Fundamentals of Physics

David Halliday, Robert Resnick

Chapter 13

Gravitation - all with Video Answers

Educators

Chapter Questions

Problem 1

A mass $M$ is split into two parts, $m$ and $M-m,$ which are then separated by a certain distance. What ratio m/M maximizes the magnitude of the gravitational force between the parts?

Penny Riley

Penny Riley

Numerade Educator

Problem 2

Some people believe that the Moon controls their activities. If the Moon moves from being directly on the opposite side of Earth from you to being directly overhead, by what percent does (a) the Moon's gravitational pull on you increase and (b) your weight (as measured on a scale) decrease? Assume that the Earth-Moon (center-to-center) distance is $3.82 \times 10^{8} \mathrm{~m}$ and Earth's radius is $6.37 \times 10^{6} \mathrm{~m} .$

Averell Hause

Carnegie Mellon University

Problem 3

What must the separation be between a $5.2 \mathrm{~kg}$ particle and a $2.4 \mathrm{~kg}$ particle for their gravitational attraction to have a magnitude of $2.3 \times 10^{-12} \mathrm{~N} ?$

Bettina Hanlon

Numerade Educator

Problem 4

The Sun and Earth each exert a gravitational force on the age Sun-Moon distance is equal to the Sun-Earth distance.)

Keshav Singh

Numerade Educator

Problem 5

Left over from the big-bang beginning of the universe, tiny black holes might still wander through the universe. If one with a mass of $1 \times 10^{11} \mathrm{~kg}$ (and a radius of only $1 \times 10^{-16} \mathrm{~m}$ ) reached Earth, at what distance from your head would its gravitational pull on you match that of Earth's?

Keshav Singh

Numerade Educator

Problem 6

A square of edge length $20.0 \mathrm{~cm}$ is formed by four spheres of masses $m_{1}=5.00 \mathrm{~g}, m_{2}=3.00 \mathrm{~g}, \quad m_{3}=1.00 \mathrm{~g},$ and
$m_{4}=5.00 \mathrm{~g} .$ In unit-vector notation, what is the net gravitational force from them on a central sphere with mass $m_{5}=2.50 \mathrm{~g} ?$

Keshav Singh

Numerade Educator

Problem 7

Two point particles are fixed on an $x$ axis separated by distance $d .$ Particle $A$ has mass $m_{A}$ and particle $B$ has mass $3.00 m_{A}$. A third particle $C,$ of mass $75.0 m_{A},$ is to be placed on the $x$ axis and near particles $A$ and $B$. In terms of distance $d,$ at what $x$ coordinate should $C$ be placed so that the net gravitational force on particle $A$ from particles $B$ and $C$ is zero?

Keshav Singh

Numerade Educator

Problem 8

Three $5.00 \mathrm{~kg}$ spheres are located at distances $d_{1}=0.300 \mathrm{~m}$ and $d_{2}=0.400 \mathrm{~m} .$ What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the net gravitational force on sphere $B$ due to spheres $A$ and $\bar{C} ?$

Keshav Singh

Numerade Educator

Problem 9

We want to position a space probe along a line that extends directly toward the Sun in order to monitor solar flares. How far from Earth's center is the point on the line where the Sun's gravitational pull on the probe balances Earth's pull?

Bettina Hanlon

Numerade Educator

Problem 10

Three point particles are fixed in place in an $x y$ plane. Particle $A$ has mass $m_{A},$ particle $B$ has mass $2.00 m_{A}$, and particle $C$ has mass $3.00 m_{A}$. A fourth particle $D,$ with mass $4.00 m_{A},$ is to be placed near the other three particles. In terms of distance $d$, at what (a) $x$ coordinate and (b) $y$ coordinate should particle $D$ be placed so that the net gravitational force on particle $A$ from particles $B, C,$ and $D$ is zero?

Keshav Singh

Numerade Educator

Problem 11

Two spheres of mass $m$ and a third sphere of mass $M$ form an equilateral triangle, and a fourth sphere of mass $m_{4}$ is at the center of the triangle. The net gravitational force on that central sphere from the three other spheres is zero. (a) What is $M$ in terms of $m ?$ (b) If we double the value of $m_{4},$ what then is the magnitude of the net gravitational force on the central sphere?

Keshav Singh

Numerade Educator

Problem 12

A particle $A$ is fixed in place at $x=-0.20 \mathrm{~m}$ on the $x$ axis and particle $B,$ with a mass of $1.0 \mathrm{~kg},$ is fixed in place at the origin. Particle $C$ (not shown) can be moved along the $x$ axis, between particle $B$ and $x=\infty .$ Figure $13-37 b$ shows the $x$ component $F_{\text {net }, x}$ of the net gravitational force on particle $B$ due to particles $A$ and $C,$ as a function of position $x$ of particle $C$. The plot actually extends to the right, approaching an asymptote of $-4.17 \times 10^{-10} \mathrm{~N}$ as $x \rightarrow \infty .$ What are the masses of (a) particle $A$ and (b) particle $C ?$

Keshav Singh

Numerade Educator

Problem 13

Shows a spherical hollow inside a lead sphere f radius $R=4.00 \mathrm{~cm} ;$ the surface of the hollow passes through the center of the sphere and "touches" the right side of the sphere. The mass of the sphere before hollowing was $M=2.95 \mathrm{~kg} .$ With what gravitional force does the hollowed-out lead sphere attract a small sphere of mass $m=0.431 \mathrm{~kg}$ that lies at a distance $d=9.00 \mathrm{~cm}$ from the center of the lead sphere, on the straight line connecting the centers of the spheres and of the hollow?

Keshav Singh

Numerade Educator

Problem 14

Three point particles are fixed in position in an $x y$ plane. Two of them, particle $A$ of mass $6.00 \mathrm{~g}$ and particle $B$ of mass $12.0 \mathrm{~g},$ are shown in Fig. $13-39$, with a separation of $d_{A B}=0.500 \mathrm{~m}$ at angle $\theta=30^{\circ} .$ Particle $C,$ with mass $8.00 \mathrm{~g},$ is not shown. The net gravitational force acting on particle $A$ due to particles $B$ and $C$ is $2.77 \times 10^{-14} \mathrm{~N}$ at an angle of $-163.8^{\circ}$ from the positive direction of the $x$ axis. What are
(a) the $x$ coordinate and (b) the $y$ coordinate of particle $C ?$

Keshav Singh

Numerade Educator

Problem 15

Three point particles are fixed in place in an $x y z$ coordinate system. Particle $A,$ at the origin, has mass $m_{A} .$ Particle $B,$ at $x y z$ coordinates $(2.00 d, 1.00 d, 2.00 d),$ has mass $2.00 m_{A},$ and particle $C,$ at coordinates $(-1.00 d, 2.00 d,-3.00 d),$ has mass $3.00 m_{A}$. A fourth particle $D,$ with mass $4.00 m_{A},$ is to be placed near the other particles. In terms of distance $d,$ at what (a) $x,(\mathrm{~b}) y,$ and
(c) $z$ coordinate should $D$ be placed so that the net gravitational force on $A$ from $B, C,$ and $D$ is zero?

Paul A.

California State Polytechnic University, Pomona

Problem 16

A Particle of mass $m_{1}=0.67 \mathrm{~kg}$ is a distance $d=23 \mathrm{~cm}$ from one end of a uniform rod with length $L=$ $3.0 \mathrm{~m}$ and mass $M=5.0 \mathrm{~kg}$. What is the magnitude of the gravitational force $F$ on the particle from the rod?

Keshav Singh

Numerade Educator

Problem 17

(a) What will an object weigh on the Moon's surface if it weighs $100 \mathrm{~N}$ on Earth's surface? (b) How many Earth radii must this same object be from the center of Earth if it is to weigh the same as it does on the Moon?

Alia Hamdan

Numerade Educator

Problem 18

A large mountain can slightly affect the direction of "down" as determined by a plumb line. Assume that we can model a mountain as a sphere of radius $R=2.00 \mathrm{~km}$ and density (mass per unit volume) $2.6 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. Assume also that we hang a $0.50 \mathrm{~m}$ plumb line at a distance of $3 R$ from the sphere's center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere?

Keshav Singh

Numerade Educator

Problem 19

At what altitude above Earth's surface would the gravitational acceleration be $4.9 \mathrm{~m} / \mathrm{s}^{2} ?$

Bettina Hanlon

Numerade Educator

Problem 20

In 1956, Frank Lloyd Wright proposed the construction of a mile-high building in Chicago. Suppose the building had been constructed. Ignoring Earth's rotation, find the change in your weight if you were to ride an elevator from the street level, where you weigh $600 \mathrm{~N},$ to the top of the building.

Averell Hause

Carnegie Mellon University

Problem 21

Certain neutron stars (extremely dense stars) are believed to be rotating at about 1 rev $/ \mathrm{s}$. If such a star has a radius of $20 \mathrm{~km},$ what must be its minimum mass so that material on its surface remains in place during the rapid rotation?

Alia Hamdan

Numerade Educator

Problem 22

The radius $R_{h}$ and mass $M_{h}$ of a black hole are related by $R_{h}=2 G M_{h} / c^{2},$ where $c$ is the speed of light. Assume that the gravitational acceleration $a_{g}$ of an object at a distance $r_{o}=1.001 R_{h}$ from the center of a black hole is given by Eq. $13-11$ (it is, for large black holes). (a) In terms of $M_{h},$ find $a_{g}$ at $r_{o} .$ (b) Does $a_{g}$ at $r_{o}$ increase or decrease as $M_{h}$ increases? (c) What is $a_{g}$ at $r_{o}$ for a very large black hole whose mass is $1.55 \times 10^{12}$ times the solar mass of $1.99 \times 10^{30} \mathrm{~kg} ?$
(d) If an astronaut of height $1.70 \mathrm{~m}$ is at $r_{o}$ with her feet down, what is the difference in gravitational acceleration between her head and feet? (e) Is the tendency to stretch the astronaut severe?

Keshav Singh

Numerade Educator

Problem 23

One model for a certain planet has a core of radius $R$ and mass $M$ surrounded by an outer shell of inner radius $R,$ outer radius $2 R$ and mass $4 M$. If $M=4.1 \times 10^{24} \mathrm{~kg}$ and $R=6.0 \times 10^{6} \mathrm{~m},$ what is the gravitational acceleration of a particle at points (a) $R$ and (b) $3 R$ from the center of the planet?

Alia Hamdan

Numerade Educator

Problem 24

Two concentric spherical shells with uniformly distributed masses $M_{1}$ and $M_{2}$ are situated as shown in Fig. $13-41 .$ Find the magnitude of the net gravitational force on a particle of mass $m,$ due to the shells, when the particle is located at radial distance (a) $a,(\mathrm{~b}) b,$ and $(\mathrm{c}) c .$

Averell Hause

Carnegie Mellon University

Problem 25

A solid sphere has a uniformly distributed mass of $1.0 \times 10^{4} \mathrm{~kg}$ and a radius of $1.0 \mathrm{~m} .$ What is the magnitude of the gravitational force due to the sphere on a particle of mass $m$ when the particle is located at a distance of (a) $1.5 \mathrm{~m}$ and (b) $0.50 \mathrm{~m}$ from the center of the sphere? (c) Write a general expression for the magnitude of the gravitational force on the particle at a distance $r \leq 1.0 \mathrm{~m}$ from the center of the sphere.

Alia Hamdan

Numerade Educator

Problem 26

A uniform solid sphere of radius $R$ produces a gravitational acceleration of $a_{g}$ on its surface. At what distance from the sphere's center are there points (a) inside and (b) outside the sphere where the gravitational acceleration is $a_{g} / 3 ?$

Averell Hause

Carnegie Mellon University

Problem 27

A cross section through the interior of Earth. Rather than being uniform throughout, Earth is divided into three zones: an outer crust, a mantle, and an inner core. The dimensions of these zones and the masses contained within them are shown on the figure. Earth has a total mass of $5.98 \times 10^{24} \mathrm{~kg}$ and a radius of $6370 \mathrm{~km} .$ Ignore rotation and assume that Earth is spherical. (a) Calculate $a_{g}$ at the surface. (b) Suppose that a bore hole (the Mohole) is driven to the crust - mantle interface at a depth of $25.0 \mathrm{~km} ;$ what would be the value of $a_{g}$ at the bottom of the hole? (c) Suppose that Earth were a uniform sphere with the same total mass and size. What would be the value of $a_{g}$ at a depth of $25.0 \mathrm{~km} ?$ (Precise measurements of $a_{g}$ are sensitive probes of the interior structure of Earth, although results can be clouded by local variations in mass distribution.)

Keshav Singh

Numerade Educator

Problem 28

Assume a planet is a uniform sphere of radius $R$ that (somehow) has a narrow radial tunnel through its center (Fig. $13-7) .$ Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let $F_{R}$ be the magnitude of the gravitational force on the apple when it is located at the planet's surface. How far from the surface is there a point where the magnitude is $\frac{1}{2} F_{R}$ if we move the apple
(a) away from the planet and
(b) into the tunnel?

Averell Hause

Carnegie Mellon University

Problem 29

Gives the potential energy function $U(r)$ of a projectile, plotted outward from the surface of a planet of radius $R_{s} .$ What least kinetic energy is required of a projectile launched at the surface if the projectile is to "escape" the planet?

Keshav Singh

Numerade Educator

Problem 30

In Problem $1,$ what ratio $m / M$ gives the least gravitational potential energy for the system?

Averell Hause

Carnegie Mellon University

Problem 31

The mean diameters of Mars and Earth are $6.9 \times 10^{3} \mathrm{~km}$ and $1.3 \times 10^{4} \mathrm{~km},$ respectively. The mass of Mars is 0.11 times Earth's mass. (a) What is the ratio of the mean density (mass per unit volume) of Mars to that of Earth? (b) What is the value of the gravitational acceleration on Mars?
(c) What is the escape speed on Mars?

Averell Hause

Carnegie Mellon University

Problem 32

(a) What is the gravitational potential energy of the twoparticle system in Problem 3 ? If you triple the separation between the particles, how much work is done (b) by the gravitational force between the particles and (c) by you?

Averell Hause

Carnegie Mellon University

Problem 33

What multiple of the energy needed to escape from Earth gives the energy needed to escape from (a) the Moon and
(b) Jupiter?

Alia Hamdan

Numerade Educator

Problem 34

Gives the potential energy function $U(r)$ of a projectile, plotted outward from the surface of a planet of radius $R_{s}$. If the projectile is launched radially outward from the surface with a mechanical energy of $-2.0 \times 10^{9} \mathrm{~J},$ what are (a) its kinetic energy at radius $r=1.25 R_{s}$ and (b) its turning point (see Module $8-3$ ) in terms of $R_{s} ?$

Keshav Singh

Numerade Educator

Problem 35

Shows four particles, each of mass $20.0 \mathrm{~g}$, that form a square with an edge length of $d=0.600 \mathrm{~m}$. If $d$ is reduced to $0.200 \mathrm{~m},$ what is the change in the gravitational potential energy of the four-particle system?

Keshav Singh

Numerade Educator

Problem 36

Zero, a hypothetical planet, has a mass of $5.0 \times 10^{23} \mathrm{~kg},$ a radius of $3.0 \times 10^{6} \mathrm{~m}$ and no atmosphere. A $10 \mathrm{~kg}$ space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial energy of $5.0 \times 10^{7} \mathrm{~J},$ what will be its kinetic energy when it is $4.0 \times 10^{6} \mathrm{~m}$ from the center of Zero?
(b) If the probe is to achieve a maximum distance of $8.0 \times 10^{6} \mathrm{~m}$ from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

Keshav Singh

Numerade Educator

Problem 37

The three spheres in Fig. $13-45,$ with masses $m_{A}=80 \mathrm{~g}$ $m_{B}=10 \mathrm{~g},$ and $m_{C}=20 \mathrm{~g},$ have their centers on a common line, with $L=12 \mathrm{~cm}$ and $d=4.0 \mathrm{~cm} .$ You move sphere $B$ along the line until its center-to-center separation from $C$ is $d=4.0 \mathrm{~cm} .$ How much work is done on sphere $B$ (a) by you and (b) by the net gravitational force on $B$ due to spheres $A$ and $C ?$

Alia Hamdan

Numerade Educator

Problem 38

In deep space, sphere $A$ of mass $20 \mathrm{~kg}$ is located at the origin of an $x$ axis and sphere $B$ of mass $10 \mathrm{~kg}$ is located on the axis at $x=0.80 \mathrm{~m} .$ Sphere $B$ is released from rest while sphere $A$ is held at the origin. (a) What is the gravitational potential energy of the two-sphere system just as $B$ is released? (b) What is the kinetic energy of $B$ when it has moved $0.20 \mathrm{~m}$ toward $A ?$

Keshav Singh

Numerade Educator

Problem 39

(a) What is the escape speed on a spherical asteroid whose radius is $500 \mathrm{~km}$ and whose gravitational acceleration at the surface is $3.0 \mathrm{~m} / \mathrm{s}^{2} ?$ (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of $1000 \mathrm{~m} / \mathrm{s} ?$
(c) With what speed will an object hit the asteroid if it is dropped from $1000 \mathrm{~km}$ above the surface?B

Linda Winkler

Numerade Educator

Problem 40

A projectile is shot directly away from Earth's surface. Neglect the rotation of Earth. What multiple of Earth's radius $R_{E}$ gives the radial distance a projectile reaches if (a) its initial speed is 0.500 of the escape speed from Earth and (b) its initial kinetic energy is 0.500 of the kinetic energy required to escape Earth?
(c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth?

Averell Hause

Carnegie Mellon University

Problem 41

Two neutron stars are separated by a distance of $1.0 \times 10^{10} \mathrm{~m} .$ They each have a mass of $1.0 \times 10^{30} \mathrm{~kg}$ and a radius of $1.0 \times 10^{5} \mathrm{~m} .$ They are initially at rest with respect to each other. As measured from that rest frame, how fast are they moving when
(a) their separation has decreased to one-half its initial value and
(b) they are about to collide?

Keshav Singh

Numerade Educator

Problem 42

Shows a particle $A$ that can be moved along a $y$ axis from an infinite distance to the origin. That origin lies at the midpoint between particles $B$ and $C,$ which have identical masses, and the $y$ axis is a perpendicular bisector between them. Distance $D$ is $0.3057 \mathrm{~m} .$ Figure $13-46 b$ shows the potential energy $U$ of the three-particle system as a function of the position of particle $A$ along the $y$ axis. The curve actually extends rightward and approaches an asymptote of $-2.7 \times 10^{-11} \mathrm{~J}$ as $y \rightarrow \infty .$ What are the masses of (a) particles $B$ and $C$ and (b) particle $A ?$

Keshav Singh

Numerade Educator

Problem 43

(a) What linear speed must an Earth satellite have to be in a circular orbit at an altitude of $160 \mathrm{~km}$ above Earth's surface?
(b) What is the period of revolution?

Alia Hamdan

Numerade Educator

Problem 44

A satellite is put in a circular orbit about Earth with a radius equal to one-half the radius of the Moon's orbit. What is its period of revolution in lunar months? (A lunar month is the period of revolution of the Moon.)

Averell Hause

Carnegie Mellon University

Problem 45

The Martian satellite Phobos travels in an approximately circular orbit of radius $9.4 \times 10^{6} \mathrm{~m}$ with a period of $7 \mathrm{~h} 39 \mathrm{~min}$. Calculate the mass of Mars from this information.

Alia Hamdan

Numerade Educator

Problem 46

The first known collision between space debris and a functioning satellite occurred in 1996 : At an altitude of $700 \mathrm{~km}$, a yearold French spy satellite was hit by a piece of an Ariane rocket. A stabilizing boom on the satellite was demolished, and the satellite was sent spinning out of control. Just before the collision and in kilometers per hour, what was the speed of the rocket piece relative to the satellite if both were in circular orbits and the collision was (a) head-on and (b) along perpendicular paths?

Bettina Hanlon

Numerade Educator

Problem 47

The Sun, which is $2.2 \times 10^{20} \mathrm{~m}$ from the center of the Milky Way galaxy, revolves around that center once every $2.5 \times 10^{8}$ years. Assuming each star in the Galaxy has a mass equal to the Sun's mass of $2.0 \times 10^{30} \mathrm{~kg},$ the stars are distributed uniformly in a sphere about the galactic center, and the Sun is at the edge of that sphere, estimate the number of stars in the Galaxy.

Bettina Hanlon

Numerade Educator

Problem 48

The mean distance of Mars from the Sun is 1.52 times that of Earth from the Sun. From Kepler's law of periods, calculate the number of years required for Mars to make one revolution around the Sun; compare your answer with the value given in Appendix C.

Averell Hause

Carnegie Mellon University

Problem 49

A comet that was seen in April 574 by Chinese astronomers on a day known by them as the Woo Woo day was spotted again in May $1994 .$ Assume the time between observations is the period of the Woo Woo day comet and its eccentricity is $0.9932 .$ What are
(a) the semimajor axis of the comet's orbit and (b) its greatest distance from the Sun in terms of the mean orbital radius $R_{P}$ of Pluto?

Alia Hamdan

Numerade Educator

Problem 50

An orbiting satellite stays over a certain spot on the equator of (rotating) Earth. What is the altitude of the orbit (called a geosynchronous orbit)?

Prashant Bana

Numerade Educator

Problem 51

A satellite, moving in an elliptical orbit, is $360 \mathrm{~km}$ above Earth's surface at its farthest point and $180 \mathrm{~km}$ above at its closest point. Calculate (a) the semimajor axis and (b) the eccentricity of the orbit.

Averell Hause

Carnegie Mellon University

Problem 52

The Sun's center is at one focus of Earth's orbit. How far from this focus is the other focus, (a) in meters and (b) in terms of the solar radius, $6.96 \times 10^{8} \mathrm{~m}$ ? The eccentricity is 0.0167 , and the semimajor axis is $1.50 \times 10^{11} \mathrm{~m} .$

Bettina Hanlon

Numerade Educator

Problem 53

A $20 \mathrm{~kg}$ satellite has a circular orbit with a period of $2.4 \mathrm{~h}$ and a radius of $8.0 \times 10^{6} \mathrm{~m}$ around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is $8.0 \mathrm{~m} / \mathrm{s}^{2},$ what is the radius of the planet?

Alia Hamdan

Numerade Educator

Problem 54

Observations of the light from a certain star indicate that it is part of a binary (two-star) system. This visible star has orbital speed $v=270 \mathrm{~km} / \mathrm{s}$ orbital period $T=1.70$ days, and approximate mass $m_{1}=6 M_{s},$ where $M_{s}$ is the Sun's mass, $1.99 \times 10^{30} \mathrm{~kg}$. Assume that the visible star and its companion star, which is dark and unseen, are both in circular orbits (Fig. $13-47$ ). What integer multiple of $M_{s}$ gives the approximate $\operatorname{mass} m_{2}$
of the dark star?

Averell Hause

Carnegie Mellon University

Problem 55

In $1610,$ Galileo used his telescope to discover four moons around Jupiter, with these mean orbital radii $a$ and periods $T$ : (a) Plot log $a(y$ axis) against log $T(x$ axis) and show that you get a straight line. (b) Measure the slope of the line and compare it with the value that you expect from Kepler's third law. (c) Find the mass of Jupiter from the intercept of this line with the $y$ axis.

Alia Hamdan

Numerade Educator

Problem 56

In 1993 the spacecraft Galileo sent an image (Fig. $13-48)$ of asteroid 243 Ida and a tiny orbiting moon (now known as Dactyl), the first confirmed example of an asteroid-moon system. In the image, the moon, which is $1.5 \mathrm{~km}$ wide, is $100 \mathrm{~km}$ from the center of the asteroid, which is $55 \mathrm{~km}$ long. Assume the moon's orbit is circular with a period of $27 \mathrm{~h}$. (a) What is the mass of the asteroid?
(b) The volume of the asteroid, measured from the Galileo images, is $14100 \mathrm{~km}^{3}$. What is the density (mass per unit volume) of the asteroid?

Averell Hause

Carnegie Mellon University

Problem 57

In a certain binary-star system, each star has the same mass as our Sun, and they revolve about their center of mass. The distance between them is the same as the distance between Earth and the Sun. What is their period of revolution in years?

Alia Hamdan

Numerade Educator

Problem 58

The presence of an unseen planet orbiting a distant star can sometimes be inferred from the motion of the star as we see it. As the star and planet orbit the center of mass of the starplanet system, the star moves toward and away from us with what is called the line of sight velocity, a motion that can be detected. Figure $13-49$ shows a graph of the line of sight velocity versus time for the star 14 Herculis. The star's mass is believed to be 0.90 of the mass of our Sun. Assume that only one planet orbits the star and that our view is along the plane of the orbit. Then approximate (a) the planet's mass in terms of Jupiter's mass $m_{J}$ and (b) the planet's orbital radius in terms of Earth's orbital radius $r_{E}$

Keshav Singh

Numerade Educator

Problem 59

Three identical stars of mass $M$ form an equilateral triangle that rotates around the triangle's center as the stars move in a common circle about that center. The triangle has edge length $L .$ What is the speed of the stars?

Alia Hamdan

Numerade Educator

Problem 60

Two satellites, $A$ and $B,$ both of mass $m=125 \mathrm{~kg},$ move in the same circular orbit of radius $r=7.87 \times 10^{6} \mathrm{~m}$ around Earth but in opposite senses of rotation and therefore on a collision course. (a) Find the total mechanical energy $E_{A}+E_{B}$ of the $t w o$ satellites + Earth system before the collision. (b) If the collision is completely inelastic so that the wreckage remains as one piece of tangled material (mass $=2 \mathrm{~m}$ ), find the total mechanical energy immediately after the collision. (c) Just after the collision, is the wreckage falling directly toward Earth's center or orbiting around Earth?

Keshav Singh

Numerade Educator

Problem 61

(a) At what height above Earth's surface is the energy required to lift a satellite to that height equal to the kinetic energy required for the satellite to be in orbit at that height? (b) For greater heights, which is greater, the energy for lifting or the kinetic energy for orbiting?

Alia Hamdan

Numerade Educator

Problem 62

Two Earth satellites, $A$ and $B,$ each of mass $m,$ are to be launched into circular orbits about Earth's center. Satellite $A$ is to orbit at an altitude of $6370 \mathrm{~km} .$ Satellite $B$ is to orbit at an altitude of $19110 \mathrm{~km} .$ The radius of Earth $R_{E}$ is $6370 \mathrm{~km} .$ (a) $\mathrm{What}$ is the ratio of the potential energy of satellite $B$ to that of satellite $A,$ in orbit? (b) What is the ratio of the kinetic energy of satellite $B$ to that of satellite $A,$ in orbit? (c) Which satellite has the greater total energy if each has a mass of $14.6 \mathrm{~kg}$ ? (d) By how much?

Averell Hause

Carnegie Mellon University

Problem 63

An asteroid, whose mass is $2.0 \times 10^{-4}$ times the mass of Earth, revolves in a circular orbit around the Sun at a distance that is twice Earth's distance from the Sun. (a) Calculate the period of revolution of the asteroid in years. (b) What is the ratio of the kinetic energy of the asteroid to the kinetic energy of Earth?

Alia Hamdan

Numerade Educator

Problem 64

A satellite orbits a planet of unknown mass in a circle of radius $2.0 \times 10^{7} \mathrm{~m} .$ The magnitude of the gravitational force on the satellite from the planet is $F=80 \mathrm{~N}$. (a) What is the kinetic energy of the satellite in this orbit? (b) What would $F$ be if the orbit radius were increased to $3.0 \times 10^{7} \mathrm{~m} ?$

Bettina Hanlon

Numerade Educator

Problem 65

A satellite is in a circular Earth orbit of radius $r .$ The area $A$ enclosed by the orbit depends on $r^{2}$ because $A=\pi r^{2} .$ Determine how the following properties of the satellite depend on $r$ : (a) period,
(b) kinetic energy, (c) angular momentum, and (d) speed.

Bettina Hanlon

Numerade Educator

Problem 66

One way to attack a satellite in Earth orbit is to launch a swarm of pellets in the same orbit as the satellite but in the opposite direction. Suppose a satellite in a circular orbit $500 \mathrm{~km}$ above Earth's surface collides with a pellet having mass $4.0 \mathrm{~g}$. (a) What is the kinetic energy of the pellet in the reference frame of the satellite just before the collision? (b) What is the ratio of this kinetic energy to the kinetic energy of a $4.0 \mathrm{~g}$ bullet from a modern army rifle with a muzzle speed of $950 \mathrm{~m} / \mathrm{s} ?$

Averell Hause

Carnegie Mellon University

Problem 67

What are (a) the speed and (b) the period of a $220 \mathrm{~kg}$ satellite in an approximately circular orbit $640 \mathrm{~km}$ above the surface of Earth? Suppose the satellite loses mechanical energy at the average rate of $1.4 \times 10^{5} \mathrm{~J}$ per orbital revolution. Adopting the reasonable approximation that the satellite's orbit becomes a "circle of slowly diminishing radius," determine the satellite's (c) altitude, (d) speed, and (e) period at the end of its 1500 th revolution. (f) What is the magnitude of the average retarding force on the satellite? Is angular momentum around Earth's center conserved for (g) the satellite and (h) the satellite-Earth system (assuming that system is isolated)?

Alia Hamdan

Numerade Educator

Problem 68

Two small spaceships, each with mass $m=2000 \mathrm{~kg}$, are in the circular Earth orbit of Fig. $13-51,$ at an altitude $h$ of $400 \mathrm{~km}$. Igor, the commander of one of the ships, arrives at any fixed point in the orbit $90 \mathrm{~s}$ ahead of Picard, the commander of the other ship. What are the (a) period $T_{0}$ and (b) speed $v_{0}$ of the ships? At point $P$ in Fig. $13-51$, Picard fires an instantaneous burst in the forward direction, reducing his ship's speed by $1.00 \% .$ After this burst, he follows the elliptical orbit shown dashed in the figure. What are the (c) kinetic energy and
(d) potential energy of his ship immediately after the burst? In Picard's new elliptical orbit, what are (e) the total energy $E,$ (f) the semimajor axis $a$, and $(\mathrm{g})$ the orbital period $T ?$
(h) How much earlier than Igor will Picard return to $P ?$

Averell Hause

Carnegie Mellon University

Problem 69

The scale on which the $60 \mathrm{~kg}$ physicist stands reads $220 \mathrm{~N}$. How long will the cantaloupe take to reach the floor if the physicist drops it (from rest relative to himself) at a height of $2.1 \mathrm{~m}$ above the floor?

Keshav Singh

Numerade Educator

Problem 70

The radius $R_{h}$ of a black hole is the radius of a mathematical sphere, called the event horizon, that is centered on the black hole. Information from events inside the event horizon cannot reach the outside world. According to Einstein's general theory of relativity, $R_{h}=2 G M / c^{2},$ where $M$ is the mass of the black hole and $c$ is the speed of light. Suppose that you wish to study a black hole near it, at a radial distance of $50 R_{h}$. However, you do not want the difference in gravitational acceleration between your feet and your head to exceed $10 \mathrm{~m} / \mathrm{s}^{2}$ when you are feet down (or head down) toward the black hole. (a) As a multiple of our Sun's mass $M_{S},$ approximately what is the limit to the mass of the black hole you can tolerate at the given radial distance? (You need to estimate your height.) (b) Is the limit an upper limit (you can tolerate smaller masses) or a lower limit (you can tolerate larger masses)?

Averell Hause

Carnegie Mellon University

Problem 71

Several planets (Jupiter, Saturn, Uranus) are encircled by rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ring-like structures. Consider a homogeneous thin ring of mass $M$ and outer radius $R$ (Fig. 13-52). (a) What gravitational attraction does it exert on a particle of mass $m$ located on the ring's central axis a distance $x$ from the ring center? (b) Suppose the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring?

Keshav Singh

Numerade Educator

Problem 72

A typical neutron star may have a mass equal to that of the Sun but a radius of only $10 \mathrm{~km}$. (a) What is the gravitational acceleration at the surface of such a star?
(b) How fast would an object be moving if it fell from rest through a distance of $1.0 \mathrm{~m}$ on such a star? (Assume the star does not rotate.)

Chasen Shaw

Numerade Educator

Problem 73

A graph of the kinetic energy $K$ of an asteroid versus its distance $r$ from Earth's center, as the asteroid falls directly in toward that center. (a) What is the (approximate) mass of the asteroid? (b) What is its speed at $r=1.945 \times 10^{7} \mathrm{~m} ?$

Keshav Singh

Numerade Educator

Problem 74

The mysterious visitor that appears in the enchanting story. The Little Prince was said to come from a planet that "was scarcely any larger than a house!" Assume that the mass per unit volume of the planet is about that of Earth and that the planet does not appreciably spin. Approximate (a) the free-fall acceleration on the planet's surface and (b) the escape speed from the planet.

Averell Hause

Carnegie Mellon University

Problem 75

The masses and coordinates of three spheres are as follows:
$20 \mathrm{~kg}, x=0.50 \mathrm{~m}, y=1.0 \mathrm{~m} ; 40 \mathrm{~kg}, x=-1.0 \mathrm{~m}, \quad y=-1.0 \mathrm{~m} ; 60 \mathrm{~kg}$
$x=0 \mathrm{~m}, y=-0.50 \mathrm{~m} .$ What is the magnitude of the gravitational force on a $20 \mathrm{~kg}$ sphere located at the origin due to these three spheres?

Keshav Singh

Numerade Educator

Problem 76

A very early, simple satellite consisted of an inflated spherical aluminum balloon $30 \mathrm{~m}$ in diameter and of mass $20 \mathrm{~kg} .$ Suppose a meteor having a mass of $7.0 \mathrm{~kg}$ passes within $3.0 \mathrm{~m}$ of the surface of the satellite. What is the magnitude of the gravitational force on the meteor from the satellite at the closest approach?

Averell Hause

Carnegie Mellon University

Problem 77

Four uniform spheres, with masses $m_{A}=40 \mathrm{~kg}, m_{B}=35 \mathrm{~kg}$, $m_{C}=200 \mathrm{~kg},$ and $m_{D}=50 \mathrm{~kg},$ have $(x, y)$ coordinates of $(0,50 \mathrm{~cm})$
$(0,0),(-80 \mathrm{~cm}, 0),$ and $(40 \mathrm{~cm}, 0),$ respectively. In unit-vector notation, what is the net gravitational force on sphere $B$ due to the other spheres?

Alia Hamdan

Numerade Educator

Problem 78

(a) In Problem 77, remove sphere $A$ and calculate the gravitational potential energy of the remaining three-particle system.
(b) If $A$ is then put back in place, is the potential energy of the four-particle system more or less than that of the system in (a)?
(c) In (a), is the work done by you to remove $A$ positive or negative? (d) In (b), is the work done by you to replace $A$ positive or negative?

Keshav Singh

Numerade Educator

Problem 79

A certain triple-star system consists of two stars, each of mass $m,$ revolving in the same circular orbit of radius $r$ around a central star of mass $M$ (Fig. $13-54) .$ The two orbiting stars are always at opposite ends of a diameter of the orbit. Derive an expression for the period of revolution of the stars.

Alia Hamdan

Numerade Educator

Problem 80

The fastest possible rate of rotation of a planet is that for which the gravitational force on material at the equator just barely provides the centripetal force needed for the rotation. (Why?) (a) Show that the corresponding shortest period of rotation is $$T=\sqrt{\frac{3 \pi}{G \rho}}$$ where $\rho$ is the uniform density (mass per unit volume) of the spherical planet. (b) Calculate the rotation period assuming a density of $3.0 \mathrm{~g} / \mathrm{cm}^{3},$ typical of many planets, satellites, and asteroids. No astronomical object has ever been found to be spinning with a period shorter than that determined by this analysis.

Averell Hause

Carnegie Mellon University

Problem 81

In a double-star system, two stars of mass $3.0 \times 10^{30} \mathrm{~kg}$ each rotate about the system's center of mass at radius $1.0 \times 10^{11} \mathrm{~m}$.
(a) What is their common angular speed? (b) If a meteoroid passes through the system's center of mass perpendicular to their orbital plane, what minimum speed must it have at the center of mass if it is to escape to "infinity" from the two-star system?

Alia Hamdan

Numerade Educator

Problem 82

A satellite is in elliptical orbit with a period of $8.00 \times 10^{4} \mathrm{~s}$ about a planet of mass $7.00 \times 10^{24} \mathrm{~kg} .$ At aphelion, at radius $4.5 \times 10^{7} \mathrm{~m},$ the satellite's angular speed is $7.158 \times 10^{-5} \mathrm{rad} / \mathrm{s} .$ What is its angular speed at perihelion?

Averell Hause

Carnegie Mellon University

Problem 83

In a shuttle craft of mass $m=3000 \mathrm{~kg}$, Captain Janeway orbits a planet of mass $M=9.50 \times 10^{25} \mathrm{~kg},$ in a circular orbit of radius $r=4.20 \times 10^{7} \mathrm{~m} .$ What are (a) the period of the orbit and
(b) the speed of the shuttle craft? Janeway briefly fires a forwardpointing thruster, reducing her speed by $2.00 \%$. Just then, what are
(c) the speed, (d) the kinetic energy, (e) the gravitational potential energy, and (f) the mechanical energy of the shuttle craft?
(g) What is the semimajor axis of the elliptical orbit now taken by the craft? (h) What is the difference between the period of the original circular orbit and that of the new elliptical orbit? (i) Which orbit has the smaller period?

Alia Hamdan

Numerade Educator

Problem 84

Consider a pulsar, a collapsed star of extremely high density, with a mass $M$ equal to that of the Sun $\left(1.98 \times 10^{30} \mathrm{~kg}\right),$ a radius $R$ of only $12 \mathrm{~km}$, and a rotational period $T$ of 0.041 s. By what percentage does the free-fall acceleration $g$ differ from the gravitational acceleration $a_{g}$ at the equator of this spherical star?

Averell Hause

Carnegie Mellon University

Problem 85

A projectile is fired vertically from Earth's surface with an initial speed of $10 \mathrm{~km} / \mathrm{s}$. Neglecting air drag, how far above the surface of Earth will it go?

Bettina Hanlon

Numerade Educator

Problem 86

An object lying on Earth's equator is accelerated (a) toward the center of Earth because Earth rotates, (b) toward the Sun because Earth revolves around the Sun in an almost circular orbit, and
(c) toward the center of our galaxy because the Sun moves around the galactic center. For the latter, the period is $2.5 \times 10^{8} \mathrm{y}$ and the radius is $2.2 \times 10^{20} \mathrm{~m} .$ Calculate these three accelerations as multiples of $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$

Bettina Hanlon

Numerade Educator

Problem 87

(a) If the legendary apple of Newton could be released from rest at a height of $2 \mathrm{~m}$ from the surface of a neutron star with a mass 1.5 times that of our Sun and a radius of $20 \mathrm{~km},$ what would be the apple's speed when it reached the surface of the star? (b) If the apple could rest on the surface of the star, what would be the approximate difference between the gravitational acceleration at the top and at the bottom of the apple? (Choose a reasonable size for an apple; the answer indicates that an apple would never survive near a neutron star.)

Averell Hause

Carnegie Mellon University

Problem 88

With what speed would mail pass through the center of Earth if falling in a tunnel through the center?

Averell Hause

Carnegie Mellon University

Problem 89

The orbit of Earth around the Sun is almost circular: The closest and farthest distances are $1.47 \times 10^{8} \mathrm{~km}$ and $1.52 \times 10^{8} \mathrm{~km}$ respectively. Determine the corresponding variations in (a) total energy, (b) gravitational potential energy, (c) kinetic energy, and
(d) orbital speed. (Hint: Use conservation of energy and conservation of angular momentum.)

Bettina Hanlon

Numerade Educator

Problem 90

A $50 \mathrm{~kg}$ satellite circles planet Cruton every $6.0 \mathrm{~h}$. The magnitude of the gravitational force exerted on the satellite by Cruton is $80 \mathrm{~N}$. (a) What is the radius of the orbit?
(b) What is the kinetic energy of the satellite? (c) What is the mass of planet Cruton?

Averell Hause

Carnegie Mellon University

Problem 91

We watch two identical astronomical bodies $A$ and $B$, each of mass $m,$ fall toward each other from rest because of the gravitational force on each from the other. Their initial center-to-center separation is $R_{i}$. Assume that we are in an inertial reference frame that is stationary with respect to the center of mass of this twobody system. Use the principle of conservation of mechanical energy $\left(K_{f}+U_{f}=K_{i}+U_{i}\right)$ to find the following when the centerto-center separation is $0.5 R_{i}:$ (a) the total kinetic energy of the system, (b) the kinetic energy of each body, (c) the speed of each body relative to us, and (d) the speed of body $B$ relative to body $A$. Next assume that we are in a reference frame attached to body $A$ (we ride on the body). Now we see body $B$ fall from rest toward us. From this reference frame, again use $K_{f}+U_{f}=K_{i}+U_{i}$ to find the following when the center-to-center separation is $0.5 R_{i}$ : (e) the kinetic energy of body $B$ and (f) the speed of body $B$ relative to body $A$. (g) Why are the answers to (d) and (f) different? Which answer is correct?

Averell Hause

Carnegie Mellon University

Problem 92

A $150.0 \mathrm{~kg}$ rocket moving radially outward from Earth has a speed of $3.70 \mathrm{~km} / \mathrm{s}$ when its engine shuts off $200 \mathrm{~km}$ above Earth's surface. (a) Assuming negligible air drag acts on the rocket, find the rocket's kinetic energy when the rocket is $1000 \mathrm{~km}$ above Earth's surface. (b) What maximum height above the surface is reached by the rocket?

Averell Hause

Carnegie Mellon University

Problem 93

Planet Roton, with a mass of $7.0 \times 10^{24} \mathrm{~kg}$ and a radius of $1600 \mathrm{~km},$ gravitationally attracts a meteorite that is initially at rest relative to the planet, at a distance great enough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, find the speed of the meteorite when it reaches the planet's surface.

Bettina Hanlon

Numerade Educator

Problem 94

Two $20 \mathrm{~kg}$ spheres are fixed in place on a $y$ axis, one at $y=0.40 \mathrm{~m}$ and the other at $y=-0.40 \mathrm{~m}$. A $10 \mathrm{~kg}$ ball is then released from rest at a point on the $x$ axis that is at a great distance (effectively infinite) from the spheres. If the only forces acting on the ball are the gravitational forces from the spheres, then when the ball reaches the $(x, y)$ point $(0.30 \mathrm{~m}, 0),$ what are
(a) its kinetic energy and (b) the net force on it from the spheres, in unit-vector notation?

Averell Hause

Carnegie Mellon University

Problem 95

Sphere $A$ with mass $80 \mathrm{~kg}$ is located at the origin of an $x y$ coordinate system; sphere $B$ with mass $60 \mathrm{~kg}$ is located at coordinates $(0.25 \mathrm{~m}, 0) ;$ sphere $C$ with mass $0.20 \mathrm{~kg}$ is located in the first quadrant $0.20 \mathrm{~m}$ from $A$ and $0.15 \mathrm{~m}$ from $B$. In unit-vector notation, what is the gravitational force on $C$ due to $A$ and $B ?$nitrocellulose to a speed of $11 \mathrm{~km} / \mathrm{s}$ along the gun barrel's length of $220 \mathrm{~m}$. (a) In $g$ units, what is the average acceleration of the capsule and astronauts in the gun barrel? (b) Is that acceleration tolerable or deadly to the astronauts?

A modern version of such gun-launched spacecraft (although without passengers) has been proposed. In this modern version, called the SHARP (Super High Altitude Research Project) gun, ignition of methane and air shoves a piston along the gun's tube, compressing hydrogen gas that then launches a rocket. During this launch, the rocket moves $3.5 \mathrm{~km}$ and reaches a speed of $7.0 \mathrm{~km} / \mathrm{s}$ Once launched, the rocket can be fired to gain additional speed.
(c) In $g$ units, what would be the average acceleration of the rocket within the launcher? (d) How much additional speed is needed (via the rocket engine) if the rocket is to orbit Earth at an altitude of $700 \mathrm{~km} ?$

Keshav Singh

Numerade Educator

Problem 97

An object of mass $m$ is initially held in place at radial distance $r=3 R_{E}$ from the center of Earth, where $R_{E}$ is the radius of Earth. Let $M_{E}$ be the mass of Earth. A force is applied to the object to move it to a radial distance $r=4 R_{E},$ where it again is held in place. Calculate the work done by the applied force during the move by integrating the force magnitude.

Alia Hamdan

Numerade Educator

Problem 98

To alleviate the traffic congestion between two cities such as Boston and Washington, D.C., engineers have proposed building a rail tunnel along a chord line connecting the cities (Fig. 13-55). A train, unpropelled by any engine and starting from rest, would fall through the first half of the tunnel and then move up the second half. Assuming Earth is a uniform sphere and ignoring air drag and friction, find the city-to-city travel time.

Bettina Hanlon

Numerade Educator

Problem 99

A thin rod with mass $M=5.00 \mathrm{~kg}$ is bent in a semicircle of radius $R=0.650 \mathrm{~m}$ (Fig. $13-56$ ).
(a) What is its gravitational force (both magnitude and direction) on a particle with mass $m=3.0 \times 10^{-3} \mathrm{~kg}$ at $P,$ the center of curvature?
(b) What would be the force on the particle if the rod were a complete circle?

Alia Hamdan

Numerade Educator

Problem 100

Identical blocks with identical masses $m=2.00 \mathrm{~kg}$ hang from strings of different lengths on a balance at Earth's surface. The strings have negligible mass and differ in length by $h=5.00 \mathrm{~cm} .$ Assume Earth is spherical with a uniform density $\rho=5.50 \mathrm{~g} / \mathrm{cm}^{3} .$ What is the difference in the weight of the blocks due to one being closer to Earth than the other?

Keshav Singh

Numerade Educator

Problem 101

A spaceship is on a straight-line path between Earth and the Moon. At what distance from Earth is the net gravitational force on the spaceship zero?

Alia Hamdan

Numerade Educator