[S_{0}] = {LT}^{−4}

and units are m/s⁴; f.

[c] = {LT}^{−5}

and units are m/s⁵.

1-83 §1.AP source ↗

A marathon runner completes a 42.188-km course in 2 h, 30 min, and 12 s. There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the percent uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?

Answer

a. 0.059%; b. 0.01%; c. 4.681 m/s; d. 0.07%, 0.003 m/s

1-85 §1.AP source ↗

When nonmetric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was used, where 1 lbm = 0.4539 kg. (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?

Answer

a. 0.02%; b. 1×10⁴ lbm

1-87 §1.AP source ↗

A car engine moves a piston with a circular cross-section of 7.500 ± 0.002 cm in diameter a distance of 3.250 ± 0.001 cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.

Answer

a. 143.6 cm³; b. 0.1 cm³ or 0.084%

1-89 §1.CP source ↗

The purpose of this problem is to show the entire concept of dimensional consistency can be summarized by the old saying “You can’t add apples and oranges.” If you have studied power series expansions in a calculus course, you know the standard mathematical functions such as trigonometric functions, logarithms, and exponential functions can be expressed as infinite sums of the form

\sum_{n = 0}^{\infty} a_{n} x^{n} =

a_{0} +

a_{1} x +

a_{2} x^{2} +

a_{3} x^{3} +

\dots,

where the

a_{n}

are dimensionless constants for all

n = 0, 1, 2, \dots

and x is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency. That is, it actually implies the arguments of standard mathematical functions must be dimensionless, so it is not really necessary to make this latter condition a separate requirement of the definition of dimensional consistency as we have done in this section.

Answer

Since each term in the power series involves the argument raised to a different power, the only way that every term in the power series can have the same dimension is if the argument is dimensionless. To see this explicitly, suppose [x] = L^aM^bT^c. Then, [xⁿ] = [x]ⁿ = L^anM^bnT^cn. If we want [x] = [xⁿ], then an = a, bn = b, and cn = c for all n. The only way this can happen is if a = b = c = 0.

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Problems and answers are adapted from University Physics Volume 1 by Samuel J. Ling, Jeff Sanny and William Moebs, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.