## Mastering Physics Solutions Chapter 12 Gravity

**Chapter 12 Gravity Q.1CQ**

It is often said that astronauts in orbit experience weightlessness because they are beyond the pull of Earth’s gravity. Is this statement correct? Explain.

**Solution:**

No The force of Earth’s gravity is practically as strong in orbit as it is on the surface of Earth The astronauts experience weightlessness because they are in constant free fall.

**Chapter 12 Gravity Q.1P**

CE System A has masses m and m separated by a distance r; system B has masses m and 2m separated by a distance 2r; system C has masses 2m and 3m separated by a distance 2r, and system D has masses 4m and 5m separated by a distance 3r. Rank these systems in order of increasing gravitational force. Indicate ties where appropriate.

**Solution:**

**Chapter 12 Gravity Q.2CQ**

When a person passes you on the street, you do not feel a gravitational tug. Explain.

**Solution:**

**Chapter 12 Gravity Q.2P**

In each hand you hold a 0.16-kg apple. What is the gravitational force exerted by each apple on the other when their separation is (a) 0.25 m and (b) 0.50 m?

**Solution:**

**Chapter 12 Gravity Q.3CQ**

Two objects experience a gravitational attraction. Give a reason why the gravitational force between them does not depend on the sum of their masses.

**Solution:**

The force of gravity between two point masses m1 and m2, separated by a distance r, is attractive and of magnitude

where G is the universal gravitational constant.

Gravity exerts an action-reaction pair of forces on m1 and m2. That is, the force exerted by gravity on m1 is equal in magnitude but opposite in direction to the force exerted on m2. It is dependent on the product of masses. If the gravitational force depended on the sum of the two masses, it would predict a non-zero force even when one of the masses is zero. That is, there would be a gravitational force between a mass and a point in empty space, which is certainly not what we observed.

**Chapter 12 Gravity Q.3P**

A 6.1-kg bowling ball and a 7.2-kg bowling ball rest on a rack 0.75 m apart. (a) What is the force of gravity exerted on each of the balls by the other ball? (b) At what separation is the force of gravity between the balls equal to 2.0 × 10?9N?

**Solution:**

**Chapter 12 Gravity Q.4CQ**

Imagine bringing the tips of your index fingers together. Each finger contains a certain finite mass, and the distance between them goes to zero as they come into contact. From the force law F = Gm1m2/r2 one might conclude that the attractive force between the fingers is infinite, and, therefore, that your fingers must remain forever stuck together. What is wrong with this argument?

**Solution:**

As the tips of the fingers approach one another, we can think of them as being two small spheres that touch each other. Even though the two spheres touch each other, the distance between the centers is not zero. This is always a finite number. Therefore, the force between the spheres is always finite, even they touch each other. As such, the two fingers simply experience the finite force of two point masses separated by a finite distance.

**Chapter 12 Gravity Q.4P**

A communications satellite with a mass of 480 kg is in a circular orbit about the Earth. The radius of the orbit is 35,000 km as measured from the center of the Earth. Calculate (a) the weight of the satellite on the surface of the Earth and (b) the gravitational force exerted on the satellite by the Earth when it is in orbit.

**Solution:**

**Chapter 12 Gravity Q.5CQ**

Does the radius vector of Mars sweep out the same amount of area per time as that of the Earth? Why or why not?

**Solution:**

No. The amount of area swept out per time varies from planet to planet because the linear speeds of planets are different.

**Chapter 12 Gravity Q.5P**

The Attraction of Ceres Ceres, the largest asteroid known, has a mass of roughly 8.7 × 1020 kg. If Ceres passes within 14,000 km. of the spaceship in which you are traveling, what force does it exert on you? (Use an approximate value for your mass, and treat yourself and the asteroid as point objects.)

**Solution:**

**Chapter 12 Gravity Q.6CQ**

When a communications satellite is placed in a geosynchronous orbit above the equator. it remains fixed over a given point on the ground. Is it possible to put a satellite into an orbi t so that it remains fixed above the North Pole? Explain

**Solution:**

INot possiblel because a satellite will appear stationary only when it revolves in an orbit that is concentric and coplanar with the equatorial plane, has a period of revolution of 24 hours, and

has a sense of revolution from the west to the east of Earth. As the north pole is away from the equatorial plane. it will not be possible to put a geostationary satellite over the north pole.

**Chapter 12 Gravity Q.6P**

In one hand you hold a 0.11-kg apple, in the other hand a 0.24-kg orange. The apple and orange are separated by 0.85 m. What is the magnitude of the force of gravity that (a) the orange exerts on the apple and (b) the apple exerts on the orange?

**Solution:**

**Chapter 12 Gravity Q.7CQ**

The Mass of Pluto On June 22, 1978, James Christy made the first observation of a moon orbiting Pluto. Until that lime the mass of Pluto was not known, but with the discovery of its moon, Charon, its mass could be calculated with some accuracy. Explain.

**Solution:**

**Chapter 12 Gravity Q.7P**

IP A spaceship of mass m travels from the Earth to the Moon along a line that passes through the center of the Earth and the center of the Moon. (a) At what distance from the center of the Earth is the force due to the Earth twice the magnitude of the force due to the Moon? (b) How does your answer to part (a) depend on the mass of the spaceship? Explain.

**Solution:**

**Chapter 12 Gravity Q.8CQ**

Rockets arc launched into space from Cape Canaveral in an easterly direction Is there an advantage to launching to the east versus launching to the west? Explain

**Solution:**

Earth revolves from west to east (counterclockwise) about its polar axis. Therefore, all the particles on Earth have a velocity from west to east. This velocity is at a maximum along the

equatorial line, as y = Rw,where R is the radius of Earth and w is the angular velocity of Earth’s revolution about its polar axis.

ICape Canaveral is situated at the equator so when a rocket is launched from west to east in this place. the maximum linear velocity is added to the launching velocity of the rocket Because of this. launching becomes easied

**Chapter 12 Gravity Q.8P**

**Solution:**

**Chapter 12 Gravity Q.9CQ**

One day in the future you may take a pleasure cruise to the Moon While there you might climb a lunar mountain and throw a rock horizontally from its summit If. in principle, you could throw the rock fast enough, it might end up hitting you in the back Explain.

**Solution:**

Ion the Moon. where there is no atmosphere, a rock can orbit at any altitudel where it clears the mountains — as long as it has sufficient speed If we could give a rock enough speed. it would orbit the Moon and return to us from the other side (behind).

**Chapter 12 Gravity Q.9P**

**Solution:**

**Chapter 12 Gravity Q.10CQ**

Apollo astronauts orbiting the Moon at low altitude noticed occasional changes ¡n their orbit that they attributed to localized concentrations of mass below the lunar surface. Just what effect would such ‘Thascons” have on their orbit?

**Solution:**

As the astronauts approach a mass concentration, its increased gravitational attraction would increase the speed of the craft Similarly, as they pass the mass concentration, its ravitationaI attraction is in the backward direction, which decreases their speed I

**Chapter 12 Gravity Q.10P**

**Solution:**

**Chapter 12 Gravity Q.11CQ**

If you light a candle on the space shuttle—which would not be a good idea—would it burn the same as on the Earth? Explain

**Solution:**

No. In the weightless environment of the shuttle, there is no convection which is required to bring fresh oxygen to the flame. Without convection, a flame usually goes out very quickly. In carefully controlled experiments on the shuttle, however, small flames have been maintained for considerable times These “weightless” flames are spherical in shape. as opposed to the tear- shaped flames on Earth

**Chapter 12 Gravity Q.11P**

IP Three 6.75-kg masses are at the corners of an equilateral triangle and located in space far from any other masses. (a) If the sides of the triangle are 1.25 m long, find the magnitude of the net force exerted on each of the three masses. (b) How does your answer to part (a) change if the sides of the triangle are doubled in length?

**Solution:**

**Chapter 12 Gravity Q.12CQ**

The force exerted by the Sun on the Moon is more than twice the force exerted by the Earth on the Moon. Should the Moon be thought of as orbiting the Earth or the Sun? Explain.

**Solution:**

**Chapter 12 Gravity Q.12**

**Solution:**

**Chapter 12 Gravity Q.13CQ**

**Solution:**

The net force acting on the moon is always directed toward the Sun, never away from the Sun. Therefore, the Moon’s orbit must always curve toward the Sun. It curves sharply toward the Sun when Earth is between the Moon and the Sun, and curves only slightly toward the Sun when the Moon is between the Sun and Earth.

**Chapter 12 Gravity Q.13P**

Suppose that three astronomical objects (1, 2, and 3) are observed to lie on a line, and that the distance from object 1 to object 3 is D. Given that object 1. has four times the mass of object 3 and seven times the mass of object 2, find the distance between objects 1 and 2 for which the net force on object 2 is zero.

**Solution:**

**Chapter 12 Gravity Q.14P**

Find the acceleration due to gravity on the surface of (a) Mercury and (b) Venus.

**Solution:**

**Chapter 12 Gravity Q.15P**

At what altitude above the Earth’s surface is the acceleration due to gravity equal to g/2?

**Solution:**

**Chapter 12 Gravity Q.16P**

Two 6-7-kg bowling balls, each with a radius of 0.11 m, are in contact with one another. What is the gravitational attraction between the bowling balls?

**Solution:**

**Chapter 12 Gravity Q.17P**

What is the acceleration due to Earth’s gravity at a distance from the center of the Earth equal to the orbital radius of the Moon?

**Solution:**

**Chapter 12 Gravity Q.18P**

Gravity on Titan Titan is the larges t moon o f Saturn and the only moon in the solar system known to have a substantial atmosphere. Find the acceleration due to gravity on Titan’s surface, given that its mass is 1.35 × 1023 kg and its radius is 2570 km.

**Solution:**

**Chapter 12 Gravity Q.19P**

IP At a certain distance from the center of the Earth, a 4.6-kg object has a weight of 2.2 N. (a) Find this distance, (b) If the object is released at this location and allowed to falï toward the Earth, what is its initial acceleration? (c) If the object is now moved twice as far from the Earth, by what factor does its weight change? Explain, (d) By what factor does its initial acceleration change? Explain.

**Solution:**

**Chapter 12 Gravity Q.20P**

Tine acceleration due to gravity on the Moon’s surface is known to be about one-sixth the acceleration due to gravity on the Earth. Given that the radius of the Moon is roughly one-quarter that of the Earth, find the mass of the Moon in terms of the mass of the Earth.

**Solution:**

**Chapter 12 Gravity Q.21P**

IP An Extraterrestrial Volcano Several volcanoes have been observed erupting on the surface of Jupiter’s closest Galilean moon, lo. Suppose that material ejected from one of these volcanoes reaches a height of 5.00 km a fter being projected straight upward with an initial speed of 134 m/s. Given that the radius of lo is 1820 km, (a) outlinca strategy thatallows you to calculate the mass of To. (b) Use your strategy to calculate Io’s mass.

**Solution:**

**Chapter 12 Gravity Q.22P**

IP Verne’s Trip to the Moon In his novel From the Earth to the Moon, Jules Verne imagined that astronauts inside a spaceship would walk on the floor of the cabin when the force exerted on the ship by the Earth was greater than the force exerted by the Moon. When the force exerted by the Moon was greater, he thought the astronauts would walk on the ceiling of the cabin, (a) At what distance from the center of the Earth would the forces exerted on the spaceship by the Earth and the Moon be equal? (b) Explain why Verne’s description of gravitational effects is incorrect.

**Solution:**

**Chapter 12 Gravity Q.23P**

Consider an asteroid with a radius of 19 km and a mass of 3.35 X 1015 kg. Assume the asteroid is roughly spherical, (a) What is the acceleration due to gravity on the surface of the asteroid? (b) Suppose the asteroid spins about an axis through its center, like the Earth, with a rotational period T. What is the smallest value T can have before loose rocks on the asteroid’s equator begin to fly off the surface?

**Solution:**

**Chapter 12 Gravity Q.24P**

CE Predict/Explain The Speed of the Earth The orbital speed of the Earth is greatest around January 4 and least around July 4. (a) Is the distance from the Earth to the Sun on January 4 greater than, less than, or equal to its distance from the Sun on July 4? (b) Choose the best explanation from among the following:

I. The Earth’s orbit is circular, with equal distance from, the Sun at all times.

II. The Earth sweeps out equal area in equal time, thus it must be closer to the Sun when it is moving faster.

III. The greater the speed of the Earth, the greater its distance from the Sun.

**Solution:**

a) The distance from the Earth to the Sun on January 4, is less than the distance from the Sun on July 4.

b) The Earth sweeps out equal area in equal time, thus it must be closer to the sun when it is moving faster.

**Chapter 12 Gravity Q.25P**

C E A satellite orbits the Earth in a circular orbit of radius r. At some point its rocket engine is fired in such a way that its speed increases rapidly by a small amount. As a result, do the (a) apogee distance and (b) perigee distance increase, decrease, or stay the same?

**Solution:**

Use the concept of orbital transfer to place the satellite into a new orbit.

(a)

The decelerating or accelerating rockets at some point in the circular orbit of the satellite would allow the satellite into a new orbit which is not a circle. The new orbit is an ellipse. The largest distance between the Earth and the satellite in an elliptical orbit is called the apogee distance. In the case of transfer of orbits, the apogee distance increases if the speed of the rocket increases a while in the original orbit.

(b)

The smallest distance between the Earth and the satellite in an elliptical orbit is nothing but the perigee distance. In case of transfer of orbits, the perigee distance doesn’t change and equal to the radius of the original circular orbit.

**Chapter 12 Gravity Q.26P**

g Repeat the previous problem., only this time with the rocket engine of the satellite fired in such a way as to slow the satellite.

**Solution:**

(A) The satellite drops into an elliptical orbit that brings it closer to Earth.

(B) The apogee distance remains unchanged.

(C) The perigee distance is reduced.

**Chapter 12 Gravity Q.27P**

CE Predict/Explain The Earth-Moon Distance Is Increasing Laser reflectors left on the surface of the Moon by the Apollo astronauts show that the average distance from the Earth to the Moon is increasing at the rate of 3.8 cm per year. (a) As a result, will the length of the month increase, decrease, or remain the same? (b) Choose the best expianation from among the following: I. The greater the radius of an orbit, the greater the period,

which implies a longer month.

II. The length of the month will remain the same due to conservation of angular momentum,

III. The speed of the Moon is greater with increasing radius; therefore, the length of the month will be less.

**Solution:**

a) If the average distance increases, then the length of the month also increases.

b) The period depends upon the radius. Greater the radius, greater will be the period. Option (1) is correct.

**Chapter 12 Gravity Q.28P**

Apollo Missions On Apollq missions to the Moon, the command module orbited at an altitude of 110 km above the lunar surface. How long did it take for the command module to complete one orbit?

**Solution:**

**Chapter 12 Gravity Q.29P**

Find the orbital speed of a satellite in a geosynchronous circular orbit 3.58 X 107 m above the surface of the Earth.

**Solution:**

**Chapter 12 Gravity Q.30P**

An Extrasolar Planet In July of 1999 a planet was reported to be orbiting the Sun-like star Iota Horologii with a period of 320 days. Find the radius of the planet’s orbi t, assuming that iota Horologii has the same mass as the Sun. (This planet is presumably similar to Jupiter, but it may have large, rocky moons that enjoy a relatively pleasant climate.)

**Solution:**

**Chapter 12 Gravity Q.31P**

Phobos, one of the moons of Mars, orbits at a distance of 9378 km from the center of the red planet. What is the orbital period of Phobos?

**Solution:**

**Chapter 12 Gravity Q.32P**

· The largest moon in the solar system is Ganymede, a moon of Jupiter. Ganymede orbits at a distance of 1.07 X 109 m from the center of Jupiter with an orbital period of about 6.18 X 10′ s. Using this information, find the mass of Jupiter.

**Solution:**

**Chapter 12 Gravity Q.33P**

IP Am Asteroid with Its Own Moon The asteroid 243 Ida has its own small moon, Dactyl. (See the photo on p. 390) (a) Outline a strategy to find the mass of 243 Ida, given that the orbital radius of Dactyl is 89 km arid its period is 19 hr. (b) Use your strategy to calculate the mass of 243 Ida.

**Solution:**

**Chapter 12 Gravity Q.34P**

GPS Satellites GPS (Global Positioning System) satellites orbit at an altitude of 2.0 x 107 m. Find (a) the orbital period, and (b) the orbital speed of such a satellite.

**Solution:**

**Chapter 12 Gravity Q.35P**

IP Two satellites orbit the Earth, with satellite 1 at a greater altitude than satellite 2. (a) Which satellite has the greater orbital speed? Explain, (b) Calculate the orbital speed of a satellite that orbits at an altitude of one Earth radius above the surface of the Earth, (c) Calculate the orbital speed of a satellite that orbits at an altitude of two Earth radii above the surface of the Earth.

**Solution:**

**Chapter 12 Gravity Q.36P**

IP Calculate the orbital periods of satellites that orbit (a) one Earth radius above the surface of the Earth and (b) two Earth radii above the surface of the Earth, (c) How do your answers to parts (a) and (b) depend on the mass of the satellites? Explain, (d) How do your answers to parts (a) and (b) depend on the mass of the Earth? Explain.

**Solution:**

**Chapter 12 Gravity Q.37P**

S P The Martian moon Deimos has an orbital period that is greater than the other Martian moon, Phobos. Both moons have approximately circular orbits, (a) Is Deimos closer to or farther from Mars than Phobos? Explain, (b) Calculate the distance from the center of Mars to Deimos given that its orbital period is 1.10 × 105 s.

**Solution:**

**Chapter 12 Gravity Q.38P**

Binary Stars Centauri A and Centauri B are binary stars with a separation of 3.45 × 1012 m and an orbital period of 2.52 × 109 s. Assuming the two stars are equally massive (which is approximately the case), determine their mass.

**Solution:**

**Chapter 12 Gravity Q.39P**

Find the speed of Centauri A and Centauri B, using the information given in the previous problem.

**Solution:**

**Chapter 12 Gravity Q.40P**

Sputnik The first artificial satellite to orbit the Earth was Sputnik I, Saunched October 4,1957. The mass of Sputnik 1 was 83.5 kg, and its distances from the center of the Earth at apogee and perigee were 7330 km-and 6610 km, respectively. Find the difference in gravitational potential energy for Sputnik I as it moved from apogee to perigee.

**Solution:**

**Chapter 12 Gravity Q.41P**

CE Predict/Explain (a) Is the amount of energy required to get a spacecraft from the Earth to the Moon greater than, less than, or equal to the energy required to get the same spacecraft from the Moon to the Earth? (b) Choose the best explanation from among the following:

I. The escape speed of the Moon is less than that of the Earth; therefore, less energy is required to leave the Moon.

II. The situation is symmetric, and hence the same amount of energy is required to travel in either direction.

III. It takes more energy to go from the Moon to the Earth because the Moon is orbiting the Earth.

**Solution:**

Use the concept of escape speed of the planet. The escape speed of the planet is the minimum speed at which the object frees from the gravitational attraction of the planet.

(a)

The escape speed of an object launched from the planet depends only on the mass and size of the planet, but not on the mass of the object. The escape speed of the Earth is much greater than that of the Moon. Since the kinetic energy is directly proportional to the square of the velocity, the more energy is required to launch the spacecraft from the Earth to the Moon than that required to launch the spacecraft from the Moon to the Earth.

(b)

The option (I) is correct.

**Chapter 12 Gravity Q.42P**

**Solution:**

**Chapter 12 Gravity Q.43P**

Calculate the gravitational potential energy of a 8.8-kg mass (a) on the surface of the Earth and (b) at an altitude of 350 km. (c) Take the difference between the results for parts (b) and (a), and compare with nigh, where h = 350 km.

**Solution:**

**Chapter 12 Gravity Q.44P**

Two 0.59-kg basketballs, each with a radius of 12 cm, are just touching. How much energy is required to change the separation between the centers of the basketballs to (a) 1.0 m and (b) 10.0 m? (Ignore any other gravitational interactions.)

**Solution:**

**Chapter 12 Gravity Q.45P**

Find the minimum kinetic energy needed for a 39,000-kg rocket to escape (a) the Moon or (b) the Earth.

**Solution:**

**Chapter 12 Gravity Q.46P**

CE Predict/Explain Suppose the Earth were to suddenly shrink to half its current diameter, with its mass remaining constant, (a) Would the escape speed of the Earth increase, decrease, or stay the same? (b) Choose the best explanation from among the following:

I. Since the radius of the Earth would be smaller, the escape speed would also be smaller.

II. The Earth would have the same amount of mass, and hence its escape speed would be unchanged.

III. The force of gravity would be much stronger on the surface of the compressed Earth, leading to a greater escape speed.

**Solution:**

a) The escape speed of the earth increases.

b) The force of gravity would be much stronger on the surface of compressed Earth, leading to a greater escape speed. Option (III) is correct.

**Chapter 12 Gravity Q.47P**

CE Is the energy required to launch a rocket vertically to a height h greater than, less than, or equal to the energy required to prit the same rocket into orbit at the height hi Explain.

**Solution:**

The energy required to launch a rocket vertically to a height h is equal to the potential energy of the rocket at that height. However, for a rocket to be put into orbit at a height h, both kinetic energy and potential energy are required. So the energy required for the rocket to be put into orbit is greater than the energy required to launch a rocket vertically to the same height.

**Chapter 12 Gravity Q.48P**

Suppose one of the Global Positioning System satellites has a speed of 4.46 km/s at perigee and a speed of 3.64 km/s at apogee. If the distance from the center of the Earth to the satellite at perigee is 2.00 × 104 lem, what is the corresponding distance at apogee?

**Solution:**

**Chapter 12 Gravity Q.49P**

**Solution:**

**Chapter 12 Gravity Q.50P**

· Referring to Example 12-1, if the Millennium Eagle is at rest at point A, what is its speed at point B?

**Solution:**

**Chapter 12 Gravity Q.51P**

What is the launch speed of a projectile that rises vertically above the Earth to an altitude equal to one Earth radius before coming to rest momentarily?

**Solution:**

**Chapter 12 Gravity Q.52P**

A projectile launched vertically from the surface of the Moon? rises to an altitude of 365 km. What was the projectile’s initial speed?

**Solution:**

**Chapter 12 Gravity Q.53P**

Find the escape velocity for (a) Mercury and (b) Ventis.

**Solution:**

**Chapter 12 Gravity Q.54P**

IP Halley’s Comet Halley’s comet, which passes around the Sun every 76 years, has an elliptical orbit. When closest to the Sun (perihelion) it is at a distance of 8.823 x 1010 m and moves with a speed of 54.6 km/s. The greatest distance between Halley’s comet and the Sun (aphelion) is 6.152 x 1012 m. (a) Is the speed of Halley’s comet greater than or less than 54.6 km/s when it is at aphelion? Explain, (b) Calculate its speed ai aphelion.

**Solution:**

**Chapter 12 Gravity Q.55P**

The End of the Lunar Module On Apollo Moon missions, the lunar module would blast off from the Moon’s surface and dock with the command module in lunar orbit. After docking, the lunar module would be jettisoned and allowed to crash back onto the lunar surface. Seismometers placed on the Moon’s surface by the astronauts would then pick up the resulting seismic waves. Find the impact speed of the lunar module, given that it is jettisoned from an orbit 110 km above the lunar surface moving with a speed of 1630 m/s.

**Solution:**

**Chapter 12 Gravity Q.56P**

If a projectile is launched vertically from the Earth with a speed equal to the escape speed, how high above the Earth’s surface is it when its speed is half the escape speed?

**Solution:**

**Chapter 12 Gravity Q.57P**

Suppose a planet is discovered orbiting a distant star. If the mass of the planet is 10 times the mass of the Earth, and its radius is one-tenth the Earth’s radius, how does the escape speed of this planet compare with that of the Earth?

**Solution:**

**Chapter 12 Gravity Q.58P**

A projectile is launched vertically from the surface of the Moon with an initiaL speed of 1050 m/s. At what altitude is the projectile’s speed one-half its initial value?

**Solution:**

**Chapter 12 Gravity Q.59P**

To what radius would the Sun have to be contracted for its escape speed to equal the speed of light? (Black holes have escape speeds greater than the speed of light; hence we see no light from them.)

**Solution:**

**Chapter 12 Gravity Q.60P**

IP Two baseballs, each with a mass of 0.148 kg, are separated by a distance of 395 m in outer space, far from any other objects. (a) If the balls are released from rest, what speed do they have when their separation has decreased to 145 m? (b) Suppose the mass of the balls is doubled. Would the speed found in part (a) increase, decrease, or stay the same? Explain.

**Solution:**

**Chapter 12 Gravity Q.61P**

On Earth, a person can jump vertically and rise to a height h. What is the radius of the largest spherical asteroid from which this person could escape by jumping straight upward? Assume that each cubic meter of the asteroid has a mass of 3500 kg.

**Solution:**

**Chapter 12 Gravity Q.62P**

As will be shown in Problem 63, the magnitude of the tidal force exerted on an object of mass m and length a is approximately 4GmMa/r3. In this expression, M is the mass of the body causing the tidal force and r is the distance from the center of m to the center of M. Suppose you are 1 million miles away from a black hole whose mass is a million times that of the Sun. (a) Estimate the tidal force exerted on your body by the black hole. (b) At what distance will the tidal force be approximately 10 times greater than your weight?

**Solution:**

**Chapter 12 Gravity Q.63P**

**Solution:**

**Chapter 12 Gravity Q.64P**

**Solution:**

**Chapter 12 Gravity Q.65GP**

CE You weigh yourself on a scale inside an airplane flying due east above the equator. If the airplane now turns around and heads due west with the same speed, will the reading on the scale increase, decrease, or stay the same? Explain.

**Solution:**

SOLUTION:

The reading on the scale is due to the force of gravity between the person on the plane and the Earth.

F = Gm1m2 / R2

Where R is the difference between the passenger on the plane and the center of the Earth. As the plane switches direction from East to West, the R value remains unchanged. Since the mass of the person and the mass of the Earth are both the same, the magnitude of gravitational force will be the same.

**Chapter 12 Gravity Q.66GP**

**Solution:**

**Chapter 12 Gravity Q.67GP**

**Solution:**

Thus the increasing order of gravitational force is given by

object C >object A >object B

**Chapter 12 Gravity Q.68GP**

**Solution:**

**Chapter 12 Gravity Q.69GP**

CE A satellite goes through one complete orbit of the Earth. (a) Is the net work done on it by the Earth’s gravitational force positive, negative, or zero? Explain, (b) Does your answer to part (a) depend on whether the orbit is circular or elliptical?

**Solution:**

(A) When a satellite goes through one complete orbit, this means the satellite returns to

the initial point at which it started. The resulting net displacement is zero. So the net work done on it by Earth’s gravitational force is zero.

(B) No, the answer to part (A) is independent of the shape of the orbit (i.e., whether the orbit is circular or elliptical). It is dependent on the displacement by the satellite.

**Chapter 12 Gravity Q.70GP**

CE The Crash of Skylab Skylab, the largest spacecraft ever to fall back to the Earth, met its fiery end on July 11,1979, after flying directly over Everett, WA, on its last orbit. On the CBS Evening News the night before the crash, anchorman Walter Cronkite, in his rich baritone voice, made the following statement: “NASA says there is a little chance that Skylab will land in a populated area.” After the commercial, he immediately corrected himself by saying,”I meant to say ‘there is little chance’ Skylab will hita populated area.” In fact, it landed primarily in the Indian Ocean off the west coast of Australia, though several pieces were recovered near the town of Espérance, Australia, which later sent the U.S. State Department a $400 bill for littering. The cause of Skylab’s crash was the friction it experienced in the upper reaches of the Earth’s atmosphere. As the radius of Skylab’s orbit decreased, did its speed increase, decrease, or stay the same? Explain.

**Solution:**

The speed of the Skylab increases with decreasing radius. We might think that friction would slow Skylab just like other objects are slowed by friction – but by dropping Skylab to a lower orbit, friction is ultimately responsible for an increase in speed.

**Chapter 12 Gravity Q.71GP**

Consider a system consisting of three masses on the x axis. Mass m1 = 1.00 kg is at x = 1.00 m; mass m2 = 2.00 kg is at x = 2.00 m; and mass m3 = 3.00 kg is at x = 3.00 m. What is the total gravitational potential energy of this system?

**Solution:**

**Chapter 12 Gravity Q.72GP**

An astronaut exploring a distant solar system lands on an unnamed planet with a radius of 3860 km. When the astronaut jumps upward with an initial speed of 3.10 m/s, she rises to a height of 0.580 m. What is the mass of the planet?

**Solution:**

**Chapter 12 Gravity Q.73GP**

IP When the Moon is in its third-quarter phase, the Earth, Moon, and Sun form a right triangle, as shown in Figure 12-22. Calculate the magnitude of the force exerted on the Moon by (a) the Earth and (b) the Sun. (c) Does it make more sense to think of the Moon as orbiting the Sun, with a small effect due to the Earth, or as orbiting the Earth, with a small effect due to the Sun?

**Solution:**

**Chapter 12 Gravity Q.74GP**

**Solution:**

**Chapter 12 Gravity Q.75GP**

**Solution:**

**Chapter 12 Gravity Q.76GP**

A Near Miss! In the early morning hours of June 14, 2002, the Earth had a remarkably close encounter with an asteroid the size of a small city. The previously unknown asteroid, now designated 2002 MN, remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was 73,600 miles from the center of the Earth?about a third of the distance to the Moon. (a) Find the speed of the asteroid at closest approach, assuming its speed at infinite distance to be zero and considering only its interaction with the Barth. (b) Observations indica te the asteroid to have a diameter of about 2.0 km. Estimate the kinetic energy of the asteroid at closest approach, assuming it has an average density of 3.33 g/cm3 (For comparison, a 1-megaton nuclear weapon releases about 5.6 × 1015J of energy.)

**Solution:**

**Chapter 12 Gravity Q.77GP**

IP Suppose a planet is discovered that has the same amount of mass in a given volume as the Earth, but has half its radius. (a) Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on the Earth? Explain. (b) Calculate the acceleration due to gravity on this planet.

**Solution:**

**Chapter 12 Gravity Q.78GP**

IP Suppose a planet is discovered that has the same total mass as the Earth, but half its radius. (a) Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on the Earth? Explain. (b) Calculate the acceleration due to gravity on this planet.

**Solution:**

**Chapter 12 Gravity Q.79GP**

**Solution:**

**Chapter 12 Gravity Q.80GP**

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**Chapter 12 Gravity Q.81GP**

**Solution:**

**Chapter 12 Gravity Q.82GP**

Using the results from Problem 54. find the angular momentum of Halley’s comet (a) at perihelion and (b) at aphelion (Take the mass of Halley’s comet to be 9.8 x 1014 kg.)

**Solution:**

**Chapter 12 Gravity Q.83GP**

Exploring Mars Inthe not-too-distant future astronauts will travel to Mars to carry out scientific explorations. As part of their mission, it is likely that a “geosynchronous” satellite will be placed above a given point on the Martian equator to facilitate communications. At what altitude above the surface of Mars should such a satellite orbit? (Note: The Martian “day” is 24.6229 hours, Other relevant information can be found in Appendix C.)

**Solution:**

**Chapter 12 Gravity Q.84GP**

IP A satellite is placed in Earth orbit 1000 miles higher than the altitude of a geosynchronous satellite. Referringto Active Example 12-1, we see that the altitude of the satellite is 23,300 mi. (a) Is the period of this satellite greater than or less than 24 hours? (b) As viewed from the surface of the Earth, does the satellite move eastward or westward? Explain. (c) Find the orbital period of this satellite.

**Solution:**

**Chapter 12 Gravity Q.85GP**

Find the speed of the Millennium Eagle at point A in Example 12-1 if its speed at point B is 0.905 m/s.

**Solution:**

**Chapter 12 Gravity Q.86GP**

Show that the force of gravity between the Moon and the Sun is always greater than the force of gravity between the Moon and the Earth.

**Solution:**

**Chapter 12 Gravity Q.87GP**

**Solution:**

**Chapter 12 Gravity Q.88GP**

(a) Find the kinetic energy of a 1720-kg satellite in a circular orbit about the Earth, given that the radius of the orbit is 12,600 miles. (b) How much energy is required to move this satellite to a circular orbit with a radius of 25,200 miles?

**Solution:**

**Chapter 12 Gravity Q.89GP**

IP Space Shuttle Orbit On a typical mission, the space shuttle (m = 2.00 × 106 kg) orbits at an altitude of 250 km above the Earth’s surface. (a) Does the orbital speed of the shuttle depend on its mass? Explain. (b) Find the speed of the shuttle in its orbit. (c) How long does it take for the shuttle to complete one orbit of the Earth?

**Solution:**

**Chapter 12 Gravity Q.90GP**

IP Consider an object of mass m orbiting the Earth at a radius r. (a) Find the speed of the object. (b) Show that the total mechanical energy of this object is equal to (?1) times its kinetic energy. (c) Does the result of part (b) apply to an object orbiting the Sun? Explain.

**Solution:**

**Chapter 12 Gravity Q.91GP**

In a binary star system two stars orbit about their common center of mass. Find the orbital period of such a system, given that the stars are separated by a distance d and have masses m and 2m.

**Solution:**

**Chapter 12 Gravity Q.92GP**

**Solution:**

**Chapter 12 Gravity Q.93GP**

Find an expression for the kinetic energy of a satellite of mass m in an orbit of radius r about a planet of mass M.

**Solution:**

**Chapter 12 Gravity Q.94GP**

Referring to Example 12-1, find the x component of the net force acting on the Millennium Eagle as a function of x. Plot your result, showing both negative and positive values of x.

**Solution:**

**Chapter 12 Gravity Q.95GP**

A satellite orbits the Earth in an elliptical orbit. At perigee its distance from the center of the Earth is 22,500 km and its speed is 4280 m/s. At apogee its distance from the center of the Earth is 24,100 km and its speed is 3990 m/s. Using this information, calculate the mass of the Earth.

**Solution:**

**Chapter 12 Gravity Q.96PP**

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**Chapter 12 Gravity Q.97PP**

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**Chapter 12 Gravity Q.98PP**

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**Chapter 12 Gravity Q.99PP**

**Solution:**

**Chapter 12 Gravity Q.100IP**

Find the orbital radius that corresponds to a “year” of 150 days.

**Solution:**

**Chapter 12 Gravity Q.101IP**

Suppose the mass of the Sun is suddenly doubled, but the Earth’s orbital radius remains the same. (a) Would the length of an Earth year increase, decrease, or stay the same? (b) Find the length of a year for the case of a Sun with twice the mass. (c) Suppose the Sun retains its present mass, but the mass of the Earth is doubled instead. Would the length of the year increase, decrease, or stay the same?

**Solution:**

**Chapter 12 Gravity Q.102IP**

(a) If the mass of the Earth were doubled, would the escape speed of a rocket increase, decrease, or stay the same? (b) Calculate the escape speed of a rocket for the case of an Earth with twice its present mass. (c) If the mass of the Earth retains its present value, but the mass of the rocket is doubled, does the escape speed increase, decrease, or stay the same?

**Solution:**

The escape speed depends on the mass of Earth, radius of Earth, and the universal gravitational constant. However, it does not depend on the mass of the rocket.

**Chapter 12 Gravity Q.103IP**

Suppose the Earth is suddenly shrunk to half its present radius without losing any of its mass. (a) Would the escape speed of a rocket increase, decrease, or stay the same? (b) Find the escape speed for an Earth with half its present radius.

**Solution:**