University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 12

Gravitation - all with Video Answers

Educators

Chapter Questions

Problem 1

A planet is in a circular orbit about a remote star, far from any other object in the universe. Which of the following statements is true?
a) There is only one force acting on the planet.
b) There are two forces acting on the planet and their resultant is zero.
c) There are two forces acting on the planet and their resultant is not zero.
d) None of the above statements are true.

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Two 30.0 -kg masses are held at opposite corners of a square of sides $20.0 \mathrm{~cm}$. If one of the masses is released and allowed to fall toward the other mass, what is the acceleration of the first mass just as it is released? Assume that the only force acting on the mass is the gravitational force of the other mass.
a) $1.52 \cdot 10^{-7} \mathrm{~m} / \mathrm{s}^{2}$
c) $7.50 \cdot 10^{-7} \mathrm{~m} / \mathrm{s}^{2}$
b) $2.50 \cdot 10^{-7} \mathrm{~m} / \mathrm{s}^{2}$
d) $3.73 \cdot 10^{-7} \mathrm{~m} / \mathrm{s}^{2}$

Narayan Hari

Numerade Educator

Problem 3

With the usual assumption that the gravitational potential energy goes to zero at infinite distance, the gravitational potential energy due to the Earth at its center is
a) positive.
c) zero.
b) negative.
d) undetermined.

Narayan Hari

Numerade Educator

Problem 4

12.4 A man inside a sturdy box is fired out of a cannon. Which of the following statements regarding the man's sensation of weightlessness is correct?
a) The man senses weightlessness only when he and the box are traveling upward.
b) The man senses weightlessness only when he and the box are traveling downward.
c) The man senses weightlessness when he and the box are traveling both upward and downward.
d) The man does not sense weightlessness at any time of the flight.

Ajay Singhal

Numerade Educator

Problem 5

In a binary star system consisting of two stars of equal mass, where is the gravitational potential equal to zero?
a) exactly halfway between the stars
b) along a line bisecting the line connecting the stars
c) infinitely far from the stars
d) none of the above

Narayan Hari

Numerade Educator

Problem 6

Two planets have the same mass, $M,$ but one of them is much denser than the other. Identical objects of mass $m$ are placed on the surfaces of the planets. Which object will have the gravitational potential energy of larger magnitude?
a) Both objects will have the same gravitational potential energy.
b) The object on the surface of the denser planet will have the larger gravitational potential energy.
c) The object on the surface of the less dense planet will have the larger gravitational potential energy.
d) It is impossible to tell.

Narayan Hari

Numerade Educator

Problem 7

Two planets have the same mass, $M .$ Each planet has a constant density, but the density of planet 2 is twice as high as that of planet $1 .$ Identical objects of mass $m$ are placed on the surfaces of the planets. What is the relationship of the gravitational potential energy on planet $1\left(U_{1}\right)$ to that on planet $2\left(U_{2}\right) ?$
a) $U_{1}=U_{2}$
d) $U_{1}=1.26 U_{2}$
b) $U_{1}=\frac{1}{2} U_{2}$
e) $U_{1}=0.794 U_{2}$
c) $U_{1}=2 U_{2}$

Ajay Singhal

Numerade Educator

Problem 8

For two identical satellites in circular motion around the Earth, which statement is true?
a) The one in the lower orbit has less total energy.
b) The one in the higher orbit has more kinetic energy.
c) The one in the lower orbit has more total energy.
d) Both have the same total energy.

Ajay Singhal

Numerade Educator

Problem 9

Which condition do all geostationary satellites orbiting the Earth have to fulfill?
a) They have to orbit above the Equator.
b) They have to orbit above the poles.
c) They have to have an orbital radius that locates them less than $30,000 \mathrm{~km}$ above the surface.

Narayan Hari

Numerade Educator

Problem 10

An object is placed between the Earth and the Moon, along the straight line that joins their centers. About how far away from the center of the Earth should the object be placed so that the net gravitational force on the object from the Earth and the Moon is zero?
a) halfway to the center of the Moon
b) $60 \%$ of the way to the center of the Moon
c) $70 \%$ of the way to the center of the Moon
d) $85 \%$ of the way to the center of the Moon
e) $90 \%$ of the way to the center of the Moon

Ajay Singhal

Numerade Educator

Problem 11

A man of mass $100 .$ kg feels a gravitational force, $F_{\mathrm{m}}$, from a woman of mass $50.0 \mathrm{~kg}$ sitting $1 \mathrm{~m}$ away. The gravitational force, $F_{w}$ experienced by the woman will be ___________ that experienced by the man.
a) more than
b) less than
c) the same as
d) not enough information given

Narayan Hari

Numerade Educator

Problem 12

Halfway between Earth's center and its surface, the gravitational acceleration is
a) zero.
b) $g / 4$.
c) $g / 2$.
d) $2 g$.
e) $4 g$.

Narayan Hari

Numerade Educator

Problem 13

The best estimate of the orbital period of the Solar System around the center of the Milky Way is between 220 and 250 million years. How much mass (in terms of solar masses) is enclosed by the 26,000 light-years (1.7 $\cdot 10^{9} \mathrm{AU}$ ) radius of the Solar System's orbit? (An orbital period of $1 \mathrm{yr}$ for an orbit of radius 1 AU corresponds to 1 solar mass.)
a) 90 billion solar masses
b) 7.2 billion solar masses
c) 52 million solar masses
d) 3.7 million solar masses
e) 432,000 solar masses

Narayan Hari

Numerade Educator

Problem 14

Imagine a large hollow sphere with mass $M$ and outer radius $R$ located in outer space. The hollow sphere has a thickness $t$, where $t \ll R$. What is the gravitational force on an object with mass $m$ on the outer and inner surfaces of the hollow sphere, respectively?
a) zero, zero
b) $m M G / R^{2}$, zero
c) zero, $m M G /(R-t)^{2}$
d) $m M G / R^{2}, m M G /(R-t)^{2}$
e) zero, $m M G / R^{2}$

Narayan Hari

Numerade Educator

Problem 15

The ratio of the mass of the Earth to the mass of the Moon is $81 .$ The
magnitude of the gravitational force that the Earth exerts on the Moon is
a) the same as the magnitude of the gravitational force that the Moon exerts on the Earth.
b) 81 times the gravitational force exerted by the Moon on the Earth.
c) $1 / 81$ of the gravitational force exerted by the Moon on the Earth.
d) 9 times the gravitational force exerted by the Moon on the Earth.
e) $1 / 9$ of the gravitational force exerted by the Moon on the Earth.

Narayan Hari

Numerade Educator

Problem 16

Can the expression for gravitational potential energy $U_{g}(y)=m g y$ be used to analyze high-altitude motion? Why or why not?

Ajay Singhal

Numerade Educator

Problem 17

Even though the Moon does not have an atmosphere, the trajectory of a projectile near its surface is only approximately a parabola. This is because the acceleration due to gravity near the surface of the Moon is only approximately constant. Describe as precisely as you can the actual shape of a projectile's path on the Moon, even one that travels a long distance over the surface of the Moon.

Ajay Singhal

Numerade Educator

Problem 18

A scientist working for a space agency has noticed that a Russian satellite of mass $250 . \mathrm{kg}$ is on collision course with an American satellite of mass $600 .$ kg orbiting at $1000 . \mathrm{km}$ above the surface. Both satellites are moving in circular orbits but in opposite directions. If the two satellites collide and stick together, will they continue to orbit or crash to the Earth? Explain.

Lydia Guertin

Numerade Educator

Problem 19

Three asteroids, located at points $P_{1}, P_{2}$, and $P_{3}$, which are not in a line, and having known masses $m_{1}, m_{2}$, and $m_{3}$, interact with one another through their mutual gravitational forces only; they are isolated in space and do not interact with any other bodies. Let $\sigma$ denote the axis going through the center of mass of the three asteroids, perpendicular to the triangle $P_{1} P_{2} P_{3}$. What conditions should the angular velocity $\omega$ of the system (around the axis $\sigma$ ) and the distances
$$
P_{1} P_{2}=a_{12}, \quad P_{2} P_{3}=a_{23}, \quad P_{1} P_{3}=a_{13},
$$
fulfill to allow the shape and size of the triangle $P_{1} P_{2} P_{3}$ to remain unchanged during the motion of the system? That is, under what conditions does the system rotate around the axis $\sigma$ as a rigid body?

Lydia Guertin

Numerade Educator

Problem 20

The more powerful the gravitational force of a planet, the greater its escape speed, $v$, and the greater the gravitational acceleration, $g$, at its surface. However, in Table 12.1 , the value for $v$ is much greater for Uranus than for Earth-but $g$ is smaller on Uranus than on Earth! How can this be?

Narayan Hari

Numerade Educator

Problem 21

Is the orbital speed of the Earth when it is closest to the Sun greater than, less than, or equal to the orbital speed when it is farthest from the Sun? Explain.

Ajay Singhal

Numerade Educator

Problem 22

Point out any flaw in the following physics exam statement: "Kepler's First Law states that all planets move in elliptical orbits with the Sun at one focal point. It follows that during one complete revolution around the Sun (1 year), the Earth will pass through a closest point to the Sun-the perihelion-as well as through a furthest point from the Sun-the aphelion. This is the main cause of the seasons (summer and winter) on Earth."

Ajay Singhal

Numerade Educator

Problem 23

A comet orbiting the Sun moves in an elliptical orbit. Where is its kinetic energy, and therefore its speed, at a maximum $-$ at perihelion or aphelion? Where is its gravitational potential energy at a maximum?

Ajay Singhal

Numerade Educator

Problem 24

Where the International Space Station orbits, the gravitational acceleration is just $11.5 \%$ less than its value on the surface of the Earth. Nevertheless, astronauts in the space station float. Why is this so?

Ajay Singhal

Numerade Educator

Problem 25

Satellites in low orbit around the Earth lose energy from colliding with the gases of the upper atmosphere, causing them to slowly spiral inward. What happens to their kinetic energy as they fall inward?

Ajay Singhal

Numerade Educator

Problem 26

Compare the magnitudes of the gravitational force that the Earth exerts on the Moon and the gravitational force that the Moon exerts on the Earth. Which is larger?

Narayan Hari

Numerade Educator

Problem 27

Imagine that two tunnels are bored completely through the Earth, passing through the center. Tunnel 1 is along the Earth's axis of rotation, and tunnel 2 is in the equatorial plane, with both ends at the Equator. Two identical balls, each with a mass of $5.00 \mathrm{~kg}$, are simultaneously dropped into the tunnels. Neglect air resistance and friction from the tunnel walls. Do the balls reach the
center of the Earth (point $C$ ) at the same time? If not, which ball reaches the center of the Earth first?

Ajay Singhal

Numerade Educator

Problem 28

Imagine that a tunnel is bored in the Earth's equatorial plane, going completely through the center of the Earth with both ends at the Equator. A mass of $5.00 \mathrm{~kg}$ is dropped into the tunnel at one end, as shown in the figure. The tunnel has a radius that is slightly larger than that of the mass. The mass is dropped into the center of the tunnel. Neglect air resistance and friction from the tunnel wall. Does the mass ever touch the wall of the tunnel as it falls? If so, which side does it touch first, north, east, south, or west? (Hint: The angular momentum of the mass is conserved if the only forces acting on it are radial.)

Ajay Singhal

Numerade Educator

Problem 29

A plumb bob located at latitude $55.0^{\circ} \mathrm{N}$ hangs motionlessly with respect to the ground beneath it. A straight line from the string supporting the bob does not go exactly through the Earth's center. axis of rotation south or north of the Earth's center?

Ajay Singhal

Numerade Educator

Problem 30

The Moon causes tides because the gravitational force it exerts differs between the side of the Earth nearer to it and the side farther from it. Find the difference in the accelerations toward the Moon of objects on the nearer and farther sides of the Earth.

Narayan Hari

Numerade Educator

Problem 31

After a spacewalk, a $1.00-\mathrm{kg}$ tool is left $50.0 \mathrm{~m}$ from the center of gravity of a 20.0 -metric ton space station, orbiting along with it. How much closer to the space station will the tool drift in an hour due to the gravitational attraction of the space station?

Narayan Hari

Numerade Educator

Problem 32

a) What is the total force on $m_{1}$ due to $m_{2}, m_{3},$ and $m_{4}$ if all four masses are located at the corners of a square of side $a$ ? Let $m_{1}=m_{2}=m_{3}=m_{4}$
b) Sketch all the forces acting on $m_{1}$.

Narayan Hari

Numerade Educator

Problem 33

Assume that $L$ is much larger than the radius of either planet. What is the position, $x$, of the spacecraft (given as a function of $L, M_{1}$, and $M_{2}$ ) if the net force on the spacecraft is zero?A spaceship of mass $m$ is located between two planets of masses $M_{1}$ and $M_{2}$; the distance between the two planets is $L$, as shown in the figure.

Narayan Hari

Numerade Educator

Problem 34

A carefully designed experiment can measure the gravitational force between masses of $1 \mathrm{~kg}$. Given that the density of iron is $7860 \mathrm{~kg} / \mathrm{m}^{3}$ what is the gravitational force between two 1.00 -kg iron spheres that are touching?

Narayan Hari

Numerade Educator

Problem 35

A uniform rod of mass $333 \mathrm{~kg}$ is in the shape of a semicircle of radius $5.00 \mathrm{~m}$ Calculate the magnitude of the force on a $77.0-\mathrm{kg}$ point mass placed at the center of the semicircle, as shown in the figure.

Ajay Singhal

Numerade Educator

Problem 36

The figure shows a system of four masses. The center-to-center distance between any two of the masses is $10.0 \mathrm{~cm} .$ The base of the pyramid is in the $x z$ -plane and the 20.0 -kg mass is on the $y$ -axis. What is the magnitude and direction of the gravitational force acting on the 10.0 -kg mass? Give direction of the net force with respect to the $x y z$ -coordinates shown.

Lydia Guertin

Numerade Educator

Problem 37

Suppose a new extrasolar planet is discovered. Its mass is double the mass of the Earth, but it has the same density and spherical shape as the Earth. How would the weight of an object at the new planet's surface differ from its weight on Earth?

Narayan Hari

Numerade Educator

Problem 38

What is the magnitude of the free-fall acceleration of a ball (mass $m)$ due to the Earth's gravity at an altitude of $2 R,$ where $R$ is the radius of the Earth? Ignore the rotation of the Earth.

Narayan Hari

Numerade Educator

Problem 39

Some of the deepest mines in the world are in South Africa and are roughly $3.5 \mathrm{~km}$ deep. Consider the Earth to be a uniform sphere of radius $6370 \mathrm{~km}$
a) How deep would a mine shaft have to be for the gravitational acceleration at the bottom to be reduced by a factor of 2 from its value on the Earth's surface?
b) What is the percentage difference in the gravitational acceleration at the bottom of the $3.5-\mathrm{km}$ -deep shaft relative to that at the Earth's mean radius? That is, what is the value of $\left(a_{\text {surf }}-a_{3.5 \mathrm{~km}}\right) / a_{\text {surf }} ?$

Narayan Hari

Numerade Educator

Problem 40

In an experiment performed at the bottom of a very deep vertical mine shaft, a ball is tossed vertically in the air with a known initial velocity of $10.0 \mathrm{~m} / \mathrm{s}$, and the maximum height the ball reaches (measured from its launch point) is determined to be $5.113 \mathrm{~m}$. Knowing the radius of the Earth, $R_{\mathrm{E}}=6370 \mathrm{~km},$ and the gravitational acceleration at the surface of the Earth, $g(0)=9.81 \mathrm{~m} / \mathrm{s}^{2},$ calculate the depth of the shaft.

Narayan Hari

Numerade Educator

Problem 41

Careful measurements of local variations in the acceleration due to gravity can reveal the locations of oil deposits. Assume that the Earth is a uniform sphere of radius $6370 \mathrm{~km}$ and density $5500 \mathrm{~kg} / \mathrm{m}^{3},$ except that there is a spherical region of radius $1.00 \mathrm{~km}$ and density $900 . \mathrm{kg} / \mathrm{m}^{3},$ whose center is at a depth of $2.00 \mathrm{~km}$. Suppose you are standing on the surface of the Earth directly above the anomaly with an instrument capable of measuring the acceleration due to gravity with great precision. What is the fractional deviation of the acceleration due to gravity that you measure compared to what you would have measured had the density been $5500 . \mathrm{kg} / \mathrm{m}^{3}$ everywhere? (Hint: Think of this as a superposition problem involving two uniform spherical masses, one with a negative density.

Lydia Guertin

Numerade Educator

Problem 42

A spaceship is launched from the Earth's surface with a speed $v$. The radius of the Earth is $R$. What will its speed be when it is very far from the Earth?

Narayan Hari

Numerade Educator

Problem 43

What is the ratio of the escape speed to the orbital speed of a satellite at the surface of the Moon, where the gravitational acceleration is about a sixth of that on Earth?

Ajay Singhal

Numerade Educator

Problem 44

Standing on the surface of a small spherical moon whose radius is $6.30 \cdot 10^{4} \mathrm{~m}$ and whose mass is $8.00 \cdot 10^{18} \mathrm{~kg}$, an astronaut throws a rock of mass $2.00 \mathrm{~kg}$ straight upward with an initial speed $40.0 \mathrm{~m} / \mathrm{s}$. (This moon is too small to have an atmosphere.) What maximum height above the surface of the moon will the rock reach?

Narayan Hari

Numerade Educator

Problem 45

object of mass $m$ is launched from the surface of the Earth. Show that the minimum speed required to send the projectile to a height of $4 R_{\mathrm{E}}$ above the surface of the Earth is $v_{\min }=\sqrt{8 G M_{\mathrm{E}} / 5 R_{\mathrm{E}}} \cdot M_{\mathrm{E}}$ is the mass of the Earth and $R_{\mathrm{E}}$ is the radius of the Earth. Neglect air resistance.

Narayan Hari

Numerade Educator

Problem 46

For the satellite in Solved Problem $12.3,$ orbiting the Earth at a distance of $3.75 R_{\mathrm{E}}$ with a speed of $4.08 \mathrm{~km} / \mathrm{s}$, with what speed would the satellite hit the Earth's surface if somehow it suddenly stopped and fell to Earth? Ignore air resistance.

Narayan Hari

Numerade Educator

Problem 47

Estimate the radius of the largest asteroid from which you could escape by jumping. Assume spherical geometry and a uniform density equal to the Earth's average density.

Narayan Hari

Numerade Educator

Problem 48

Eris, the largest dwarf planet known in the Solar System, has a radius $R=1200 \mathrm{~km}$ and an acceleration due to gravity on its surface of magnitude $g=0.77 \mathrm{~m} / \mathrm{s}^{2}$.
a) Use these numbers to calculate the escape speed from the surface of Eris.
b) If an object is fired directly upward from the surface of Eris with half of this escape speed, to what maximum height above the surface will the object rise? (Assume that Eris has no atmosphere and negligible rotation.)

Ajay Singhal

Numerade Educator

Problem 49

Two identical $20.0-\mathrm{kg}$ spheres of radius $10.0 \mathrm{~cm}$ are $30.0 \mathrm{~cm}$ apart (center-to-center distance).
a) If they are released from rest and allowed to fall toward one another, what is their speed when they first make contact?
b) If the spheres are initially at rest and just touching, how much energy is required to separate them to $1.00 \mathrm{~m}$ apart? Assume that the only force acting on each mass is the gravitational force due to the other mass.

Narayan Hari

Numerade Educator

Problem 50

Imagine that a tunnel is bored completely through the Earth along its axis of rotation. A ball with a mass of $5.00 \mathrm{~kg}$ is dropped from rest into the tunnel at the North Pole, as shown in the figure. Neglect air resistance and friction from the tunnel wall. Calculate the potential energy of the ball as a function of its distance from the center of the Earth. What is the speed of the ball when it arrives at the center of the Earth (point $C) ?$

Ajay Singhal

Numerade Educator

Problem 51

The Apollo 8 mission in 1968 included a circular orbit at an altitude of $111 \mathrm{~km}$ above the Moon's surface. What was the period of this orbit? (You need to look up the mass and radius of the Moon to answer this question!)

Ajay Singhal

Numerade Educator

Problem 52

Halley's comet orbits the Sun with a period of 75.3 yr.
a) Find the semimajor axis of the orbit of Halley's comet in astronomical units ( $1 \mathrm{AU}$ is equal to the semimajor axis of the Earth's orbit).
b) If Halley's comet is 0.586 AU from the Sun at perihelion, what is its maximum distance from the Sun, and what is the eccentricity of its orbit?

Ajay Singhal

Numerade Educator

Problem 53

A satellite of mass $m$ is in an elliptical orbit (that satisfies Kepler's laws) about a body of mass $M,$ with $m$ negligible compared to $M$.
a) Find the total energy of the satellite as a function of its speed, $v$, and distance, $r,$ from the body it is orbiting.
b) At the maximum and minimum distance between the satellite and the body, and only there, the angular momentum is simply related to the speed and distance. Use this relationship and the result of part (a) to eliminate $v$ and obtain a relationship between the extreme distance $r$ and the satellite's energy and angular momentum.
c) Solve the result of part (b) for the maximum and minimum radii of the orbit in terms of the energy and angular momentum per unit mass of the satellite.
d) Transform the results of part (c) into expressions for the semimajor axis, $a,$ and eccentricity of the orbit, $e$, in terms of the energy and angular momentum per unit mass of the satellite.

Linda Winkler

Numerade Educator

Problem 54

Consider the Sun to be at the origin of an $x y$ -coordinate system. A telescope spots an asteroid in the $x y$ -plane at a position given by $\left(2.00 \cdot 10^{11} \mathrm{~m},\right.$ $\left.3.00 \cdot 10^{11} \mathrm{~m}\right)$ with a velocity given by $\left(-9.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s},-7.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\right)$. What will the asteroid's speed and distance from the Sun be at closest approach?

Linda Winkler

Numerade Educator

Problem 55

A spy satellite was launched into a circular orbit with a height of $700 . \mathrm{km}$ above the surface of the Earth. Determine its orbital speed and period.

Narayan Hari

Numerade Educator

Problem 56

Express algebraically the ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun. Why, since the ratio is so small, doesn't the Sun pull the Moon away from the Earth?

Narayan Hari

Numerade Educator

Problem 57

A space shuttle is initially in a circular orbit at a radius of $r=6.60 \cdot 10^{6} \mathrm{~m}$ from the center of the Earth. A retrorocket is fired forward reducing the total energy of the space shuttle by $10.0 \%$ (that is, increasing the magnitude of the negative total energy by $10.0 \%$ ), and the space shuttle moves to a new circular orbit with a radius that is smaller than $r$. Find the speed of the space shuttle (a) before and (b) after the retrorocket is fired.

Ajay Singhal

Numerade Educator

Problem 58

A 200.-kg satellite is in circular orbit around the Earth and moving at a speed of $5.00 \mathrm{~km} / \mathrm{s}$. How much work must be done to move the satellite into another circular orbit that is twice as high above the surface of the Earth?

Narayan Hari

Numerade Educator

Problem 59

The radius of a black hole is the distance from the black hole's center at which the escape speed is the speed of light.
a) What is the radius of a black hole with a mass twice that of the Sun?
b) At what radius from the center of the black hole in part (a) would the orbital speed be equal to the speed of light?
c) What is the radius of a black hole with the same mass as that of the Earth?

Ajay Singhal

Numerade Educator

Problem 60

A satellite is in a circular orbit around a planet. The ratio of the satellite's kinetic energy to its gravitational potential energy, $K / U_{g}$, is a constant whose value is independent of the masses of the satellite and planet and of the radius and velocity of the orbit. Find the value of this constant. (Potential energy is taken to be zero at infinite separation.)

Narayan Hari

Numerade Educator

Problem 61

Determine the minimum amount of energy that a projectile of mass $100.0 \mathrm{~kg}$ must gain to reach a circular orbit $10.00 \mathrm{~km}$ above the Earth's surface if launched from (a) the North Pole or (b) the Equator (keep answers to four significant figures). Do not be concerned about the direction of the launch or of the final orbit. Is there an advantage or disadvantage to launching from the Equator? If so, how significant is the difference? Do not neglect the rotation of the Earth when calculating the initial energies. Use $5.974 \cdot 10^{24} \mathrm{~kg}$ for the mass of the Earth and $6357 \mathrm{~km}$ as the radius of the Earth.

Narayan Hari

Numerade Educator

Problem 62

A rocket with mass $M=12.0$ metric tons is moving around the Moon in a circular orbit at the height of $h=100 . \mathrm{km} .$ The braking engine is activated for a short time to lower the orbital height so that the rocket can make a lunar landing. The velocity of the ejected gases is $u=1.00 \cdot 10^{4} \mathrm{~m} / \mathrm{s}$ relative to the rocket's initial velocity. The Moon's radius is $R_{M}=1.74 \cdot 10^{3} \mathrm{~km} ;$ the acceleration of gravity near the Moon's surface is $g_{M}=1.62 \mathrm{~m} / \mathrm{s}^{2}$.
a) What amount of fuel will be used by the braking engine if it is activated at point $A$ of the orbit and the rocket lands on the Moon at point $B$ (see the left part of the figure)?
b) Suppose that, at point $A$, the rocket is given an impulse directed toward the center of the Moon, to put it on a trajectory that meets the Moon's surface at point $C$ (see the right part of the figure). What amount of fuel is needed in this case?

Dominador Tan

Numerade Educator

Problem 63

Calculate the magnitudes of the gravitational forces exerted on the Moon by the Sun and by the Earth when the two forces are in direct competition, that is, when the Sun, Moon, and Earth are aligned with the Moon between the Sun and the Earth. (This alignment corresponds to a solar eclipse.) Does the orbit of the Moon ever actually curve away from the Sun, toward the Earth?

Ajay Singhal

Numerade Educator

Problem 64

A projectile is shot vertically from the surface of the Earth by means of a very powerful cannon. If the projectile reaches a height of $55.0 \mathrm{~km}$ above Earth's surface, what was the speed of the projectile when it left the cannon?

Narayan Hari

Numerade Educator

Problem 65

Newton's Law of Gravity specifies the magnitude of the interaction force between two point masses, $m_{1}$ and $m_{2}$, separated by a distance $r$ as $F(r)=G m_{1} m_{2} / r^{2} .$ The gravitational constant $G$ can be determined by directly measuring the interaction force (gravitational attraction) between two sets of spheres by using the apparatus constructed in the late 18th century by the English scientist Henry Cavendish. This apparatus was a torsion balance consisting 6.00-ft wooden rod suspended fr a torsion wire, with a lead sphere having a diameter of 2.00 in and weight of $1.61 \mathrm{lb}$ attached to each end. Two 12.0 -in, 348 -lb lead ball were located near the smaller bal about 9.00 in away, and held in place with a separate suspension system. Today's accepted value for $G$ is $6.674 \cdot 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}$ Determine the force of attraction between the larger and smaller balls that had to be measured by this balance. Compare this force to the weight of the small balls.

Ajay Singhal

Numerade Educator

Problem 66

Newton was holding an apple of mass $100 . \mathrm{g}$ and thinking about the gravitational forces exerted on the apple by himself and by the Sun. Calculate the magnitude of the gravitational force acting on the apple due to (a) Newton, (b) the Sun, and (c) the Earth, assuming that the distance from the apple to Newton's center of mass is $50.0 \mathrm{~cm}$ and Newton's mass is $80.0 \mathrm{~kg}$

Narayan Hari

Numerade Educator

Problem 67

A 1000.-kg communications satellite is released from a space shuttle to initially orbit the Earth at a radius of $7.00 \cdot 10^{6} \mathrm{~m}$. After being deployed, the satellite's rockets are fired to put it into a higher altitude orbit of radius $5.00 \cdot 10^{7} \mathrm{~m} .$ What is the minimum mechanical energy supplied by the rockets to effect this change in orbit?

Narayan Hari

Numerade Educator

Problem 68

Consider a 0.300 -kg apple (a) attached to a tree and (b) falling. Does the apple exert a gravitational force on the Earth? If so, what is the magnitude of this force?

Ajay Singhal

Numerade Educator

Problem 69

At what height $h$ above the Earth will a satellite moving in a circular orbit have half the period of the Earth's rotation about its own axis?

Ajay Singhal

Numerade Educator

Problem 70

In the Earth-Moon system, there is a point where the gravitational forces balance. Assume that the mass of the Moon is $\frac{1}{81}$ that of the Earth.
a) At what point, on a line between the Earth and the Moon, is the gravitational force exerted on an object by the Earth exactly balanced by the gravitational force exerted on the object by the Moon?
b) Is this point one of stable or unstable equilibrium?
c) Calculate the ratio of the force of gravity due to the Sun acting on an object at this point to the force of gravity due to Earth and, separately, to the force of gravity due to the Moon.

Ajay Singhal

Numerade Educator

Problem 71

Consider a particle on the surface of the Earth, at a position with an angle of latitude $\lambda=30.0^{\circ} \mathrm{N}$ as shown in the figure. For this problem, assume that the Earth is a sphere with radius $R=6.37 \cdot 10^{6} \mathrm{~m}$ and that $g=9.81 \mathrm{~m} / \mathrm{s}^{2} .$ Find $(\mathrm{a})$ the magnitude, and (b) the direction of the effective gravitational force acting on the particle, taking into consideration the rotation of the Earth. (c) What angle $\lambda$ gives rise to the maximum deviation of the
gravitational acceleration?

Linda Winkler

Numerade Educator

Problem 72

An asteroid is discovered to have a tiny moon that orbits it in a circular path at a distance of $100 . \mathrm{km}$ and with a period of $40.0 \mathrm{~h}$. The asteroid is roughly spherical (unusual for such a small body) with a radius of $20.0 \mathrm{~km}$.
a) Find the acceleration of gravity at the surface of the asteroid.
b) Find the escape speed from the asteroid.

Ajay Singhal

Numerade Educator

Problem 73

a) By what percentage does the gravitational potential energy of the Earth change between perihelion and aphelion? (Assume that the Earth's potential energy would be zero if it moved to a very large distance away from the Sun.)
b) By what percentage does the kinetic energy of the Earth change between perihelion and aphelion?

Samuel Smith

Numerade Educator

Problem 74

A planet with a mass of $7.00 \cdot 10^{21} \mathrm{~kg}$ is in a circular orbit around a star with a mass of $2.00 \cdot 10^{30} \mathrm{~kg}$. The planet has an orbital radius of $3.00 \cdot 10^{10} \mathrm{~m}$
a) What is the linear orbital velocity of the planet?
b) What is the period of the planet's orbit?
c) What is the total mechanical energy of the planet?

Ajay Singhal

Numerade Educator

Problem 75

The astronomical unit (AU, equal to the mean radius of the Earth's orbit) is $1.4960 \cdot 10^{11} \mathrm{~m}$, and a year is $3.1557 \cdot 10^{7} \mathrm{~s}$. Newton's gravitational constant is $G=6.6738 \cdot 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2} .$ Calculate the mass of the Sun in kilograms. (Recalling or looking up the mass of the Sun does not constitute a solution to this problem.)

Narayan Hari

Numerade Educator

Problem 76

The distances from the Sun at perihelion and aphelion for Pluto are $4410 \cdot 10^{6} \mathrm{~km}$ and $7360 \cdot 10^{6} \mathrm{~km}$, respectively. What is the ratio of Pluto's orbital speed around the Sun at perihelion to that at aphelion?

Ajay Singhal

Numerade Educator

Problem 77

The weight of a star is usually balanced by two forces: the gravitational force, acting inward, and the force created by nuclear reactions, acting outward. Over a long period of time, the force due to nuclear reactions gets weaker, causing the gravitational collapse of the star and crushing atoms out of existence. Under such extreme conditions, protons and electrons are squeezed to form neutrons, giving birth to a neutron star. Neutron stars are massively heavy - a teaspoon of the substance of a neutron star would weigh 50 million metric tons on the Earth.
a) Consider a neutron star whose mass is twice the mass of the Sun and whose radius is $10.0 \mathrm{~km}$. If it rotates with a period of $1.00 \mathrm{~s}$, what is the speed of a point on the equator of this star? Compare this speed with the speed of a point on Earth's Equator.
b) What is the value of $g$ at the surface of this star?
c) Compare the weight of a 1.00 -kg mass on the Earth with its weight on the neutron star.
d) If a satellite is to circle $10.0 \mathrm{~km}$ above the surface of such a neutron star, how many revolutions per minute will it make?
e) What is the radius of the geostationary orbit for this neutron star?

Ajay Singhal

Numerade Educator

Problem 78

You have been sent in a small spacecraft to rendezvous with a space station that is in a circular orbit of radius $2.5000 \cdot 10^{4} \mathrm{~km}$ from the Earth's center. Due to a mishandling of units by a technician, you find yourself in the same orbit as the station but exactly halfway around the orbit from it! You do not apply forward thrust in an attempt to chase the station; that would be fatal folly. Instead, you apply a brief braking force against the direction of your motion, to put you into an elliptical orbit, whose highest point is your present position, and whose period is half that of your present orbit. Thus, you will return to your present position when the space station has come halfway around the circle to meet you. Is the minimum radius from the Earth's center-the low point-of your new elliptical orbit greater than the radius of the Earth $(6370 \mathrm{~km}),$ or have you botched your last physics problem?

Ajay Singhal

Numerade Educator

Problem 79

If you and the space station are initially in a low orbit-say, with a radius of $6720 \mathrm{~km}$, approximately that of the orbit of the International Space Station-the maneuver of Problem 12.78 will fail unpleasantly. Keeping in mind that the life-support capabilities of your small spacecraft are limited and so time is of the essence, can you perform a similar maneuver that will enable you to rendezvous with the station? Find the perihelion, aphelion, and period of the transfer orbit you should use.

Narayan Hari

Numerade Educator

Problem 80

A satellite is placed between the Earth and the Moon, along a straight line that connects their centers of mass. The satellite has an orbital period around the Earth that is the same as that of the Moon, 27.3 days. How far away from the Earth should this satellite be placed?

Ajay Singhal

Numerade Educator

Problem 81

An electromagnetic rail accelerator is used to launch a research probe vertically from the surface of the Moon. The initial speed of the projectile is $114.5 \mathrm{~m} / \mathrm{s}$. What height does it reach above the surface of the Moon? Assume that the radius of the Moon is $1737 \mathrm{~km}$ and the mass of the Moon is $7.348 \cdot 10^{22} \mathrm{~kg}$..

Ajay Singhal

Numerade Educator

Problem 82

An electromagnetic rail accelerator is used to launch a research probe vertically from the surface of the Moon. The probe reaches a height of $4.905 \mathrm{~km}$ above the surface of the Moon. What was the initial speed of the probe? Assume that the radius of the Moon is $1737 \mathrm{~km}$ and the mass of the Moon is $7.348 \cdot 10^{22} \mathrm{~kg}$.

Narayan Hari

Numerade Educator

Problem 83

A comet orbits the Sun with a period of 89.17 yr. At perihelion, the comet is 1.331 AU from the Sun. How far from the Sun (in AU) is the
comet at aphelion?

Ajay Singhal

Numerade Educator

Problem 84

A comet orbits the Sun with a period of 98.11 yr. At aphelion, the comet is 41.19 AU from the Sun. How far from the Sun (in AU) is the comet at perihelion?

Narayan Hari

Numerade Educator

Problem 85

A comet orbits the Sun. The aphelion of its orbit is $31.95 \mathrm{AU}$ from the Sun. The perihelion is $1.373 \mathrm{AU}$. What is the period (in years) of the comet's orbit?

Narayan Hari

Numerade Educator

Problem 86

A spherical asteroid has a mass of $1.869 \cdot 10^{20} \mathrm{~kg}$ and a radius of $358.9 \mathrm{~km} .$ What is the escape speed from its surface?

Narayan Hari

Numerade Educator

Problem 87

A spherical asteroid has a mass of $1.769 \cdot 10^{20} \mathrm{~kg} .$ The escape speed from its surface is $273.7 \mathrm{~m} / \mathrm{s}$. What is the radius of the asteroid?

Narayan Hari

Numerade Educator

Problem 88

A spherical asteroid has a radius of $365.1 \mathrm{~km}$. The escape speed from its surface is $319.2 \mathrm{~m} / \mathrm{s}$. What is the mass of the asteroid?

Narayan Hari

Numerade Educator