• Home
  • Textbooks
  • University Physics with Modern Physics
  • Gravitation

University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 13

Gravitation - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

03:17

Problem 1

What is the ratio of the gravitational pull of the sun on the moon to that of the earth on the moon? (Assume the distance of the moon from the sun can be approximated by the distance of the earth from the sun.) Use the data in Appendix $\mathrm{F}$. Is it more accurate to say that the moon orbits the earth, or that the moon orbits the sun?

David Gold
David Gold
Numerade Educator
09:30

Problem 2

You are sitting in the front row in your physics class. Estimate the gravitational force that the instructor exerts on you. Identify the assumptions and approximations you made to reach your answer. How does the magnitude that you estimate for this force compare with the gravity force exerted on you by the earth?

Maria Gabriela Cota Moreira
Maria Gabriela Cota Moreira
Numerade Educator
04:57

Problem 3

Rendezvous in Space! A couple of astronauts agree to rendezvous in space after hours. Their plan is to let gravity bring them together. One of them has a mass of $65 \mathrm{~kg}$ and the other a mass of $72 \mathrm{~kg}$. and they start from rest $20.0 \mathrm{~m}$ apart. (a) Make a free-body diagram of each astronaut, and use it to find his or her initial acceleration. As a rough approximation, we can model the astronauts as uniform spheres.
(b) If the astronauts' acceleration remained constant, how many days would they have to wait before reaching each other? (Careful! They both have acceleration toward cach other.) (c) Would their acceleration, in fact, remain constant? If not, would it increase or decrease? Why?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
00:51

Problem 4

Two uniform spheres, each with mass $M$ and radius $R$, touch each other. What is the magnitude of their gravitational force of attraction?

Jilin Wang
Jilin Wang
Boston University
03:56

Problem 5

Two uniform spheres, each of mass $0.260 \mathrm{~kg},$ are fixed at points $A$ and $B$ (Fig. E13.5). Find the magnitude and direction of the initial acceleration of a uniform sphere with mass $0.010 \mathrm{~kg}$ if released from rest at point $P$ and acted on only by forces of gravitational attraction of the spheres at $A$ and $\bar{B}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:01

Problem 6

Find the magnitude and direction of the net gravitational force on mass $A$ due to masses $B$ and $C$ in Fig. $\mathbf{E} 13.6 .$ Each mass is $2.00 \mathrm{~kg}$.

Sachin Rao
Sachin Rao
Numerade Educator
03:17

Problem 7

A typical adult human has a mass of about $70 \mathrm{~kg}$. (a) What force does a full moon exert on such a human when it is directly overhead with its center $378,000 \mathrm{~km}$ away? (b) Compare this force with the force exerted on the human by the earth.

David Gold
David Gold
Numerade Educator
03:04

Problem 8

An $8.00 \mathrm{~kg}$ point mass and a $12.0 \mathrm{~kg}$ point mass are held in place $50.0 \mathrm{~cm}$ apart. A particle of mass $m$ is released from a point between the two masses $20.0 \mathrm{~cm}$ from the $8.00 \mathrm{~kg}$ mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.

Jilin Wang
Jilin Wang
Boston University
03:31

Problem 9

A particle of mass $3 m$ is located $1.00 \mathrm{~m}$ from a particle of mass $m$.
(a) Where should you put a third mass $M$ so that the net gravitational force on $M$ due to the two masses is exactly zero? (b) Is the equilibrium of $M$ at this point stable or unstable (i) for points along the line connecting $m$ and $3 m,$ and (ii) for points along the line passing through $M$ and perpendicular to the line connecting $m$ and $3 m ?$

Ajay Singhal
Ajay Singhal
Numerade Educator
10:05

Problem 10

The point masses $m$ and $2 m$ lie along the $x$ -axis, with $m$ at the origin and $2 m$ at $x=L$. A third point mass $M$ is moved along the $x$ -axis. (a) At what point is the net gravitational force on $M$ due to the other two masses equal to zero? (b) Sketch the $x$ -component of the net force on $M$ due to $m$ and $2 m,$ taking quantities to the right as positive. Include the regions $x<0,0<x<L,$ and $x>L$. Be especially careful to show the behavior of the graph on either side of $x=0$ and $x=L$

NR
Nathaniel Riche
Numerade Educator
02:17

Problem 11

At what distance above the surface of the earth is the acceleration due to the earth's gravity $0.980 \mathrm{~m} / \mathrm{s}^{2}$ if the acceleration due to gravity at the surface has magnitude $9.80 \mathrm{~m} / \mathrm{s}^{2} ?$

David Gold
David Gold
Numerade Educator
05:58

Problem 12

The mass of Venus is $81.5 \%$ that of the earth, and its radius is $94.9 \%$ that of the earth. (a) Compute the acceleration due to gravity on the surface of Venus from these data. (b) If a rock weighs $75.0 \mathrm{~N}$ on earth, what would it weigh at the surface of Venus?

NR
Nathaniel Riche
Numerade Educator
08:16

Problem 13

Titania, the largest moon of the planet Uranus, has $\frac{1}{8}$ the radius of the carth and $\frac{1}{1700}$ the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

Maria Gabriela Cota Moreira
Maria Gabriela Cota Moreira
Numerade Educator
01:37

Problem 14

Rhea, one of Saturn's moons, has a radius of $764 \mathrm{~km}$ and an acceleration due to gravity of $0.265 \mathrm{~m} / \mathrm{s}^{2}$ at its surface. Calculate its mass and average density.

Jilin Wang
Jilin Wang
Boston University
09:57

Problem 15

$A} science-fiction author asks for your help. He wants to write about a newly discovered spherically symmetric planet that has the same average density as the earth but with a $25 \%$ larger radius. (a) What is $g$ on this planet? (b) If he decides to have his explorers weigh the same on this planet as on earth, how should he change its average density?

Linda Winkler
Linda Winkler
Numerade Educator
02:52

Problem 16

Volcanoes on Io. Jupiter's moon Io has active volcanoes (in fact. it is the most volcanically active object in the solar system) that eject material as high as $500 \mathrm{~km}$ (or even higher) above the surface. Io has a mass of $8.93 \times 10^{22} \mathrm{~kg}$ and a radius of $1821 \mathrm{~km} .$ For this calculation, ignore any variation in gravity over the $500 \mathrm{~km}$ range of the debris. How high would this material go on earth if it were ejected with the same speed as on lo?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:23

Problem 17

Use the results of Example 13.5 (Section 13.3 ) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft's mass?

David Gold
David Gold
Numerade Educator
01:54

Problem 18

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was $2.87 \times 10^{6} \mathrm{~km}$ from the earth and traveling at $1.20 \times 10^{4} \mathrm{~km} / \mathrm{h}$ relative to the earth. At this time, what were (a) the spacecraft's kinetic energy relative to the earth and (b) the potential energy of the earthspacecraft system?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:18

Problem 19

A planet orbiting a distant star has radius $3.24 \times 10^{6} \mathrm{~m}$. The escape speed for an object launched from this planet's surface is $7.65 \times 10^{3} \mathrm{~m} / \mathrm{s}$. What is the acceleration due to gravity at the surface of the planet?

David Gold
David Gold
Numerade Educator
07:07

Problem 20

Estimate the cruising altitude of an airplane during a transatlantic flight. If you are a passenger in the plane when it is at this altitude, what is the percentage change in the gravitational energy of the system of you and the Earth compared to when you were in the airport waiting to board the plane? For which of your two locations is the gravitational potential energy greater? (Hint: Use the power series for $(1+x)^{n}$ given in Appendix B.)

Linda Winkler
Linda Winkler
Numerade Educator
01:17

Problem 21

You are standing on the surface of a planet that has spherical symmetry and a radius of $5.00 \times 10^{6} \mathrm{~m}$. The gravitational potential encrgy $U$ of the system composed of you and the planet is $-1.20 \times 10^{+9} \mathrm{~J}$ if we choose $U$ to be zero when you are very far from the planet. What is the magnitude of the gravity force that the planet exerts on you when you are standing on its surface?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:17

Problem 22

$\cdot \mathrm{A}$ satellite of mass $m$ is in a circular orbit around a spherical planet of mass $m_{\mathrm{p}}$. The kinetic energy of the satellite is $K_{A}$ when its orbit radius is $r_{A}$. In terms of $r_{A}$, what must the orbit radius be in order for the kinetic energy of the satellite to be $2 K_{A} ?$

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:35

Problem 23

Let the gravitational potential energy of the earth-satellite system be zero in the limit that the orbit radius is very large, so the total mechanical energy of the satellite-earth system is given by Eq. (13.13). If the kinetic energy of the satellite is $2.00 \times 10^{6} \mathrm{~J}$, what are the total energy and the gravitational potential energy of the satellite-earth system?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:42

Problem 24

An earth satellite moves in a circular orbit with an orbital speed of $6200 \mathrm{~m} / \mathrm{s}$. Find (a) the time of one revolution of the satellite; (b) the radial acceleration of the satellite in its orbit.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
03:09

Problem 25

For a satellite to be in a circular orbit $890 \mathrm{~km}$ above the surface of the earth, (a) what orbital speed must it be given, and (b) what is the period of the orbit (in hours)?

David Gold
David Gold
Numerade Educator
02:59

Problem 26

Aura Mission. On July 15, 2004 NASA launched the Aura spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit $705 \mathrm{~km}$ above the earth's surface. Assume a circular orbit. (a) How many hours does it take this satellite to make one orbit? (b) How fast (in $\mathrm{km} / \mathrm{s})$ is the Aura spacecraft moving?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:49

Problem 27

Two satellites are in circular orbits around a planet that has radius $9.00 \times 10^{6} \mathrm{~m}$. One satellite has mass $68.0 \mathrm{~kg}$. orbital radius $7.00 \times 10^{7} \mathrm{~m},$ and orbital speed $4800 \mathrm{~m} / \mathrm{s}$. The second satellite has mass $84.0 \mathrm{~kg}$ and orbital radius $3.00 \times 10^{7} \mathrm{~m}$. What is the orbital speed of this second satellite?

David Gold
David Gold
Numerade Educator
04:05

Problem 28

International Space Station. In its orbit each day, the International Space Station makes 15.65 revolutions around the earth. Assuming a circular orbit, how high is this satellite above the surface of the earth?

Krystal K
Krystal K
Numerade Educator
02:37

Problem 29

Deimos, a moon of Mars, is about $12 \mathrm{~km}$ in diameter with mass $1.5 \times 10^{15} \mathrm{~kg} .$ Suppose you are stranded alone on Deimos and want to play a one-person game of baseball. You would be the pitcher, and you would be the batter! (a) With what speed would you have to throw a baseball so that it would go into a circular orbit just above the surface and return to you so you could hit it? Do you think you could actually throw it at this speed? (b) How long (in hours) after throwing the ball should you be ready to hit it? Would this be an action-packed bascball game?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
05:28

Problem 30

Hot Jupiters. In 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term "hot Jupiter"). The orbit was just $\frac{1}{9}$ the distance of Mercury from our sun, and it takes the planet only 3.09 days to make one orbit (assumed to be circular). (a) What is the mass of the star? Express your answer in kilograms and as a multiple of our sun's mass. (b) How fast (in $\mathrm{km} / \mathrm{s})$ is this planet moving?

Meghan Miholics
Meghan Miholics
Numerade Educator
03:04

Problem 31

The star Rho ${ }^{1}$ Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho ${ }^{1}$ Cancri with an orbital radius equal to 0.11 times the radius of the earth's orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho " Cancri?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:51

Problem 32

In March $2006,$ two small satellites were discovered orbiting Pluto, one at a distance of $48,000 \mathrm{~km}$ and the other at $64,000 \mathrm{~km}$. Pluto already was known to have a large satellite Charon, orbiting at $19,600 \mathrm{~km}$ with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using the mass of Pluto.

Jilin Wang
Jilin Wang
Boston University
02:55

Problem 33

The dwarf planet Pluto has an elliptical orbit with a semi-major axis of $5.91 \times 10^{12} \mathrm{~m}$ and eccentricity $0.249 .$ (a) Calculate Pluto's orbital period. Express your answer in seconds and in earth years. (b) During Pluto's orbit around the sun, what are its closest and farthest distances from the sun?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:31

Problem 34

Planets Beyond the Solar System. On October 15,2001 a planet was discovered orbiting around the star HD 68988 . Its orbital distance was measured to be 10.5 million kilometers from the center of the star, and its orbital period was estimated at 6.3 days. What is the mass of HD 68988 ? Express your answer in kilograms and in terms of our sun's mass. (Consult Appendix F.)

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:45

Problem 35

A thin spherical shell has radius $r_{A}=4.00 \mathrm{~m}$ and mass $m_{A}=20.0 \mathrm{~kg} .$ It is concentric with a second thin spherical shell that has radius $r_{B}=6.00 \mathrm{~m}$ and mass $m_{B}=40.0 \mathrm{~kg} .$ What is the net gravitational force that the two shells exert on a point mass of $0.0200 \mathrm{~kg}$ that is a distance $r$ from the common center of the two shells, for (a) $r=2.00 \mathrm{~m}$ (inside both shells), (b) $r=5.00 \mathrm{~m}$ (in the space between the two shells), and (c) $r=8.00 \mathrm{~m}$ (outside both shells)?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
00:58

Problem 36

A uniform, spherical, $1000.0 \mathrm{~kg}$ shell has a radius of $5.00 \mathrm{~m} .$
(a) Find the gravitational force this shell exerts on a $2.00 \mathrm{~kg}$ point mass placed at the following distances from the center of the shell: (i) $5.01 \mathrm{~m}$. (ii) $4.99 \mathrm{~m},$ (iii) $2.72 \mathrm{~m}$. (b) Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass $m$ as a function of the distance $r$ of $m$ from the center of the sphere. Include the region from $r=0$ to $r \rightarrow \infty$

Jilin Wang
Jilin Wang
Boston University
07:20

Problem 37

A uniform, solid, $1000.0 \mathrm{~kg}$ sphere has a radius of $5.00 \mathrm{~m}$.
(a) Find the gravitational force this sphere exerts on a $2.00 \mathrm{~kg}$ point mass placed at the following distances from the center of the sphere: (i) $5.01 \mathrm{~m}$, (ii) $2.50 \mathrm{~m}$. (b) Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass $m$ as a function of the distance $r$ of $m$ from the center of the sphere. Include the region from $r=0$ to $r \rightarrow \infty$

David Gold
David Gold
Numerade Educator
05:58

Problem 38

A thin, uniform rod has length $L$ and mass $M$. A small uniform sphere of mass $m$ is placed a distance $x$ from one end of the rod, along the axis of the rod (Fig. E13.38). (a) Calculate the gravitational potential energy of the rod-sphere system. Take the potential energy to be zero when the rod and sphere are infinitely far apart. Show that your answer reduces to the expected result when $x$ is much larger than $L$. (Hint: Use the power series expansion for $\ln (1+x)$ given in Appendix B.) (b) Use $F_{x}=-d U / d x$ to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4 ). Show that your answer reduces to the expected result when $x$ is much larger than $L$

Prabhu Ramji
Prabhu Ramji
Numerade Educator
06:41

Problem 39

Consider the ringshaped object in Fig. $\mathrm{E} 13.39 .$ A particle with mass $m$ is placed a distance $x$ from the center of the ring. along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy $U$ of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when $x$ is much larger than the radius $a$ of the ring. (c) Use $F_{x}=-d U / d x$ to find the magnitude and direction of the force on the particle (see Section 7.4 ). (d) Show that your answer to part (c) reduces to the expected result when $x$ is much larger than $a$. (e) What are the values of $U$ and $F_{x}$ when $x=0 ?$ Explain why these results make sense.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
04:15

Problem 40

Define the gravitational field $\vec{g}$ at some point to be equal to the gravitational force $F_{e}$ on a small object placed at that point divided by the mass $m$ of the object, so $\bar{g}=\vec{F}_{e} / m .$ A spherical shell has mass $M$ and radius $R .$ What is the magnitude of the gravitational field at the following distances from the center of the shell: (a) $r<R$ and (b) $r>R ?$

Linda Winkler
Linda Winkler
Numerade Educator
01:59

Problem 41

The acceleration due to gravity at the north pole of Neptune is approximately $11.2 \mathrm{~m} / \mathrm{s}^{2} .$ Neptune has mass $1.02 \times 10^{26} \mathrm{~kg}$ and radius $2.46 \times 10^{4} \mathrm{~km}$ and rotates once around its axis in about $16 \mathrm{~h}$. (a) What is the gravitational force on a $3.00 \mathrm{~kg}$ object at the north pole of Neptune? (b) What is the apparent weight of this same object at Neptune's equator? (Note that Neptune's "surface" is gaseous, not solid, so it is impossible to stand on it.)

Prabhu Ramji
Prabhu Ramji
Numerade Educator
03:53

Problem 42

A Visit to Santa. You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of $395.0 \mathrm{~N}$ in the rope. If Sneezy hangs from a similar rope while delivering presents at the earth's equator, what will the tension in it be? (Recall that the earth is rotating about an axis through its north and south poles.) Consult Appendix $\mathrm{F}$ and start with a free-body diagram of Sneezy at the equator.

Jilin Wang
Jilin Wang
Boston University
08:53

Problem 43

At the Galaxy's Core. Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 13.8). A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about $200 \mathrm{~km} / \mathrm{s}$ (a) Determine the mass of the object at the center of the Milky Way galaxy. Give your answer both in kilograms and in solar masses (one solar mass is the mass of the sun). (b) Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star? (c) Many astronomers believe that the massive object at the center of the Milky Way galaxy is a black hole. If so, what must the Schwarzschild radius of this black hole be? Would a black hole of this size fit inside the earth's orbit around the sun?

Linda Winkler
Linda Winkler
Numerade Educator
01:44

Problem 44

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at $30,000 \mathrm{~km} / \mathrm{s}$. (a) How far are these clumps from the center of the black hole? (b) What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass. (c) What is the radius of its event horizon?

Jilin Wang
Jilin Wang
Boston University
07:11

Problem 45

Three uniform spheres are fixed at the positions shown in Fig. $P 13.45$. (a) What are the magnitude and direction of the force on a $0.0150 \mathrm{~kg}$ particle placed at $P ?$ (b) If the spheres are in deep outer space and a $0.0150 \mathrm{~kg}$ particle is released from rest $300 \mathrm{~m}$ from the origin along a line $45^{\circ}$ below the $-x$ -axis, what will the particle's speed be when it reaches the origin?

David Gold
David Gold
Numerade Educator
01:13

Problem 46

There is strong evidence that Europa, a satellite of Jupiter, has a liquid ocean beneath its icy surface. Many scientists think we should land a vehicle there to search for life. Before launching it, we would want to test such a lander under the gravity conditions at the surface of Europa. One way to do this is to put the lander at the end of a rotating arm in an orbiting earth satellite. If the arm is $4.25 \mathrm{~m}$ long and pivots about one end, at what angular speed (in rpm) should it spin so that the acceleration of the lander is the same as the acceleration due to gravity at the surface of Europa? The mass of Europa is $4.80 \times 10^{22} \mathrm{~kg}$ and its diameter is $3120 \mathrm{~km}$

Jilin Wang
Jilin Wang
Boston University
07:28

Problem 47

An experiment is performed in deep space with two uniform spheres, one with mass $50.0 \mathrm{~kg}$ and the other with mass $100.0 \mathrm{~kg}$. They have equal radii, $r=0.20 \mathrm{~m}$. The spheres are released from rest with their centers $40.0 \mathrm{~m}$ apart. They accelerate toward each other because of their mutual gravitational attraction. You can ignore all gravitational forces other than that between the two spheres.
(a) Explain why linear momentum is conserved. (b) When their centers are $20.0 \mathrm{~m}$ apart, find (i) the speed of each sphere and (ii) the magnitude of the relative velocity with which one sphere is approaching the other. (c) How far from the initial position of the center of the $50.0 \mathrm{~kg}$ sphere do the surfaces of the two spheres collide?

David Gold
David Gold
Numerade Educator
09:19

Problem 48

Mission to Titan. On December $25,2004,$ th probe separated from the Cassini spacecraft orbiting Saturn 22 day journey to Saturn's moon Titan, on whose surface Besides the data in Appendix $\mathrm{F},$ it is useful to know is $1.22 \times 10^{6} \mathrm{~km}$ from the center of Saturn and has $1.35 \times 10^{23} \mathrm{~kg}$ and a diameter of $5150 \mathrm{~km}$. At what distance should the gravitational pull of Titan the gravitational pull of Saturn?

Linda Winkler
Linda Winkler
Numerade Educator
05:56

Problem 49

Many satcllites are moving in a circle in the earth's equatorial plane. They are at such a height above the earth's surface that they always remain above the same point. (a) Find the altitude of these satellites above the earth's surface. (Such an orbit is said to be geosynchronous. signals from these satellites cannot directly reach receivers on earth that are north of $81.3^{\circ} \mathrm{N}$ latitude.

Linda Winkler
Linda Winkler
Numerade Educator
02:46

Problem 50

Two spherically symmetric planets with no atmosphere have the same average density, but planet $B$ has twice the radius of planet $A$. A small satellite of mass $m_{A}$ has period $T_{A}$ when it orbits planet $A$ in a circular orbit that is just above the surface of the planet. A small satellite of mass $m_{B}$ has period $T_{B}$ when it orbits planet $B$ in a circular orbit that is just above the surface of the planet. How does $T_{B}$ compare to $T_{A} ?$
with a density of $2500 \mathrm{~kg} / \mathrm{m}^{3} ?$

Prabhu Ramji
Prabhu Ramji
Numerade Educator
03:56

Problem 51

What is the escape speed from a $300-\mathrm{km}$ -diameter asteroid with a density of $2500 \mathrm{~kg} / \mathrm{m}^{3} ?$

Meghan Miholics
Meghan Miholics
Numerade Educator
02:43

Problem 52

A landing craft with mass $12,500 \mathrm{~kg}$ is in a circular orbit $5.75 \times 10^{5} \mathrm{~m}$ above the surface of a planet. The period of the orbit is 5800 s. The astronauts in the lander measure the diameter of the planet to be $9.60 \times 10^{6} \mathrm{~m}$. The lander sets down at the north pole of the planet. What is the weight of an $85.6 \mathrm{~kg}$ astronaut as he steps out onto the planet's surface?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
08:30

Problem 53

Planet X rotates in the same manner as the earth, around an axis through its north and south poles, and is perfectly spherical. An astronaut who weighs $943.0 \mathrm{~N}$ on the earth weighs $915.0 \mathrm{~N}$ at the north pole of Planet $X$ and only $850.0 \mathrm{~N}$ at its equator. The distance from the north pole to the equator is $18,850 \mathrm{~km}$, measured along the surface of Planet X. (a) How long is the day on Planet X? (b) If a $45,000 \mathrm{~kg}$ satellite is placed in a circular orbit $2000 \mathrm{~km}$ above the surface of Planet $\mathrm{X}$. what will be its orbital period?

David Gold
David Gold
Numerade Educator
08:27

Problem 54

Example 13.10 assumes a constant density for the earth, but Fig. 13.9 shows that this is not accurate. (a) Approximate the density graph in Fig. 13.9 by a straight line running from $13,000 \mathrm{~kg} / \mathrm{m}^{3}$ at $r=0$ to $3000 \mathrm{~kg} / \mathrm{m}^{3}$ at $r=R_{\mathrm{E}}$. Divide the earth into concentric shells of width $d r$ and therefore volume $d V=4 \pi r^{2} d r .$ Integrate to find the mass enclosed as a function of the distance from the earth's center.
(b) Find the expression for the magnitude of the gravitational force on a mass $m$ at a distance $r$ from the earth's center.

Linda Winkler
Linda Winkler
Numerade Educator
03:33

Problem 55

An astronaut, whose mission is to go where no one has gone before, lands on a spherical planet in a distant galaxy. As she stands on the surface of the planet, she releases a small rock from rest and finds that it takes the rock $0.480 \mathrm{~s}$ to fall $1.90 \mathrm{~m}$. If the radius of the planet is $8.60 \times 10^{7} \mathrm{~m},$ what is the mass of the planet?

PR
Patrick Ramsey
Numerade Educator
03:39

Problem 56

Your starship, the Aimless Wanderer, lands on the mysterious planet Mongo. As chicf scientist-engineer, you make the following measurements: A $2.50 \mathrm{~kg}$ stone thrown upward from the ground at $12.0 \mathrm{~m} / \mathrm{s}$ returns to the ground in $4.80 \mathrm{~s} ;$ the circumference of Mongo at the equator is $2.00 \times 10^{5} \mathrm{~km} ;$ and there is no appreciable atmosphere on Mongo. The starship commander, Captain Confusion, asks for the following information: (a) What is the mass of Mongo? (b) If the Aimless Wanderer goes into a circular orbit $30,000 \mathrm{~km}$ above the surface of Mongo. how many hours will it take the ship to complete one orbit?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
03:44

Problem 57

You are exploring a distant planet. When your spaceship is in a circular orbit at a distance of $630 \mathrm{~km}$ above the planet's surface, the ship's orbital speed is $4900 \mathrm{~m} / \mathrm{s}$. By observing the planet, you determine its radius to be $4.48 \times 10^{6} \mathrm{~m}$. You then land on the surface and, at a place where the ground is level, launch a small projectile with initial speed $12.6 \mathrm{~m} / \mathrm{s}$ at an angle of $30.8^{\circ}$ above the horizontal. If resistance due to the planet's atmosphere is negligible, what is the horizontal range of the projectile?

David Gold
David Gold
Numerade Educator
06:39

Problem 58

The $0.100 \mathrm{~kg}$ sphere in Fig. $\mathrm{P} 13.58$ is released from rest at the position shown in the sketch, with its center $0.400 \mathrm{~m}$ from the center of the $5.00 \mathrm{~kg}$ mass. Assume that the only forces on the $0.100 \mathrm{~kg}$ sphere are the gravitational forces exerted by the other two spheres and that the $5.00 \mathrm{~kg}$ and $10.0 \mathrm{~kg}$ spheres are held in place at their initial positions. What is the speed of the $0.100 \mathrm{~kg}$ sphere when it has moved $0.400 \mathrm{~m}$ to the right from its initial position?

Linda Winkler
Linda Winkler
Numerade Educator
02:38

Problem 59

The magnitude of the gravitational flux $\Phi_{\mathrm{g}}$ through a surface with area $A$ is defined such that it is equal to $g A$ when the gravitational field $\vec{g}$ (see Exercise 13.40 ) is at all points on the surface constant and perpendicular to the surface. For a spherically symmetric object with radius $R$ and mass $M$, this is the case for a surface that is spherical with radius $r$ and concentric with the object. Consider the case where $r>R .$ For this surface calculate the magnitude of the gravitational flux through the surface due to the gravitation field of the object. Does your answer depend on the radius $r$ of the spherical surface? (We'll consider a similar result for the electric field in Chapter $22 .$ )

Prabhu Ramji
Prabhu Ramji
Numerade Educator
17:07

Problem 60

A narrow uniform rod has length $2 a$. The linear mass density of the rod is $\rho,$ so the mass $m$ of a length $l$ of the rod is $\rho l$. (a) A point mass is located a perpendicular distance $r$ from the center of the rod. Calculate the magnitude and direction of the force that the rod exerts on the point mass. (Hint: Let the rod be along the $y$ -axis with the center of the rod at the origin, and divide the rod into infinitesimal segments that have length $d y$ and that are located at coordinate $y$. The mass of the segment is $d m=\rho d y$. Write expressions for the $x$ - and $y$ -components of the force on the point mass, and integrate from $-a$ to $+a$ to find the components of the total force. Use the integrals in Appendix B.) (b) What does your result become for $a \gg r ?$ (Hint: Use the power series for $(1+x)^{n}$ given in Appendix B.) (c) For $a \gg r,$ what is the gravitational field $g=\boldsymbol{F}_{g} / m$ at a distance $r$ from the rod? (d) Consider a cylinder of radius $r$ and length $L$ whose axis is along the rod. As in part (c), let the length of the rod be much greater than both the radius and length of the cylinder. Then the gravitational ficld is constant on the curved side of the cylinder and perpendicular to it, so the gravitational flux $\Phi_{g}$ through this surface is cqual to $g A$, where $A=2 \pi r L$ is the area of the curved side of the cylinder (see Problem 13.59 ). Calculate this flux. Write your result in terms of the mass $M$ of the portion of the rod that is inside the cylindrical surface. How does your result depend on the radius of the cylindrical surface?

Linda Winkler
Linda Winkler
Numerade Educator
03:07

Problem 61

Falling Hammer. A hammer with mass $m$ is dropped from rest from a height $h$ above the earth's surface. This height is not necessarily small compared with the radius $R_{\mathrm{E}}$ of the earth. Ignoring air resistance, derive an expression for the speed $v$ of the hammer when it reaches the earth's surface. Your expression should involve $h, R_{\mathrm{E}},$ and $m_{\mathrm{E}}$ (the earth's mass).

Prabhu Ramji
Prabhu Ramji
Numerade Educator
10:19

Problem 62

(a) Calculate how much work is required to launch a spacecraft of mass $m$ from the surface of the earth (mass $m_{E}$, radius $R_{\mathrm{E}}$ ) and place it in a circular low earth orbit - that is, an orbit whose altitude above the earth's surface is much less than $R_{\mathrm{E}}$. (As an example, the International Space Station is in low earth orbit at an altitude of about $400 \mathrm{~km},$ much less than $R_{\mathrm{E}}=6370 \mathrm{~km} .$ ) Ignore the kinetic energy that the spacecraft has on the ground due to the earth's rotation.
(b) Calculate the minimum amount of additional work required to move the spacecraft from low earth orbit to a very great distance from the earth. Ignore the gravitational effects of the sun, the moon, and the other planets. (c) Justify the statement "In terms of energy, low earth orbit is halfway to the edge of the universe."

Linda Winkler
Linda Winkler
Numerade Educator
09:54

Problem 63

Binary Star-Equal Masses. Two identical stars with mass $M$ orbit around their center of mass. Each orbit is circular and has radius $R,$ so that the two stars are always on opposite sides of the circle. (a) Find the gravitational force of one star on the other. (b) Find the orbital speed of each star and the period of the orbit. (c) How much energy would be required to separate the two stars to infinity?

Constance Wall
Constance Wall
Numerade Educator
19:46

Problem 64

Binary Star-Different Masses. Two stars, with masses $M_{1}$ and $M_{2}$, are in circular orbits around their center of mass. The star with mass $M_{1}$ has an orbit of radius $R_{1}$; the star with mass $M_{2}$ has an orbit of radius $R_{2}$. (a) Show that the ratio of the orbital radii of the two stars equals the reciprocal of the ratio of their masses-that is, $R_{1} / R_{2}=M_{2} / M_{1}$. (b) Explain why the two stars have the same orbital period, and show that the period $T$ is given by $T=2 \pi\left(R_{1}+R_{2}\right)^{3 / 2} / \sqrt{G\left(M_{1}+M_{2}\right)} .$ (c) The two stars in a certain binary star system move in circular orbits. The first star, Alpha, has an orbital speed of $36.0 \mathrm{~km} / \mathrm{s}$. The second star, Beta, has an orbital speed of $12.0 \mathrm{~km} / \mathrm{s}$. The orbital period is $137 \mathrm{~d}$. What are the masses of each of the two stars? (d) One of the best candidates for a black hole is found in the binary system called $\mathrm{A} 0620-0090$. The two objects in the binary system are an orange star, V6 16 Monocerotis, and a compact object believed to be a black hole (see Fig. 13.28 ). The orbital period of $\mathrm{A} 0620-0090$ is 7.75 hours, the mass of $\mathrm{V} 616$ Monocerotis is estimated to be 0.67 times the mass of the sun, and the mass of the black hole is estimated to be 3.8 times the mass of the sun. Assuming that the orbits are circular, find the radius of each object's orbit and the orbital speed of each object. Compare these answers to the orbital radius and orbital speed of the earth in its orbit around the sun.

Linda Winkler
Linda Winkler
Numerade Educator
02:52

Problem 65

Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed $2.0 \times 10^{4} \mathrm{~m} / \mathrm{s}$ when at a distance of $2.5 \times 10^{11} \mathrm{~m}$ from the center of the sun, what is its speed when at a distance of $5.0 \times 10^{10} \mathrm{~m} ?$

David Gold
David Gold
Numerade Educator
02:33

Problem 66

The planet Uranus has a radius of $25,360 \mathrm{~km}$ and a surface acceleration due to gravity of $9.0 \mathrm{~m} / \mathrm{s}^{2}$ at its poles. Its moon Miranda (discovered by Kuiper in 1948 ) is in a circular orbit about Uranus at an altitude of $104,000 \mathrm{~km}$ above the planet's surface. Miranda has a mass of $6.6 \times 10^{19} \mathrm{~kg}$ and a radius of $236 \mathrm{~km}$. (a) Calculate the mass of Uranus from the given data. (b) Calculate the magnitude of Miranda's acceleration due to its orbital motion about Uranus. (c) Calculate the acceleration due to Miranda's gravity at the surface of Miranda. (d) Do the answers to parts (b) and (c) mean that an object released $1 \mathrm{~m}$ above Miranda's surface on the side toward Uranus will fall up relative to Miranda? Explain.

Jilin Wang
Jilin Wang
Boston University
12:48

Problem 67

Consider a spacecraft in an elliptical orbit around the earth. At the low point, or perigee, of its orbit, it is $400 \mathrm{~km}$ above the earth's surface; at the high point, or apogee, it is $4000 \mathrm{~km}$ above the carth's surface. (a) What is the period of the spacecraft's orbit? (b) Using conservation of angular momentum, find the ratio of the spacecraft's speed at perigee to its speed at apogee. (c) Using conservation of energy, find the speed at perigee and the speed at apogee. (d) It is necessary to have the spacecraft escape from the earth completely. If the spacecraft's rockets are fired at perigee, by how much would the speed have to be increased to achieve this? What if the rockets were fired at apogee? Which point in the orbit is more efficient to use?

Linda Winkler
Linda Winkler
Numerade Educator
02:13

Problem 68

A rocket with mass $5.00 \times 10^{3} \mathrm{~kg}$ is in a circular orbit of radius $7.20 \times 10^{6} \mathrm{~m}$ around the earth. The rocket's engines fire for a period of time to increase that radius to $8.80 \times 10^{6} \mathrm{~m},$ with the orbit again circular. (a) What is the change in the rocket's kinetic energy? Does the kinetic energy increase or decrease? (b) What is the change in the rocket's gravitational potential energy? Does the potential energy increase or decrease? (c) How much work is done by the rocket engines in changing the orbital radius?

Jilin Wang
Jilin Wang
Boston University
03:17

Problem 69

A $5000 \mathrm{~kg}$ spacecraft is in a circular orbit $2000 \mathrm{~km}$ above the surface of Mars. How much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is $4000 \mathrm{~km}$ above
the surface?

David Gold
David Gold
Numerade Educator
02:34

Problem 70

A satellite with mass $848 \mathrm{~kg}$ is in a circular orbit with an orbital speed of $9640 \mathrm{~m} / \mathrm{s}$ around the earth. What is the new orbital speed after friction from the earth's upper atmosphere has done $-7.50 \times 10^{9} \mathrm{~J}$ of work on the satellite? Does the speed increase or decrease?

Jilin Wang
Jilin Wang
Boston University
08:34

Problem 71

Planets are not uniform inside. Normally, they are densest at the center and have decreasing density outward toward the surface. Model a spherically symmetric planet, with the same radius as the earth, as having a density that decreases linearly with distance from the center. Let the density be $15.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ at the center and $2.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ at the surface. What is the acceleration due to gravity at the surface of this planet?

David Gold
David Gold
Numerade Educator
05:21

Problem 72

One of the brightest comets of the 20 th century was Comet Hyakutake, which passed close to the sun in early 1996 . The orbital period of this comet is estimated to be about 30,000 years. Find the semimajor axis of this comet's orbit. Compare it to the average sun-Pluto distance and to the distance to Alpha Centauri, the nearest star to the sun, which is 4.3 light-years distant.

Linda Winkler
Linda Winkler
Numerade Educator
06:41

Problem 73

An object in the shape of a thin ring has radius $a$ and mass $M$. A uniform sphere with mass $m$ and radius $R$ is placed with its center at a distance $x$ to the right of the center of the ring, along a line through the center of the ring. and perpendicular to its plane (see Fig. E13.39). What is the gravitational force that the sphere exerts on the ring-shaped object? Show that your result reduces to the expected result when $x$ is much larger than $a$.

David Gold
David Gold
Numerade Educator
06:57

Problem 75

A shaft is drilled from the surface to the center of the earth (see Fig. 13.25 ). As in Example 13.10 (Section 13.6 ), make the unrealistic assumption that the density of the earth is uniform. With this approximation, the gravitational force on an object with mass $m$ that is inside the earth at a distance $r$ from the center, has magnitude $F_{g}=G m_{\mathrm{E}} m r / R_{\mathrm{E}}^{3}$ (as shown in Example 13.10 ) and points toward the center of the earth. (a) Derive an expression for the gravitational potential energy $U(r)$ of the object-carth system as a function of the object's distance from the center of the earth. Take the potential energy to be zero when the object is at the center of the earth. (b) If an object is released in the shaft at the earth's surface, what speed will it have when it reaches the center of the earth?

Linda Winkler
Linda Winkler
Numerade Educator
11:58

Problem 76

DATA For each of the cight planets Mercury to Neptune, the semi-major axis $a$ of their orbit and their orbital period $T$ are as follows:
$$
\begin{array}{lcc}
& \begin{array}{c}
\text { Semi-major } \\
\text { Axis }\left(10^{6} \mathrm{~km}\right)
\end{array} & \begin{array}{c}
\text { Orbital Period } \\
(\text { days })
\end{array} \\
\hline \text { Planet } & 57.9 & 88.0 \\
\text { Mercury } & 108.2 & 224.7 \\
\text { Venus } & 149.6 & 365.2 \\
\text { Earth } & 227.9 & 687.0 \\
\text { Mars } & 778.3 & 4331 \\
\text { Jupiter } & 1426.7 & 10,747 \\
\text { Saturn } & 2870.7 & 30,589 \\
\text { Uranus } & 4498.4 & 59,800
\end{array}
$$
(a) Explain why these values, when plotted as $T^{2}$ versus $a^{3},$ fall close to a straight line. Which of Kepler's laws is being tested? However, the values of $T^{2}$ and $a^{3}$ cover such a wide range that this plot is not a very practical way to graph the data. (Try it.) Instead, plot log( $T$ ) (with $T$ in seconds) versus $\log (a)$ (with $a$ in meters). Explain why the data should also fall close to a straight line in such a plot. (b) According to Kepler's laws, what should be the slope of your log( $T$ ) versus $\log (a)$ graph in part (a)? Does your graph have this slope? (c) Using $G=6.674 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$, calculate the mass of the sun from the
$y$ -intercept of your graph. How does your calculated value compare with the value given in Appendix F? (d) The only asteroid visible to the naked eye (and then only under ideal viewing conditions) is Vesta, which has an orbital period of 1325.4 days. What is the length of the semi-major axis of Vesta's orbit? Where does this place Vesta's orbit relative to the orbits of the eight major planets? Some scientists argue that Vesta should be called a minor planet rather than an asteroid.

Linda Winkler
Linda Winkler
Numerade Educator
14:04

Problem 77

DATA For a spherical planet with mass $M,$ volume $V,$ and radius $R,$ derive an expression for the acceleration due to gravity at the planet's surface, $g$, in terms of the average density of the planet, $\rho=M / V,$ and the planet's diameter, $D=2 R .$ The table gives the values of $D$ and $g$ for the eight major planets:
$$
\begin{array}{lrc}
\text { Planet } & D(\mathrm{~km}) & g\left(\mathrm{~m} / \mathrm{s}^{2}\right) \\
\hline \text { Mercury } & 4879 & 3.7 \\
\text { Venus } & 12,104 & 8.9 \\
\text { Earth } & 12,756 & 9.8 \\
\text { Mars } & 6792 & 3.7 \\
\text { Jupiter } & 142,984 & 23.1 \\
\text { Saturn } & 120,536 & 9.0 \\
\text { Uranus } & 51,118 & 8.7 \\
\text { Neptune } & 49.528 & 11.0
\end{array}
$$
(a) Treat the planets as spheres. Your equation for $g$ as a function of $\rho$ and $D$ shows that if the average density of the planets is constant, a graph of $g$ versus $D$ will be well represented by a straight line. Graph 8 as a function of $D$ for the eight major planets. What does the graph tell you about the variation in average density?
(b) Calculate the average density for each major planet. List the planets in order of decreasing density, and give the calculated average density of each. (c) The earth is not a uniform sphere and has greater density near its center. It is reasonable to assume this might be true for the other planets. Discuss the effect this has on your analysis. (d) If Saturn had the same average density as the earth, what would be the value of $g$ at Saturn's surface?

Brandy Heflin
Brandy Heflin
Numerade Educator
View

Problem 78

DATA For a planet in our solar system, assume that the axis of orbit is at the sun and is circular. Then the angular momentum about that axis due to the planet's orbital motion is $L=M v R$. (a) Derive an expression for $L$ in terms of the planet's mass $M,$ orbital radius $R,$ and period $T$ of the orbit. (b) Using Appendix F, calculate the magnitude of the orbital angular momentum for each of the eight major planets. (Assume a circular orbit.) Add these values to obtain the total angular momentum of the major planets due to their orbital motion. (All the major planets orbit in the same direction in close to the same plane, so adding the magnitudes to get the total is a reasonable approximation.) (c) The rotational period of the sun is 24.6 days. Using Appendix $\mathrm{F}$, calculate the angular momentum the sun has due to the rotation about its axis. (Assume that the sun is a uniform sphere.) (d) How does the rotational angular momentum of the sun compare with the total orbital angular momentum of the planets? How does the mass of the sun compare with the total mass of the planets? The fact that the sun has most of the mass of the solar system but only a small fraction of its total angular momentum must be accounted for in models of how the solar system formed. (e) The sun has a density that decreases with distance from its center. Does this mean that your calculation in part (c) overestimates or underestimates the rotational angular momentum of the sun? Or doesn't the nonuniform density have any effect?

JB
Joshua Bell
Numerade Educator
07:06

Problem 79

The most efficient way to send a spacecraft from the earth to another planet is to use a Hohmann transfer orbit (Fig. $\mathrm{P} 13.79$ ). If the orbits of the departure and destination planets are circular, the Hohmann transfer orbit is an elliptical orbit whose perihelion and aphelion are tangent to the orbits of the two planets. The rockets are fired briefly at the departure planet to put the spacecraft into the transfer orbit; the spacecraft then coasts until it reaches the destination planet. The rockets are then fired again to put the spacecraft into the same orbit about the sun as the destination planet.
(a) For a flight from earth to Mars, in what direction must the rockets be fired at the earth and at Mars: in the direction of motion or opposite the direction of motion? What about for a flight from Mars to the earth? (b) How long does a one-way trip from the earth to Mars take, between the firings of the rockets? (c) To reach Mars from the earth, the launch must be timed so that Mars will be at the right spot when the spacecraft reaches Mars's orbit around the sun. At launch, what must the angle between a sun-Mars line and a sun-earth line be? Use Appendix F.

David Gold
David Gold
Numerade Educator
14:08

Problem 80

Tidal Forces near a Black Hole. An astronaut inside a spacecraft, which protects her from harmful radiation, is orbiting a black hole at a distance of $120 \mathrm{~km}$ from its center. The black hole is 5.00 times the mass of the sun and has a Schwarzschild radius of $15.0 \mathrm{~km} .$ The astronaut is positioned inside the spaceship such that one of her $0.030 \mathrm{~kg}$ ears is $6.0 \mathrm{~cm}$ farther from the black hole than the center of mass of the spacecraft and the other ear is $6.0 \mathrm{~cm}$ closer. (a) What is the tension between her ears? Would the astronaut find it difficult to keep from being torn apart by the gravitational forces? (Since her whole body orbits with the same angular velocity, one ear is moving too slowly for the radius of its orbit and the other is moving too fast. Hence her head must exert forces on her cars to keep them in their orbits.) (b) Is the center of gravity of her head at the same point as the center of mass? Explain.

Linda Winkler
Linda Winkler
Numerade Educator
12:00

Problem 81

Mass $M$ is distributed uniformly over a disk of radius $a .$ Find the gravitational force (magnitude and direction) between this disk-shaped mass and a particle with mass $m$ located a distance $x$ above the center of the disk (Fig. $\mathbf{P} 13.81$ ). Does your result reduce to the correct expression as $x$ becomes very large? (Hint: Divide the disk into infinitesimally thin con-centric rings, use the expression derived in Exercise 13.39 for the gravitational force due to cach ring, and integrate to find the total force.)

Linda Winkler
Linda Winkler
Numerade Educator
01:00

Problem 82

Based on these data, what is the most likely composition of this planet? (a) Mostly iron; (b) iron and rock; (c) iron and rock with some lighter elements; (d) hydrogen and helium gases.

Jilin Wang
Jilin Wang
Boston University
02:09

Problem 83

How many times the acceleration due to gravity $g$ near the earth's surface is the acceleration due to gravity near the surface of this exoplanet? (a) About $0.29 \mathrm{~g} ;$ (b) about $0.65 \mathrm{~g} ;$ (c) about $1.5 \mathrm{~g} ;$ (d) about $7.9 g$

David Gold
David Gold
Numerade Educator
04:12

Problem 84

Observations of this planet over time show that it is in a nearly circular orbit around its star and completes one orbit in only $9.5$ days. How many times the orbital radius $r$ of the earth around our sun is this exoplanet's orbital radius around its sun? Assume that the earth is also
in a nearly circular orbit. (a) $0.026 r$, (b) $0.078 r$, (c) $0.70 r$; (d) $2.3 r$.

Linda Winkler
Linda Winkler
Numerade Educator