Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 14

Gravitation - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Bettina Hanlon

Numerade Educator

Problem 2

Some believe that the positions of the planets at the time of birth influence the newborn. Others deride this belief and claim that the gravitational force exerted on a baby by the obstetrician is greater than that exerted by the planets. To check this claim, calculate and compare the magnitude of the gravitational force exerted on a $3 \mathrm{~kg}$ baby (a) by a $70 \mathrm{~kg}$ obstetrician who is $1 \mathrm{~m}$ away and roughly approximated as a point mass, (b) by the massive planet Jupiter $\left(m=2 \times 10^{27} \mathrm{~kg}\right)$ at its closest approach to Earth $\left(=6 \times 10^{11} \mathrm{~m}\right)$, and $(\mathrm{c})$ by Jupiter at its greatest distance from Earth $\left(=9 \times 10^{11} \mathrm{~m}\right)$. (d) Is the claim correct?

Alfjad Alfjad

Numerade Educator

Problem 3

One of the Echo satellites 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 gravita-
tional force on the meteor from the satellite at the closest approach?

Alfjad Alfjad

Numerade Educator

Problem 4

The Sun and Earth each exert a gravitational force on the Moon. What is the ratio $F_{\text {Sun } \rightarrow \text { Moon }} / F_{\text {Earth } \rightarrow \text { Moon }}$ of the magnitudes of these two forces? (The average Sun-Moon distance is equal to the Sun-Earth distance.)

Alfjad Alfjad

Numerade Educator

Problem 5

Split into Two $A$ 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

Numerade Educator

Problem 6

A spaceship is on a straight-line path between Earth and its moon. At what distance from Earth is the net gravitational force (due to the Earth and the Moon only) on the spaceship zero?

Alfjad Alfjad

Numerade Educator

Problem 7

How far from Earth must a space probe be along a line toward the Sun so that the Sun's gravitational pull on the probe balances Earth's pull?

Alfjad Alfjad

Numerade Educator

Problem 8

Three $5.0 \mathrm{~kg}$ spheres are located in the $x y$ plane as shown in Fig. $14-17 .$ What is the magnitude of the net gravitational force on the sphere at the origin due to the other two spheres?

Alfjad Alfjad

Numerade Educator

Problem 9

In Fig. $14-18 a$, four spheres form the corners of a square whose side is $2.0 \mathrm{~cm}$ long. What are the magnitude and direction of the net gravitational force from them on a central sphere with mass $m_{A}=250 \mathrm{~kg}$ ?

Alfjad Alfjad

Numerade Educator

Problem 10

In Fig. $14-18 b$, two spheres of mass $m$ and a third sphere of mass $M$ form an equilateral triangle, and a fourth sphere of mass $m_{B}$ 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 $m_{B}$, what then is the magnitude of the net gravitational force on the central sphere?

Alfjad Alfjad

Numerade Educator

Problem 11

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

Averell Hause

Carnegie Mellon University

Problem 12

Four uniform spheres, with masses $m_{A}=400 \mathrm{~kg}, m_{B}=350 \mathrm{~kg}, m_{C}=2000 \mathrm{~kg}$, and $m_{D}=500 \mathrm{~kg}$, have
$(x, y)$ coordinates of $(0,50) \mathrm{cm},(0,0) \mathrm{cm},(-80,0) \mathrm{cm}$, and $(40,0) \mathrm{cm}$,
respectively. What is the net gravitational force on sphere $B$ due to the other spheres?

Alfjad Alfjad

Numerade Educator

Problem 13

13. Spherical Hollow Figure $14-19$ shows a spherical hollow inside a lead sphere of radius $R ;$ 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 $M$. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass $m$ that lies at a distance $d$ from the center of the lead sphere, on the straight line connecting the centers of the spheres and of the hollow?

Alfjad Alfjad

Numerade Educator

Problem 14

Empire State Building You weigh $530 \mathrm{~N}$ at sidewalk level outside the Empire State Building in New York City. Suppose that you ride from this level to the 102 nd floor tower, a height of $373 \mathrm{~m}$. Ig-
noring Earth's rotation, how much less would you weigh there (because you are slightly farther from the center of Earth)?

Alfjad Alfjad

Numerade Educator

Problem 15

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

Bettina Hanlon

Numerade Educator

Problem 16

(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 17

Rate of Rotation 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 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.

Linda Winkler

Numerade Educator

Problem 18

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 19

A body is suspended from a spring scale in a ship sailing along the equator with speed $v .$ (a) Show that the scale reading will be very close to $W_{0}(1 \pm 2 \omega v / g)$, where $\omega$ is the rotational speed of Earth and $W_{0}$ is the scale reading when the ship is at rest.
(b) Explain the $\pm$ sign.

Linda Winkler

Numerade Educator

Problem 20

Certain neutron stars (extremely dense stars) are believed to be rotating at about 1 rev/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 21

Assume that a planet is a sphere of radius $R$ with a uniform density and (somehow) has a narrow radial tunnel through its center. Also assume that 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 a point where the magnitude of the gravitational force on the apple is $\frac{1}{2} F_{R}$ if we move the apple (a) away from the planet and (b) into the tunnel?

Alfjad Alfjad

Numerade Educator

Problem 22

Two concentric shells of uniform density having masses $M_{1}$ and $M_{2}$ are situated as shown in Fig. $14-20$. Find the magnitude of the net gravitational force on a particle of mass $m$, due to the shells, when the particle is located at (a) point $A$, at distance $r=a$ from the center, (b) point $B$ at $r=b$, and $(c)$ point $C$ at $r=c .$ The distance $r$ is measured from the center of the shells.

Alfjad Alfjad

Numerade Educator

Problem 23

A solid sphere of uniform density has a 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 $m$ 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.

Alfjad Alfjad

Numerade Educator

Problem 24

A uniform solid sphere of radius $R$ has a gravitational strength $g_{\text {local }}$ at its surface. At what two distances from the center of the sphere is the gravitational strength $g_{\text {local }} / 3 ?$ (Hint: Consider distances both inside and outside the sphere.)

Alfjad Alfjad

Numerade Educator

Problem 25

Figure $14-21$ shows, not to scale, 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 the local gravitational strength $g$ at the surface. (b) Suppose that a bore hole (the Mohole) is driven to the crust-mantle interface at a depth of $25 \mathrm{~km} .$ What would be the value of $g$ 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 $g$ at a depth of $25 \mathrm{~km}$ ? (Precise measurements of $g$ are sensitive probes of the interior structure of Earth, although results can be clouded by local density variations.)

Alfjad Alfjad

Numerade Educator

Problem 26

(A) What is the gravitational potential energy of the two-particle system in Problem $1 ?$ 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?

Alfjad Alfjad

Numerade Educator

Problem 27

(A) In Problem 12, remove sphere $A$ and calculate the gravitational potential energy of the remaining threeparticle 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?

Linda Winkler

Numerade Educator

Problem 28

In Problem 5, what ratio $m / M$ gives the least gravitational potential energy for the system?

Alfjad Alfjad

Numerade Educator

Problem 29

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 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?

Alfjad Alfjad

Numerade Educator

Problem 30

Calculate the amount of energy required to escape from (a) Earth's moon and (b) Jupiter relative to that required to escape from Earth.

Alfjad Alfjad

Numerade Educator

Problem 31

The three spheres in Fig. 14-22, with masses $m_{A}=800 \mathrm{~g}, m_{B}=100 \mathrm{~g}$, and $m_{C}=200 \mathrm{~g}$, 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 $C$ ?

Linda Winkler

Numerade Educator

Problem 32

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?

Linda Winkler

Numerade Educator

Problem 33

A rocket is accelerated to speed $v=$ $2 \sqrt{g R_{E}}$ near Earth's surface (where Earth's radius is $R_{E}$ ), and it then coasts upward. (a) Show that it will escape from Earth. (b) Show that very far from Earth its speed will be $v=\sqrt{2 g R_{E}}$.

Alfjad Alfjad

Numerade Educator

Problem 34

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 great enough distance 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 35

(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 $1000 \mathrm{~m} / \mathrm{s} ?(\mathrm{c})$ With what speed will an object hit the asteroid if it is dropped from $1000 \mathrm{~km}$ above the surface?

Linda Winkler

Numerade Educator

Problem 36

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, 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?

Alfjad Alfjad

Numerade Educator

Problem 37

Two neutron stars are separated by a distance of $10^{10} \mathrm{~m}$. They each have a mass of $10^{30} \mathrm{~kg}$ and a radius of $10^{5} \mathrm{~m}$. 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?

Linda Winkler

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 as $B$ is released? (b) What is the kinetic energy of $B$ when it has moved $0.20 \mathrm{~m}$ toward $A$ ?

Linda Winkler

Numerade Educator

Problem 39

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 40

In Fig. $14-13 b$, 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), $2.1 \mathrm{~m}$ from the floor?

Linda Winkler

Numerade Educator

Problem 41

Figure 14 23 shows two identical spheres, each with mass $2.00 \mathrm{~kg}$ and radius $R=$ $0.0200 \mathrm{~m}$, that initially touch, somewhere in deep space. Suppose the spheres are blown apart such that they initially separate at the relative speed $1.05 \times 10^{-4} \mathrm{~m} / \mathrm{s}$. They then slow due to the gravitational force between them.
Center-of-mass frame: Assume that we are in an inertial reference frame that is stationary with respect to the center of mass of the two-sphere system. Use the principle of conservation of mechanical energy $\left(K_{2}+U_{2}=K_{1}+U_{1}\right)$ to find the following when the center-to-center separation is $10 R:$ (a) the kinetic energy of each sphere and (b) the speed of sphere $B$ relative to sphere $A$. Sphere frame: Next assume that we are in a reference frame attached to sphere $A$ (we ride on the body). Now we see sphere $B$ move away from us. From this reference frame, again use $K_{2}+$ $U_{2}=K_{1}+U_{1}$ to find the following when the center-to-center separation is $10 R:$ (c) the kinetic energy of sphere $B$ and (d) the speed of sphere $B$ relative to sphere $A$. (e) Why are the answers to (b) and (d) different? Which answer is correct?

Victor Salazar

Numerade Educator

Problem 42

Black Hole 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 black holes near them, at a radial distance of $50 R_{k} .$ 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, 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 43

Two schoolmates, Romeo and Juliet, catch each other's eye across a crowded dance floor at a school dance. Estimate the gravitational attraction they exert on each other.

Alfjad Alfjad

Numerade Educator

Problem 44

The Alignment of the Planets Some authors seeking public attention have suggested that when many planets are "aligned" (i.e., are close together in the sky) their gravitational pull on the Earth all acting together might produce earthquakes and other disasters. To get an idea of whether this is plausible, set up the following calculation: (a) Draw a sketch of the solar system and arrange the planets so that Mars, Jupiter, and Saturn are on the same side of the Sun as the Earth. Look up (there is a table in the back of Under. standing Physics) the radii of the planetary orbits and their masses.
(b) Infer the distances these planets would be from Earth in this arrangement. (c) Without doing all the calculations, decide which of the three planets would exert the strongest gravitational force on the Earth. (Hint: Use the dependence of Newton's universal gravitation law on mass and distance.) (d) Calculate the gravitational force of the most important planet on the Earth. (e) Calculate how this compares to the gravitational force the Moon exerts on the Earth. Note: In fact, it is not the gravitational force itself that produces the possibly dangerous effects, but the tidal forces - the derivative of the gravitational force. This reduces the effect by another factor of the distance. That is, the tidal force goes like $1 / r^{3}$ instead of like $1 / r^{2}$. This weakens the planet's gravitational effect compared to the Moon's by an additional factor of $r_{\text {Earth-moon }} / r_{\text {Earth-planet }}$, a number much less than 1 .

Ben Nicholson

Numerade Educator

Problem 45

Is Newton's Law of Gravity Wrong? A professional scientist (not a physicist) stops you in the hall and says: "I can prove Newton's theory of gravity is wrong. The Sun is 320,000 times as massive as the Earth, but only 400 times as far from the Moon as is the Earth. Therefore, the force of the Sun's gravity on the Moon should be twice as big as the Earth's and the Moon should go around the Sun instead of around the Earth. Since it doesn't, Newton's theory of gravity must be wrong!" What's the matter with this reasoning?

Linda Winkler

Numerade Educator

Problem 46

In the Shuttle When we see the astronauts in orbit in the space shuttle on TV, they seem to float. If they let go of something, it just stays where they put it. It doesn't fall. What happens to gravity for objects in orbit? Does gravity stop at the Earth's atmosphere? Explain what's happening in terms of the physics you have learned.

Linda Winkler

Numerade Educator