Physics For Scientist and Engineers with Modern Physics

Raymond A. Serway, John W. Jenwett Jr.

Chapter 1

Physics and Measurement - all with Video Answers

Educators

Chapter Questions

Problem 1

(a) Use information on the endpapers of this book to calculate the average density of the Earth. (b) Where does the value fit among those listed in Table 14.1 in Chapter 14? Look up the density of a typical surface rock like granite in another source and compare it with the density of the Earth.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 2

The standard kilogram (Fig. $1.1 \mathrm{a}$ ) is a platinum-iridium cylinder $39.0 \mathrm{mm}$ in height and $39.0 \mathrm{mm}$ in diameter. What is the density of the material?

Mayukh Banik

Numerade Educator

Problem 3

An automobile company displays a die-cast model of its first car, made from $9.35 \mathrm{kg}$ of iron. To celebrate its hundredth year in business, a worker will recast the model in solid gold from the original dies. What mass of gold is needed to make the new model?

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 4

A proton, which is the nucleus of a hydrogen atom, can be modeled as a sphere with a diameter of $2.4 \mathrm{fm}$ and a mass of $1.67 \times 10^{-27} \mathrm{kg} .$ (a) Determine the density of the proton. (b) State how your answer to part (a) compares with the density of osmium, given in Table 14.1 in Chapter 14.

Sheh Lit Chang

University of Washington

Problem 5

Two spheres are cut from a certain uniform rock. One has radius $4.50 \mathrm{cm} .$ The mass of the other is five times greater. Find its radius.

NM

Nicholas Mesmer

Numerade Educator

Problem 6

The mass of a copper atom is $1.06 \times 10^{-25} \mathrm{kg},$ and the density of copper is $8920 \mathrm{kg} / \mathrm{m}^{3} .$ (a) Determine the number of atoms in $1 \mathrm{cm}^{3}$ of copper. (b) Visualize the one cubic centimeter as formed by stacking up identical cubes, with one copper atom at the center of each. Determine the volume of each cube. (c) Find the edge dimension of each cube, which represents an estimate for the spacing between atoms.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 7

A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown in Figure P1.7a. The atoms reside at the corners of cubes of side $L=$ $0.200 \mathrm{nm} .$ One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal as shown in Figure P1.7b. Calculate the spacing $d$ between two adjacent atomic planes that separate when the crystal cleaves.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 8

Figure $P 1.8$ shows a frustum of $a$ cone. Match each of the expressions
(a) $\pi\left(r_{1}+r_{2}\right)\left[h^{2}+\left(r_{2}-r_{1}\right)^{2}\right]^{1 / 2}$
(b) $2 \pi\left(r_{1}+r_{2}\right),$ and
(c) $\pi h\left(r_{1}^{2}+r_{1} r_{2}+r_{2}^{2}\right) / 3$ with the quantity it describes:
(d) the total circumference of the flat circular faces, (e) the volume, or (f) the area of the curved surface.
(FIGURE CANNOT COPY)

Mayukh Banik

Numerade Educator

Problem 9

Which of the following equations are dimensionally correct? (a) $v_{f}=v_{i}+a x$ (b) $y=(2 \mathrm{m}) \cos (k x),$ where $k=2 \mathrm{m}^{-1}.$

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 10

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

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 11

Kinetic energy $K$ (Chapter 7 ) has dimensions $\mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}$ It can be written in terms of the momentum $p$ (Chapter 9 ) and mass $m$ as $$K=\frac{p^{2}}{2 m}$$ (a) Determine the proper units for momentum using dimensional analysis. (b) The unit of force is the newton N, where $1 \mathrm{N}=1 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2} .$ What are the units of momentum $p$ in terms of a newton and another fundamental SI unit?

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 12

(a) Assume the equation $x=A t^{3}+B t$ describes the motion of a particular object, with $x$ having the dimension of length and $t$ having the dimension of time. Determine the dimensions of the constants $A$ and $B$. (b) Determine the dimensions of the derivative $d x / d t=3 A t^{2}+B.$

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 13

A rectangular building lot has a width of $75.0 \mathrm{ft}$ and a length of $125 \mathrm{ft}$. Determine the area of this lot in square meters.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 14

Suppose your hair grows at the rate $1 / 32$ in. 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 layers of atoms are assembled in this protein synthesis.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 15

A solid piece of lead has a mass of $23.94 \mathrm{g}$ and a volume of $2.10 \mathrm{cm}^{3} .$ From these data, calculate the density of lead in SI units (kilograms per cubic meter).

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 16

An ore loader moves 1200 tons/ $\mathrm{h}$ from a mine to the surface. Convert this rate to pounds per second, using 1 ton $=$ $2000 \mathrm{lb}.$

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 17

Why is the following situation impossible? A student's dormitory room measures $3.8 \mathrm{m}$ by $3.6 \mathrm{m},$ and its ceiling is $2.5 \mathrm{m}$ high. After the student completes his physics course, he displays his dedication by completely wallpapering the walls of the room with the pages from his copy of volume 1 (Chapters $1-22$ ) of this textbook. He even covers the door and window.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 18

A pyramid has a height of $481 \mathrm{ft},$ and its base covers an area of 13.0 acres (Fig. $\mathrm{P} 1.18$ ). The volume of a pyramid is given by the expression $V=\frac{1}{3} B h,$ where $B$ is the area of the base and $h$ is the height. Find the volume of this pyramid in cubic meters. $\left(1 \text { acre }=43560 \mathrm{ft}^{2}$ ) \right. (IMAGE CAMMOT COPY)

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 19

The pyramid described in Problem 18 contains approximately 2 million stone blocks that average 2.50 tons each. Find the weight of this pyramid in pounds.

Prashant Bana

Numerade Educator

Problem 20

Assume it takes 7.00 min to fill a 30.0 -gal gasoline tank. (a) Calculate the rate at which the tank is filled in gallons per second. (b) Calculate the rate at which the tank is filled in cubic meters per second. (c) Determine the time interval, in hours, required to fill a $1.00-\mathrm{m}^{3}$ volume at the same rate. (1 U.S. gal $=231$ in. $^{3}$ )

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 21

One cubic meter $\left(1.00 \mathrm{m}^{3}\right)$ of aluminum has a mass of $2.70 \times 10^{3} \mathrm{kg},$ and the same volume of iron has a mass of $7.86 \times 10^{3} \mathrm{kg} .$ Find the radius of a solid aluminum sphere that will balance a solid iron sphere of radius $2.00 \mathrm{cm}$ on an equal-arm balance.

Keshav Singh

Numerade Educator

Problem 22

Let $\rho_{\mathrm{AI}}$ represent the density of aluminum and $\rho_{\mathrm{Fe}}$ that of iron. Find the radius of a solid aluminum sphere that balances a solid iron sphere of radius $r_{\mathrm{Fe}}$ on an equal-arm balance.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

One gallon of paint (volume $=3.78 \times 10^{-3} \mathrm{m}^{3}$ ) covers an area of $25.0 \mathrm{m}^{2} .$ What is the thickness of the fresh paint on the wall?

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 24

An auditorium measures $40.0 \mathrm{m} \times 20.0 \mathrm{m} \times 12.0 \mathrm{m} .$ The density of air is $1.20 \mathrm{kg} / \mathrm{m}^{3} .$ What are (a) the volume of the room in cubic feet and (b) the weight of air in the room in pounds?

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 25

(a) At the time of this book's printing, the U.S. national debt is about 10 trillion dollars . If payments were made at the rate of 1000 dollars per second, how many years would it take to pay off the debt, assuming no interest were charged?
(b) A dollar bill is about $15.5 \mathrm{cm}$ long. How many dollar bills attached end to end would it take to reach the Moon? The front endpapers give the Earth-Moon distance. Note:
Before doing these calculations, try to guess at the answers. You may be very surprised.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 26

A hydrogen atom has a diameter of $1.06 \times 10^{-10} \mathrm{m} .$ The nucleus of the hydrogen atom has a diameter of approximately $2.40 \times 10^{-15} \mathrm{m} .$ (a) For a scale model, represent the diameter of the hydrogen atom by the playing length of an American football field $(100 \text { yards }=300 \mathrm{ft}$ ) and determine the diameter of the nucleus in millimeters. (b) Find the ratio of the volume of the hydrogen atom to the volume of its nucleus.

Keshav Singh

Numerade Educator

Problem 27

State the quantities you measure or estimate and the values you take for them.
Find the order of magnitude of the number of table-tennis balls that would fit into a typical-size room (without being crushed).

Keshav Singh

Numerade Educator

Problem 28

State the quantities you measure or estimate and the values you take for them.
(a) Compute the order of magnitude of the mass of a bathtub half full of water. (b) Compute the order of magnitude of the mass of a bathtub half full of copper coins.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 29

State the quantities you measure or estimate and the values you take for them.
To an order of magnitude, how many piano tuners reside in New York City? The physicist Enrico Fermi was famous for asking questions like this one on oral Ph.D. qualifying examinations.

Emily Anderson

Numerade Educator

Problem 30

State the quantities you measure or estimate and the values you take for them.
An automobile tire is rated to last for 50 000 miles. To an order of magnitude, through how many revolutions will it turn over its lifetime?

Mayukh Banik

Numerade Educator

Problem 31

Note: Appendix B.8 on propagation of uncertainty may be useful in solving some problems in this section.
The tropical year, the time interval from one vernal equinox to the next vernal equinox, is the basis for our calendar. It contains 365.242199 days. Find the number of seconds in a tropical year.

Melissa Walsh

Numerade Educator

Problem 32

Note: Appendix B.8 on propagation of uncertainty may be useful in solving some problems in this section.
How many significant figures are in the following numbers?
(a) $78.9 \pm 0.2$ (b) $3.788 \times 10^{9}$ (c) $2.46 \times 10^{-6}$ (d) 0.0053

Melissa Walsh

Numerade Educator

Problem 33

Note: Appendix B.8 on propagation of uncertainty may be useful in solving some problems in this section.
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 34

Note: Appendix B.8 on propagation of uncertainty may be useful in solving some problems in this section.
Carry out the arithmetic operations (a) the sum of the measured values $756,37.2,0.83,$ and $2 ;$ (b) the product $0.0032 \times 356.3 ;$ and $(\mathrm{c})$ the product $5.620 \times \pi.$

Eduard Sanchez

Numerade Educator

Problem 35

Call on mathematical skills from your prior education that will be useful throughout this course.
A child is surprised that because of sales tax she must pay 1.36 dollars for a toy marked 1.25 dollars . What is the effective tax rate on this purchase, expressed as a percentage?

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 36

Call on mathematical skills from your prior education that will be useful throughout this course.
Review. The average density of the planet Uranus is $1.27 \times$ $10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ The ratio of the mass of Neptune to that of Uranus is $1.19 .$ The ratio of the radius of Neptune to that of Uranus is $0.969 .$ Find the average density of Neptune.

Dominador Tan

Numerade Educator

Problem 37

Call on mathematical skills from your prior education that will be useful throughout this course.
Review. In a community college parking lot, the number of ordinary cars is larger than the number of sport utility vehicles by $94.7 \% .$ The difference between the number of cars and the number of SUVs is $18 .$ Find the number of SUVs in the lot.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 38

Call on mathematical skills from your prior education that will be useful throughout this course.
Review. The ratio of the number of sparrows visiting a bird feeder to the number of more interesting birds is $2.25 .$ On a morning when altogether 91 birds visit the feeder, what is the number of sparrows?

KS

Keyan Sheppard

Numerade Educator

Problem 39

Call on mathematical skills from your prior education that will be useful throughout this course.
Review. The ratio of the number of sparrows visiting a bird feeder to the number of more interesting birds is $2.25 .$ On a morning when altogether 91 birds visit the feeder, what is the number of sparrows?

KS

Keyan Sheppard

Numerade Educator

Problem 40

Call on mathematical skills from your prior education that will be useful throughout this course.
Find every angle $\theta$ between 0 and $360^{\circ}$ for which the ratio of $\sin \theta$ to $\cos \theta$ is -3.00.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 41

Call on mathematical skills from your prior education that will be useful throughout this course.
Prove that one solution of the equation
$$2.00 x^{4}-3.00 x^{3}+5.00 x=70.0$$ is $x=-2.22$

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 42

Call on mathematical skills from your prior education that will be useful throughout this course.
A highway curve forms a section of a circle. A car goes around the curve as shown in the helicopter view of Figure P1.42. Its dashboard compass shows that the car is initially heading due east. After it travels $d=840 \mathrm{m},$ it is heading $\theta=35.0^{\circ}$ south of east. Find the radius of curvature of its path. Suggestion: You may find it useful to learn a geometric theorem stated in Appendix B.3. (FIGURE CANNOT COPY)

Rashmi Sinha

Numerade Educator

Problem 43

Call on mathematical skills from your prior education that will be useful throughout this course.
A pet lamb grows rapidly, with its mass proportional to the cube of its length. When the lamb's length changes by $15.8 \%,$ its mass increases by $17.3 \mathrm{kg} .$ Find the lamb's mass at the end of this process.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 44

Call on mathematical skills from your prior education that will be useful throughout this course.
From the set of equations $$\begin{aligned}p &=3 q \\p r &=q s \\\frac{1}{2} p r^{2}+\frac{1}{2} q s^{2} &=\frac{1}{2} q t^{2}\end{aligned}$$ involving the unknowns $p, q, r, s,$ and $t,$ find the value of the ratio of $t$ to $r$

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 45

Call on mathematical skills from your prior education that will be useful throughout this course.
A student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. He cuts out various shapes in various sizes, calculates their areas, measures their masses, and prepares the graph of Figure P1.45. (a) Consider the fourth experimental point from the top. How far is it from the best-fit straight line? Express your answer as a difference in vertical-axis coordinate. (b) Express your answer as a percentage. (c) Calculate the slope of the line. (d) State what the graph demonstrates, referring to the shape of the graph and the results of parts (b) and (c). (e) Describe whether this result should be expected theoretically. (f) Describe the physical meaning of the slope. (GRAPH CANNOT COPY)

Mayukh Banik

Numerade Educator

Problem 46

Call on mathematical skills from your prior education that will be useful throughout this course.
Figure P1.46 shows students studying the thermal conduction of energy into cylindrical blocks of ice. As we will see in Chapter $20,$ this process is described by the equation $$\frac{Q}{\Delta t}=\frac{k \pi d^{2}\left(T_{h}-T_{c}\right)}{4 L}$$ For experimental control, in one set of trials all quantities except $d$ and $\Delta t$ are constant. (a) If $d$ is made three times larger, does the equation predict that $\Delta t$ will get larger or get smaller? By what factor? (b) What pattern of proportionality of $\Delta t$ to $d$ does the equation predict? (c) To display this proportionality as a straight line on a graph, what quantities should you plot on the horizontal and vertical axes? (d) What expression represents the theoretical slope of this graph?(IMAGE CAMMOT COPY)

Dominador Tan

Numerade Educator

Problem 47

The radius of a uniform solid sphere is measured to be $(6.50 \pm 0.20) \mathrm{cm},$ and its mass is measured to be $(1.85 \pm$ 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.

Mayukh Banik

Numerade Educator

Problem 48

(a) What is the order of magnitude of the number of microorganisms in the human intestinal tract? A typical bacterial length scale is $10^{-6} \mathrm{m}$. Estimate the intestinal volume and assume $1 \%$ of it is occupied by bacteria. (b) Does the number of bacteria suggest whether the bacteria are beneficial, dangerous, or neutral for the human body? What functions could they serve?

Dominador Tan

Numerade Educator

Problem 49

In a situation in which data are known to three significant digits, we write $6.379 \mathrm{m}=6.38 \mathrm{m}$ and $6.374 \mathrm{m}=6.37 \mathrm{m}$ When a number ends in $5,$ we arbitrarily choose to write $6.375 \mathrm{m}=6.38 \mathrm{m} .$ We could equally well write $6.375 \mathrm{m}=$ $6.37 \mathrm{m},$ "rounding down" instead of "rounding up," because we would change the number 6.375 by equal increments in both cases. Now consider an order-of-magnitude estimate, in which factors of change rather than increments are important. We write $500 \mathrm{m} \sim 10^{3} \mathrm{m}$ because 500 differs from 100 by a factor of 5 while it differs from 1000 by only a factor of $2 .$ We write $437 \mathrm{m} \sim 10^{3} \mathrm{m}$ and $305 \mathrm{m} \sim 10^{2} \mathrm{m}$. What distance differs from $100 \mathrm{m}$ and from $1000 \mathrm{m}$ by equal factors so that we could equally well choose to represent its order of magnitude as $\sim 10^{2} \mathrm{m}$ or as $\sim 10^{3} \mathrm{m} ?$

Keshav Singh

Numerade Educator

Problem 50

Collectible coins are sometimes plated with gold to enhance their beauty and value. Consider a commemorative quarter-dollar advertised for sale at $\$ 4.98 .$ It has a diameter of $24.1 \mathrm{mm}$ and a thickness of $1.78 \mathrm{mm},$ and it is completely covered with a layer of pure gold $0.180 \mu \mathrm{m}$ thick. The volume of the plating is equal to the thickness of the layer multiplied by the area to which it is applied. The patterns on the faces of the coin and the grooves on its edge have a negligible effect on its area. Assume the price of gold is $\$ 25.0$ per gram. (a) Find the cost of the gold added to the coin. (b) Does the cost of the gold significantly enhance the value of the coin? Explain your answer.

Mayukh Banik

Numerade Educator

Problem 51

The diameter of our disk-shaped galaxy, the Milky Way, is about $1.0 \times 10^{5}$ light-years (ly). The distance to the Andromeda galaxy (Fig. P1.51), which is the spiral galaxy nearest to the Milky Way, is about 2.0 million ly. If a scale model represents the Milky Way and Andromeda galaxies as dinner plates $25 \mathrm{cm}$ in diameter, determine the distance between the centers of the two plates.(IMAGE CAMMOT COPY)

Prashant Bana

Numerade Educator

Problem 52

Why is the following situation impossible? In an effort to boost interest in a television game show, each weekly winner is offered an additional1 million dollars bonus prize if he or she can personally count out that exact amount from a supply of one-dollar bills. The winner must do this task under supervision by television show executives and within one 40-hour work week. To the dismay of the show's producers, most contestants succeed at the challenge.

Rashmi Sinha

Numerade Educator

Problem 53

A high fountain of water is located at the center of a circular pool as shown in Figure $P 1.53 .$ 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 of the top of the fountain to be $\phi=55.0^{\circ} .$ How high is the fountain? (FIGURE CANNOT COPY)

Averell Hause

Carnegie Mellon University

Problem 54

A water fountain is at the center of a circular pool as shown in Figure $\mathrm{P} 1.53 .$ A student walks around the pool and measures its circumference $C$. Next, he stands at the edge of the pool and uses a protractor to measure the angle of elevation $\phi$ of his sightline to the top of the water jet. How high is the fountain?

Averell Hause

Carnegie Mellon University

Problem 55

The data in the following table represent measurements of the masses and dimensions of solid cylinders of aluminum, copper, brass, tin, and iron. (a) Use these data to calculate the densities of these substances. (b) State how your results compare with those given in Table 14.1. $$\begin{array}{lccc}\hline & \text { Mass } & \text { Diameter } & \text { Length } \\
\text { Substance } & (\mathrm{g}) & (\mathrm{cm}) & (\mathrm{cm}) \\\hline \text { Aluminum } & 51.5 & 2.52 & 3.75 \\\text { Copper } & 56.3 & 1.23 & 5.06 \\\text { Brass } & 94.4 & 1.54 & 5.69 \\\text { Tin } & 69.1 & 1.75 & 3.74 \\\text { Iron } & 216.1 & 1.89 & 9.77 \\
\hline\end{array}$$

Matthew Baker

Numerade Educator

Problem 56

The distance from the Sun to the nearest star is about $4 \times 10^{16} \mathrm{m} .$ The Milky Way galaxy (Fig. P1.56) is roughly a disk of diameter $\sim 10^{21} \mathrm{m}$ and thickness $\sim 10^{19} \mathrm{m} .$ Find the order of magnitude of the number of stars in the Milky Way. Assume the distance between the Sun and our nearest neighbor is typical. (IMAGE CAMMOT COPY)

Mayukh Banik

Numerade Educator

Problem 57

Assume there are 100 million passenger cars in the United States and the average fuel consumption is $20 \mathrm{mi} / \mathrm{gal}$ of gasoline. If the average distance traveled by each car is $10000 \mathrm{mi} / \mathrm{yr},$ how much gasoline would be saved per year if average fuel consumption could be increased to $25 \mathrm{mi} / \mathrm{gal} ?$

Averell Hause

Carnegie Mellon University

Problem 58

A spherical shell has an outside radius of $2.60 \mathrm{cm}$ and an inside radius of $a$. The shell wall has uniform thickness and is made of a material with density $4.70 \mathrm{g} / \mathrm{cm}^{3} .$ The space inside the shell is filled with a liquid having a density of $1.23 \mathrm{g} / \mathrm{cm}^{3} .$ (a) Find the mass $m$ of the sphere, including its contents, as a function of $a$. (b) For what value of the variable $a$ does $m$ have its maximum possible value? (c) What is this maximum mass? (d) Explain whether the value from part (c) agrees with the result of a direct calculation of the mass of a solid sphere of uniform density made of the same material as the shell. (e) What If? Would the answer to part (a) change if the inner wall were not concentric with the outer wall?

Rashmi Sinha

Numerade Educator

Problem 59

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) Estimate the total number of bacteria and other prokaryotes on the Earth. (b) Estimate the total mass of all such microbes.

Ajay Singhal

Numerade Educator

Problem 60

Air is blown into a spherical balloon so that, when its radius is $6.50 \mathrm{cm},$ its radius is increasing at the rate $0.900 \mathrm{cm} / \mathrm{s} .$ (a) Find the rate at which the volume of the balloon is increasing. (b) If this volume flow rate of air entering the balloon is constant, at what rate will the radius be increasing when the radius is $13.0 \mathrm{cm} ?$ (c) Explain physically why the answer to part (b) is larger or smaller than $0.9 \mathrm{cm} / \mathrm{s},$ if it is different.

Mayukh Banik

Numerade Educator

Problem 61

A rod extending between $x=0$ and $x=14.0 \mathrm{cm}$ has uniform cross-sectional area $A=9.00 \mathrm{cm}^{2} .$ Its density increases steadily between its ends from $2.70 \mathrm{g} / \mathrm{cm}^{3}$ to $19.3 \mathrm{g} / \mathrm{cm}^{3} .$
(a) Identify the constants $B$ and $C$ required in the expression $\rho=B+C x$ to describe the variable density. (b) The mass of the rod is given by $$m=\int_{\text {all material }} \rho d V=\int_{\text {all } x} \rho A d x=\int_{0}^{14.0 \mathrm{cm}}(B+C x)\left(9.00 \mathrm{cm}^{2}\right) d x$$ Carry out the integration to find the mass of the rod.

Vipender Yadav

Numerade Educator

Problem 62

In physics, it is important to use mathematical approximations. (a) Demonstrate that for small angles $\left(<20^{\circ}\right)$ $$\tan \alpha=\sin \alpha=\alpha=\frac{\pi \alpha^{\prime}}{180^{\circ}}$$ where $\alpha$ is in radians and $\alpha^{\prime}$ is in degrees. (b) Use a calculator to find the largest angle for which tan $\alpha$ may be approximated by $\alpha$ with an error less than $10.0 \%.$

Mark J

Numerade Educator

Problem 63

The consumption of natural gas by a company satisfies the empirical equation $V=1.50 t+0.00800 t^{2},$ where $V$ is the volume of gas in millions of cubic feet and $t$ is the time in months. Express this equation in units of cubic feet and seconds. Assume a month is 30.0 days.

Keshav Singh

Numerade Educator

Problem 64

A woman wishing to know the height of a mountain measures the angle of elevation of the 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) Using the symbol $y$ to represent the mountain height and the symbol $x$ to represent the woman's original distance from the mountain, label the picture. (c) Using the labeled picture, write two trigonometric equations relating the two selected variables. (d) Find the height $y.$

Keshav Singh

Numerade Educator

Problem 65

A child loves to watch as you fill a transparent plastic bottle with shampoo (Fig P1.65). Every horizontal cross section of the bottle is circular, but the diameters of the circles have different values. You pour the brightly colored shampoo into the bottle at a constant rate of $16.5 \mathrm{cm}^{3} / \mathrm{s} .$ At what rate is its level in the bottle rising (a) at a point where the diameter of the bottle is $6.30 \mathrm{cm}$ and (b) at a point where the diameter is $1.35 \mathrm{cm} ?$ (IMAGE CAMMOT COPY)

Rashmi Sinha

Numerade Educator

Problem 66

A woman stands at a horizontal distance $x$ from a mountain and measures the angle of elevation of the mountaintop above the horizontal as $\theta .$ After walking a distance $d$ closer to the mountain on level ground, she finds the angle to be $\phi .$ Find a general equation for the height y of the mountain in terms of $d, \phi,$ and $\theta,$ neglecting the height of her eyes above the ground.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

You stand in a nat meadow and observe two cows (Fig. P1.67). Cow A is due north of you and 15.0 $\mathrm{m}$ from your position. Cow $\mathrm{B}$ is $25.0 \mathrm{m}$ from your position. From your point of view, the angle between cow $\mathrm{A}$ and $\operatorname{cow} \mathrm{B}$ is $20.0^{\circ},$ with cow $\mathrm{B}$ appearing to the right of cow $\mathrm{A}$. (a) How far apart are cow $A$ and cow $B ?$ (b) Consider the view seen by cow A. According to this cow, what is the angle between you and cow B? (c) Consider the view seen by cow B. According to this cow, what is the angle between you and cow A? Hint: What does the situation look like to a hummingbird hovering above the meadow? (d) Two stars in the sky appear to be $20.0^{\circ}$ apart. Star $\mathrm{A}$ is 15.0 ly from the Earth, and star B, appearing to the right of star A, is $25.0 \mathrm{ly}$ from the Earth. To an inhabitant of a planet orbiting star A, what is the angle in the sky between star $B$ and our Sun? (IMAGE CAMMOT COPY)

Dominador Tan

Numerade Educator