Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett

Chapter 1

Introduction - all with Video Answers

Educators

Chapter Questions

Problem 1

The period of a simple pendulum, defined as the time necessary for one complete oscillation, is measured in time units and is given by $$T=2 \pi \sqrt{\frac{\ell}{g}}$$
where $\ell$ is the length of the pendulum and $g$ is the acceleration due to gravity, in units of length divided by time squared. Show that this equation is dimensionally consistent. (You might want to check the formula using your keys at the end of a string and a stopwatch.)

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

(a) Suppose that the displacement of an object is related to time according to the expression $x=B t^{2}$. What are the dimensions of $B ?$ (b) A disphcement is related to time as $x=A \sin (2 \pi f t)$, where $A$ and $f$ are constants. Find the dimensions of $A$. (Himt: A trigonometric function appearing in an equation must be dimensionless.)

Averell Hause

Carnegie Mellon University

Problem 3

A shape that covers an area $A$ and has a uniform height $h$ has a volume $V=A h$. (a) Show that $V=A h$ is dimensionally correct. (b) Show that the volumes of a cylinder and of a rectangular box can be written in the form $V=A h$ identifying $A$ in each case. (Note that $A$, sometimes called the "footprint" of the object, can have any shape and that the height can, in general, be replaced by the average thickness of the object.)

Averell Hause

Carnegie Mellon University

Problem 4

Each of the following equations was given by a student during an examination:
$\frac{1}{2} m v^{2}=\frac{1}{2} m v_{0}^{2}+\sqrt{m g h} \quad v=v_{0}+a t^{2} \quad m a=v^{2}$
Do a dimensional analysis of each equation and explain why the equation can't be correct.

Averell Hause

Carnegie Mellon University

Problem 5

Newton's law of universal gravitation is represented by
$$F=G \frac{M m}{r^{2}}$$
where $F$ is the gravitational force, $M$ and $m$ are masses, and $r$ is a length. Force has the SI units $\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}$. What are the SI units of the proportionality constant $G$ ?

Averell Hause

Carnegie Mellon University

Problem 6

ECP Kinetic energy KE (Chapter 5 ) has dimensions $\mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}$. It can be written in terms of the momentum $p$ (Chapter 6) and mass m as
$$K E=\frac{p^{2}}{2 m}$$
(a) Determine the proper units for momentum using dimensional analysis. (b) Refer to Problem 5, Given the units of force, write a simple equation relating a constant force $F$ exerted on an object, an interval of time $t$ during which the force is applied, and the resulting momentum of the object, $p$.

Averell Hause

Carnegie Mellon University

Problem 7

A fisherman catches two striped bass. The smaller of the two has a measured length of $93.46 \mathrm{~cm}$ (two decimal places, four significant figures), and the larger fish has a measured length of $135.3 \mathrm{~cm}$ (one decimal place, four significant figures). What is the total length of fish caught for the day?

Averell Hause

Carnegie Mellon University

Problem 8

A rectangular plate has a length of $(21.3 \pm 0.2) \mathrm{cm}$ and a width of $(9.8 \pm 0.1) \mathrm{cm}$. Calculate the area of the plate, including its uncertainty.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 9

How many significant figures are there in (a) $78.9 \pm 0.2$,
(b) $3.788 \times 10^{9}$, (c) $2.46 \times 10^{-6}$,
(d) $0.0032$

Averell Hause

Carnegie Mellon University

Problem 10

The speed of light is now defined to be $2.99792458 \times$ $10^{8} \mathrm{~m} / \mathrm{s}$. Express the speed of light to (a) three significant figures, (b) five significant figures, and (c) seven significant figures.

Averell Hause

Carnegie Mellon University

Problem 11

ECp A block of gold has length $5.62 \mathrm{~cm}$, width $6.35 \mathrm{~cm}$, and height $2.78 \mathrm{~cm}$. (a) Calculate the length times the width and round the answer to the appropriate number of significant figures. (b) Now multiply the rounded result of part (a) by the height and again round, obtaining the volume. (c) Repeat the process, first finding the width times the height, rounding it, and then obtaining the volume by multiplying by the length. (d) Explain why the answers don't agree in the third significant figure.

Averell Hause

Carnegie Mellon University

Problem 12

. The radius of a circle is measured to be $(10.5 \pm 0.2) \mathrm{m}$. Calculate (a) the area and (b) the circumference of the circle, and give the uncertainty in each value.

Averell Hause

Carnegie Mellon University

Problem 13

Carry out the following arithmetic operations: (a) the sum of the measured values $756,37.2,0.83$, and $2.5 ;$ (b) the product $0.0032 \times 356.3 ;$ (c) the product $5.620 \times \pi$.

Averell Hause

Carnegie Mellon University

Problem 14

(a) Using your calculator, find, in scientific notation with appropriate rounding, (a) the value of $\left(2.437 \times 10^{4}\right)$ $\left(6.5211 \times 10^{9}\right) /\left(5.37 \times 10^{4}\right)$ and (b) the value of
$\left(3.14159 \times 10^{2}\right)\left(27.01 \times 10^{4}\right) /\left(1234 \times 10^{6}\right)$

Averell Hause

Carnegie Mellon University

Problem 15

A fathom is a unit of length, usually reserved for measuring the depth of water. A fathom is approximately $6 \mathrm{ft}$ in length. Take the distance from Earth to the Moon to be 250000 miles, and use the given approximation to find the distance in fathoms.

Averell Hause

Carnegie Mellon University

Problem 16

A furlong is an old British unit of length equal to $0.125 \mathrm{mi}$, derived from the length of a furrow in an acre of ploughed land. A fortnight is a unit of time corresponding to two weeks, or 14 days and nights. Find the speed of light in megafurlongs per fortmight. (One megafurlong equals a million furlongs.)

Averell Hause

Carnegie Mellon University

Problem 17

A firkin is an old British unit of volume equal to 9 gallons. How many cubic meters are there in $6.00$ firkins?

Averell Hause

Carnegie Mellon University

Problem 18

Find the height or length of these natural wonders in kilometers, meters, and centimeters: (a) The longest cave system in the world is the Mammoth Cave system in Central Kentucky, with a mapped length of 348 miles. (b) In the United States, the waterfall with the greatest single drop is Ribbon Falls in California, which drops 1612 ft.
(c) At 20320 feet, Mount McKinley in Alaska is America's highest mountain. (d) The deepest canyon in the United States is King's Canyon in California, with a depth of $8200 \mathrm{ft}$

Averell Hause

Carnegie Mellon University

Problem 19

A rectangular building lot measures $1.00 \times 10^{2} \mathrm{ft}$ by $1.50$ $\times 10^{2} \mathrm{ft}$. Determine the area of this lot in square meters $\left(\mathrm{m}^{2}\right)$

Averell Hause

Carnegie Mellon University

Problem 20

Using the data in Table $1.3$ and the appropriate conversion factors, find the age of Earth in years.

Supratim Pal

Numerade Educator

Problem 21

Using the data in Table $1.1$ and the appropriate conversion factors, find the distance to the nearest star in feet.

Averell Hause

Carnegie Mellon University

Problem 22

1. Suppose your hair grows at the rate of $1 / 32$ inch per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of $0.1 \mathrm{~nm}$, your answer suggests how rapidly atoms are assembled in this protein synthesis.

Averell Hause

Carnegie Mellon University

Problem 23

The speed of light is about $3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$. Convert this figure to miles per hour.

Averell Hause

Carnegie Mellon University

Problem 24

A house is $50.0$ ft long and $26 \mathrm{ft}$ wide and has $8.0$ -ft-high ceilings. What is the volume of the interior of the house in cubic meters and in cubic centimeters?

Averell Hause

Carnegie Mellon University

Problem 25

The amount of water in reservoirs is often measured in acre-ft. One acre-ft is a volume that covers an area of one acre to a depth of one foot. An acre is $43560 \mathrm{ft}^{2}$. Find the volume in SI units of a reservoir containing $25.0$ acre-ft of water.

Averell Hause

Carnegie Mellon University

Problem 26

The base of a pyramid covers an area of $19.0$ acres $\left(1\right.$ acre $\left.=48560 \mathrm{ft}^{2}\right)$ and has a height of $481 \mathrm{ft}$ (Fig. $\left.\mathrm{P} 1.26\right)$. If the volume of a pyramid is given by the expression $V=b h / 3$, where $b$ is the area of the base and $h$ is the height, find the volume of this pyramid in cubic meters.

Averell Hause

Carnegie Mellon University

Problem 27

A quart container of ice cream is to be made in the form of a cube. What should be the length of a side, in centimeters? (Use the conversion 1 gallon $=3.786$ liter.)

Averell Hause

Carnegie Mellon University

Problem 28

A hamburger chain advertises that it has sold more than 50 billion hamburgers. Estimate how many pounds of hamburger meat must have been used by the chain and how many head of cattle were required to furnish the meat.

Averell Hause

Carnegie Mellon University

Problem 29

Estimate the number of Ping-Pong balls that would fit into a typical-size room (without being crushed). In your solution, state the quantities you measure or estimate and the values you take for them.

Vipender Yadav

Numerade Educator

Problem 30

Estimate the number of people in the world who are suffering from the common cold on any given day. (Answers may vary. Remember that a person suffers from a cold for about a week.)

Averell Hause

Carnegie Mellon University

Problem 31

(a) About how many microorganisms are found in the human intestinal tract? (A typical bacterial length scale is $10^{-6} \mathrm{~m}$. Estimate the intestinal volume and assume one hundredth of it is occupied by bacteria.) (b) Discuss your answer to part (a). Are these bacteria beneficial, dangerous, or neutral? What finctions could they serve?

Averell Hause

Carnegie Mellon University

Problem 32

Grass grows densely everywhere on a quarter-acre plot of land. What is the order of magnitude of the number of blades of grass? Explain your reasoning. Note that $1 \mathrm{acre}=43560 \mathrm{ft}^{2}$

Shahab Ullah

Numerade Educator

Problem 33

An automobile tire is rated to last for 50000 miles. Estimate the number of revolutions the tire will make in its lifetime.

Matthew Baker

Numerade Educator

Problem 34

Bacteria and other prokaryotes are found deep underground, in water, and in the air. One micron $\left(10^{-6} \mathrm{~m}\right)$ is a typical length scale associated with these microbes.
(a) Fstimate the total number of bacteria and other prokaryotes in the biosphere of the Earth. (b) Fstimate the total mass of all such microbes. (c) Discuss the relative importance of humans and microbes to the ecology of planet Earth. Can Homo sapiens survive without them?

Donald Albin

Numerade Educator

Problem 35

A point is located in a polar coordinate system by the coordinates $r=2.5 \mathrm{~m}$ and $\theta=35^{\circ}$. Find the $x$ - and $y$ coordinates of this point, assuming that the two coordinate systems have the same origin.

Supratim Pal

Numerade Educator

Problem 36

A certain corner of a room is selected as the origin of a rectangular coordinate system. If a fly is crawling on an adjacent wall at a point having coordinates $(2.0,1.0)$, where the units are meters, what is the distance of the fly from the corner of the room?

Averell Hause

Carnegie Mellon University

Problem 37

Express the location of the fly in Problem 36 in polar coordinates.

Averell Hause

Carnegie Mellon University

Problem 38

Two points in a rectangular coordinate system have the coordinates $(5.0,3.0)$ and $(-3.0,4.0)$, where the units are centimeters. Determine the distance between these points:

Averell Hause

Carnegie Mellon University

Problem 39

Two points are given in polar coordinates by $(r, \theta)=$ $\left(2.00 \mathrm{~m}, 50.0^{\circ}\right)$ and $\left(r, \theta=\left(5.00 \mathrm{~m},-50.0^{\circ}\right)\right.$, respectively.
What is the distance between them?

Averell Hause

Carnegie Mellon University

Problem 40

Given points $\left(r_{1}, \theta_{1}\right)$ and $\left(r_{2}, \theta_{2}\right)$ in polar coordinates, obtain a general formula for the distance between them. Simplify it as much as possible using the identity $\cos ^{2} \theta+$ $\sin ^{2} \theta=1 .$ Hint: Write the expressions for the two points in Cartesian coordinates and substitute into the usual distance formula.

Averell Hause

Carnegie Mellon University

Problem 41

For the triangle shown in Figure $\mathrm{P} 1.41$, what are (a) the length of the unknown side, (b) the tangent of $\theta$, and
(c) the sine of $\phi$ ?

Averell Hause

Carnegie Mellon University

Problem 42

A ladder $9.00 \mathrm{~m}$ long leans against the side of a building. If the ladder is inclined at an angle of $75.0^{\circ}$ to the horizontal, what is the horizontal distance from the bottom of the ladder to the building?

Averell Hause

Carnegie Mellon University

Problem 43

A high fountain of water is located at the center of a circular pool as shown in Figure $\mathrm{P} 1.43 .$ Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be $15.0 \mathrm{~m} .$ Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation at the bottom of the fountain to be $55.0^{\circ}$. How high is the fountain?

Averell Hause

Carnegie Mellon University

Problem 44

A right triangle has a hypotenuse of length $3.00 \mathrm{~m}$, and one of its angles is $30.0^{\circ}$. What are the lengths of (a) the side opposite the $80.0^{\circ}$ angle and (b) the side adjacent to the $30.0^{\circ}$ angle?

Averell Hause

Carnegie Mellon University

Problem 45

In Figure $\mathrm{P} 1,45$, find (a) the side opposite $\theta$, (b) the side adjacent to $\phi,(c) \cos \theta,(\mathrm{d}) \sin \phi$, and $(\mathrm{c}) \tan \phi .$

Averell Hause

Carnegie Mellon University

Problem 46

In a certain right uriangle, the two sides that are perpendicular to each other are $5.00 \mathrm{~m}$ and $7.00 \mathrm{~m}$ long. What is the length of the third side of the triangle?

Victor Salazar

Numerade Educator

Problem 47

In Problem 46 , what is the tangent of the angle for which $5.00 \mathrm{~m}$ is the opposite side?

Averell Hause

Carnegie Mellon University

Problem 48

A woman measures the angle of elevation of a mountaintop as $12.0^{\circ}$. After walking $1.00 \mathrm{~km}$ closer to the mountain on level ground, she finds the angle to be $14.0^{\circ} .$ (a) Draw a picture of the problem, neglecting the height of the woman's eyes above the ground. Hint: Use two triangles. (b) Select variable names for the mountain height (suggestion: $y$ ) and the woman's original distance from the mountain (suggestion: $x$ ) and label the picture. (c) Using the labeled picture and the tangent function, write two trigonometric equations relating the two sclected variables. (d) Find the height $y$ of the mountain by first solving one equation for $x$ and substituting the result into the other equation.

Prabhat Tyagi

Numerade Educator

Problem 49

A surveyor measures the distance across a straight river by the following method: Starting directly across from a tree on the opposite bank, he walks $100 \mathrm{~m}$ along the riverbank to establish a baseline. Then he sights across to the tree. The angle from his baseline to the tree is $85.0^{\circ}$. How wide is the river?

Anand Jangid

Numerade Educator

Problem 50

Refer to Problem $48 .$ Suppose the mountain height is $y$, the wonman's original distance from the mountain is
$x$, and the angle of elevation she measures from the horizontal to the top of the mountain is $\theta .$ If she moves a distance $d$ closer to the mountain and measures an angle of clevation $\phi$, find a general equation for the height of the mountain y in terms of $d, \phi$, and $\theta$, neglecting the height of her eyes above the ground.

Averell Hause

Carnegie Mellon University

Problem 51

(a) One of the fundamental laws of motion states that the acceleration of an object is directly proportional to the resultant force on it and inversely proportional to its mass. If the proportionality constant is defined to have no dimensions, determine the dimensions of force. (b) The newton is the SI unit of force. According to the results for (a), how can you express a force having units of newtons by using the fundamental units of mass, length, and time?

Averell Hause

Carnegie Mellon University

Problem 52

(a) Find a conversion factor to convert from miles per hour to kilometers per hour. (b) For a while, federal law mandated that the maximum highway speed would be 55 $\mathrm{mi} / \mathrm{h}$. Use the conversion factor from part (a) to find the speed in kilometers per hour, (c) The maximum highway speed has been raised to $65 \mathrm{mi} / \mathrm{h}$ in some places. In kilometers per hour, how much of an increase is this over the $55-\mathrm{mi} / \mathrm{h}$ limit?

Averell Hause

Carnegie Mellon University

Problem 53

Wone cubic centimeter $\left(1.0 \mathrm{~cm}^{5}\right)$ of water has a mass of $1.0 \times 10^{-3} \mathrm{~kg}$
(a) Determine the mass of $1.0 \mathrm{~m}^{3}$ of water.
(b) Assuming that biological substances are $98 \%$ water. estimate the masses of a cell with a diameter of $1.0 \mu \mathrm{m}$, a human kidney, and a fly. Take a kidney to be roughly a sphere with a radius of $4.0 \mathrm{~cm}$ and a fly to be roughly a cylinder $4.0 \mathrm{~mm}$ long and $2.0 \mathrm{~mm}$ in diameter.

Averell Hause

Carnegie Mellon University

Problem 54

Soft drinks are commonly sold in aluminum containers. To an order of magnitude, how many such containers are thrown away or recycled each year by U.S. consumers? How many tons of aluminum does this represent? In your solution, state the quantities you measure or estimate and the values you take for them.

Averell Hause

Carnegie Mellon University

Problem 55

The displacement of an object moving under uniform acceleration is some function of time and the acceleration. Suppose we write this displacement as $s=k a^{w} f^{n}$, where $k$ is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if $m=I$ and $n=2$. Can the analysis give the value of $R$ .

Averell Hause

Carnegie Mellon University

Problem 56

Compute the order of magnitude of the mass of (a) a bathtub filled with water and (b) a bathtub filled with pennies. In your solution, list the quantities you estimate and the value you estimate for each.

Donald Albin

Numerade Educator

Problem 57

You can obtain a rough estimate of the size of a molecule by the following simple experiment: Let a droplet of oil spread out on a smooth surface of water. The resulting oil slick will be approximately one molecule thick. Given an oil droplet of mass $9.00 \times 10^{-7} \mathrm{~kg}$ and density $918 \mathrm{~kg} / \mathrm{m}^{3}$ that spreads out into a circle of radius $41.8 \mathrm{~cm}$ on the water surface, what is the order of magnitude of the diameter of an oil molecule?

Averell Hause

Carnegie Mellon University

Problem 58

Sphere 1 has surface area $A_{1}$ and volume $V_{1}$, and sphere 2 has surface area $A_{2}$ and volume $V_{2}$. If the radius of sphere 2 is double the radius of sphere 1, what is the ratio of (a) the areas, $A_{2} / A_{1}$ and (b) the volumes, $V_{2} / V_{1} ?$

Vishal Gupta

Numerade Educator

Problem 59

Estimate the number of piano tuners living in New York City. This question was raised by the physicist Enrico Fermi, who was well known for making order-ofmagnitude calculations.

Averell Hause

Carnegie Mellon University

Problem 60

In 2007 , the U.S. national debt was about $$\$ 9$$ trillion.
(a) If payments were made at the rate of $$\$ 1000$$ per second, how many years would it take to pay off the debt, assuming that no interest were charged? (b) A dollar bill is about $15.5 \mathrm{~cm}$ long. If nine trillion dollar bills were laid end to end around the Earth's equator, how many times would they encircle the planet? Take the radius of the Earth at the equator to be $6378 \mathrm{~km}$.

Averell Hause

Carnegie Mellon University

Problem 61

(a) How many seconds are there in a year? (b) If one micrometeorite (a sphere with a diameter on the order of $10^{-6} \mathrm{~m}$ ) struck each square meter of the Moon each second, estimate the number of years it would take to cover the Moon with micrometeorites to a depth of one meter. (Hint: Consider a cubic box, $1 \mathrm{~m}$ on a side, on the Moon, and find how long it would take to fill the box.)

Averell Hause

Carnegie Mellon University

Problem 62

Imagine that you are the equipment manager of a professional baseball team. One of your jobs is to keep baseballs on hand for games. Balls are sometimes lost when players hit them into the stands as either home runs or foul balls. Estimate how many baseballs you have to buy per season in order to make up for such losses. Assume that your team plays an 81 -game home schedule in a season.

Averell Hause

Carnegie Mellon University