Exact solutions for gravitational problems involving more than two bodies are

What do Newtons apple and the Moon have in common?

Explain the difference between G and g.

When you stand on Earth, the distance between you and Earth is zero. So why isnt the gravitational force infinite?

The force of gravity on an object is proportional to the objects mass, yet all objects fall with the same gravitational acceleration. Why?

A friend who knows nothing about physics asks what keeps an orbiting satellite from falling to Earth. Give an answer that will satisfy your friend.

Could you put a satellite in an orbit that keeps it stationary over the south pole? Explain.

Why are satellites generally launched eastward and from low latitudes? (Hint: Think about Earths rotation.)

Given Earths mass, the Moons distance and orbital period, and the value of G, could you calculate the Moons mass? If yes, how? If no, why not?

How should a satellite be launched so that its orbit takes it over every point on the (rotating) Earth?

Does the gravitational force of the Sun do work on a planet in a circular orbit? In an elliptical orbit? Explain.

Space explorers land on a planet that has the same mass as Earth, but find they weigh twice as much as they would on Earth. Whats the planets radius?

Use data for the Moons orbit from Appendix E to compute the Moons acceleration in its circular orbit, and verify that the result is consistent with Newtons law of gravitation.

To what fraction of its current radius would Earth have to shrink (with no change in mass) for the gravitational acceleration at its surface to triple?

Calculate the gravitational acceleration at the surface of (a) Mercury and (b) Saturns moon Titan.

Two identical lead spheres with their centers 14 cm apart attract each other with a force. Find their mass.

Whats the approximate value of the gravitational force between a 67-kg astronaut and a 73,000-kg spacecraft when theyre 84 m apart?

A sensitive gravimeter is carried to the top of Chicagos Willis (formerly Sears) Tower, where its reading for the acceleration of gravity is lower than at street level. Find the buildings height.

At what altitude will a satellite complete a circular orbit of Earth in 2.0 h?

Find the speed of a satellite in geosynchronous orbit.

Marss orbit has a diameter 1.52 times that of Earths orbit. How long does it take Mars to orbit the Sun?

Calculate the orbital period for Jupiters moon Io, which orbits 4.223105 km from the planets center.

An astronaut hits a golf ball horizontally from the top of a lunar mountain so fast that it goes into circular orbit. Whats its orbital period?

The Mars Reconnaissance Orbiter circles the red planet with a 112-min period. Whats the spacecrafts altitude?

Earths distance from the Sun varies from 147 Gm at perihelion to 152 Gm at aphelion because its orbit isnt quite circular. Find the change in potential energy as Earth goes from perihelion to aphelion.

So-called suborbital missions take scientific instruments into space for brief periods without the expense of getting into orbit; their trajectories are often simple up and down vertical paths. How much energy does it take to launch a 230-kg instrument on a vertical trajectory that peaks at 1800 km altitude?

A rocket is launched vertically upward from Earths surface at 5.1 km/s. Whats its maximum altitude?

What vertical launch speed is necessary to get a rocket to an altitude of 1100 km?

Find the energy necessary to put 1 kg, initially at rest on Earths surface, into geosynchronous orbit.

Whats the total mechanical energy associated with Earths orbital motion?

The escape speed from a planet of mass is 7.1 km/s. Find the planets radius.

Determine escape speeds from (a) Jupiters moon Callisto and (b) a neutron star, with the Suns mass crammed into a sphere of radius 6.0 km. See Appendix E for relevant data.

To what radius would Earth have to shrink, with no change in mass, for escape speed at its surface to be 30 km/s?

The gravitational acceleration at a planets surface is Find the acceleration at an altitude equal to half the planets radius.

One of the longest-standing athletic records is Cuban Javier Sotomayors 2.45-m high jump. How high could Sotomayor jump on (a) Mars and (b) Earths Moon?

Youre the navigator on a spaceship studying an unexplored planet. Your ship has just gone into a circular orbit around the planet, and you determine that the gravitational acceleration at your orbital altitude is half what it would be at the surface. What do you report for your altitude, in terms of the planets radius?

If youre standing on the ground 15 m directly below the center of a spherical water tank containing of water, by what fraction is your weight reduced due to the waters gravitational attraction?

Given the Moons orbital radius of 384,400 km and period of 27.3 days, calculate its acceleration in its circular orbit, and compare with the acceleration of gravity at Earths surface. Show that the Moons acceleration is lower by the ratio of the square of Earths radius to the square of the Moons orbital radius, thus confirming the inverse-square law for the gravitational force.

Equation 7.8 relates force to the derivative of potential energy. Use this fact to differentiate Equation 8.6 for gravitational potential energy, and show that you recover Newtons law of gravitation.

During the Apollo Moon landings, one astronaut remained with the command module in lunar orbit, about 130 km above the surface. For half of each orbit, this astronaut was completely cut off from the rest of humanity as the spacecraft rounded the far side of the Moon. How long did this period last?

A white dwarf is a collapsed star with roughly the Suns mass compressed into the size of Earth. What would be (a) the orbital speed and (b) the orbital period for a spaceship in orbit just above the surface of a white dwarf?

Given that our Sun orbits the galaxy with a period of 200 My at 20 2.6 310 m from the galactic center, estimate the galaxys mass. Assume (incorrectly) that the galaxy is essentially spherical and that most of its mass lies interior to the Suns orbit.

Youre preparing an exhibit for the Golf Hall of Fame, and you realize that the longest golf shot in history was Astronaut Alan Shepards lunar drive. Shepard, swinging single-handed with a golf club attached to a lunar sample scoop, claimed his ball went miles and miles. The record for a single-handed golf shot on Earth is 257 m. Could Shepards ball really have gone miles and miles? Assume the balls initial speed is independent of gravita- tional acceleration.

Exact solutions for gravitational problems involving more than two bodies are notoriously difficult. One solvable problem involves a configuration of three equal-mass objects spaced in an equilateral triangle. Forces due to their mutual gravitation cause the configuration to rotate. Suppose three identical stars, each of mass M, form a triangle of side L. Find an expression for the period of their orbital motion.

Satellites A and B are in circular orbits, with A twice as far from Earths center as B. How do their orbital periods compare?

The asteroid Pasachoff orbits the Sun with period 1417 days. Find the semimajor axis of its orbit from Keplers third law. Use Earths orbital radius and period, respectively, as your units of distance and time.

We still dont have a permanent solution for the disposal of radioactive waste. As a nuclear waste specialist with the Department of Energy, youre asked to evaluate a proposal to shoot waste canisters into the Sun. You need to report the speed at which a canister, dropped from rest in the vicinity of Earths orbit, would hit the Sun. Whats your answer?

At perihelion in February 1986, Comet Halley was from the Sun and was moving at 54.6 km/s. What was Halleys speed when it crossed Neptunes orbit in 2006? 7 8.79 310 km

Neglecting air resistance, to what height would you have to fire a rocket for the constant-acceleration equations of Chapter 2 to give a height in error by 1%? Would those equations overestimate or underestimate the height?

Show that an object released from rest very far from Earth reaches Earths surface at essentially escape speed.

By what factor must an objects speed in circular orbit be increased to reach escape speed from its orbital altitude?

Youre in charge of tracking celestial objects that might pose a danger to Earth. Astronomers have discovered a new comet thats moving at 45 km/s as it crosses Earths orbit. Determine whether the comet will again return to Earths vicinity.

Two meteoroids are 250,000 km from Earths center and moving at 2.1 km/s. One is headed straight for Earth, while the other is on a path that will come within 8500 km of Earths center (Fig. 8.15). Find the speed (a) of the first meteoroid when it strikes Earth and (b) of the second meteoroid at its closest approach. (c) Will the second meteoroid ever return to Earths vicinity?

Neglecting Earths rotation, show that the energy needed to launch a satellite of mass m into circular orbit at altitude h is

A projectile is launched vertically upward from a planet of mass M and radius R; its initial speed is twice the escape speed. Derive an expression for its speed as a function of the distance r from the planets center.

A spacecraft is in circular orbit 5500 km above Earths surface. How much will its altitude decrease if it moves to a new circular orbit where (a) its orbital speed is 10% higher or (b) its orbital period is 10% shorter?

Two meteoroids are 160,000 km from Earths center and heading straight toward Earth, one at 10 km/s, the other at 20 km/s. At what speeds will they strike Earth?

Two rockets are launched from Earths surface, one at 12 km/s and the other at 18 km/s. How fast is each moving when it crosses the Moons orbit?

A satellite is in an elliptical orbit at altitudes ranging from 230 to 890 km. At its highest point, its moving at 7.23 km/s. How fast is it moving at its lowest point?

A missiles trajectory takes it to a maximum altitude of 1200 km. If its launch speed is 6.1 km/s, how fast is it moving at the peak of its trajectory?

A 720-kg spacecraft has total energy TJ and is in circular orbit around the Sun. Find (a) its orbital radius, (b) its kinetic energy, and (c) its speed.

Mercurys orbital speed varies from 38.8 km/s at aphelion to 59.0 km/s at perihelion. If the planet is from the Suns center at aphelion, how far is it at perihelion?

Show that the form follows from Equation 8.5 when [Hint: Write and apply the binomial approximation (Appendix A).]

Two satellites are in geosynchronous orbit but in diametrically opposite positions (Fig. 8.16). In order to catch up with the other, one satellite descends into a lower circular orbit (see Conceptual Example 8.1 for a description of this maneuver). How far should it descend if its to catch up in 10 orbits? Neglect rocket firing times and time spent moving between the two circular orbits.

We derived Equation 8.4 on the assumption that the massive gravitating center remains fixed. Now consider two objects with equal mass M orbiting each other, as shown in Fig. 8.17. Show that the orbital period is given by where d is the distance between the objects.

Tidal effects in the Earth-Moon system cause the Moons orbital period to increase at a current rate of about 35 ms per century. Assuming the Moons orbit is circular, to what rate of change in the Earth-Moon distance does this correspond? (Hint: Differentiate Keplers third law, Equation 8.4, and consult Appendix E.)

As a member of the 2040 Olympic committee, youre considering a new sport: asteroid jumping. On Earth, world-class high jumpers routinely clear 2 m. Your job is to make sure athletes jumping from asteroids will return to the asteroid. Make the simplifying assumption that asteroids are spherical, with average density For safety, make sure even a jumper capable of 3 m on Earth will return to the surface. What do you report for the minimum asteroid diameter?

The Olympic Committee is keeping you busy! Youre now asked to consider a proposal for lunar hockey. The record speed for a hockey puck is 168 km/h. Is there any danger that hockey pucks will go into lunar orbit?

Tidal forces are proportional to the variation in gravity with position. By differentiating Equation 8.1, estimate the ratio of the tidal forces due to the Sun and the Moon. Compare your answer with the ratio of the gravitational forces that the Sun and Moon exert on Earth. Use data from Appendix E.

Spacecraft that study the Sun are often placed at the so-called L1 Lagrange point, located sunward of Earth on the Sun-Earth line. L1 is the point where Earths and Suns gravity together produce an orbital period of one year, so that a spacecraft at L1 stays fixed relative to Earth as both planet and spacecraft orbit the Sun. This placement ensures an uninterrupted view of the Sun, without being periodically eclipsed by Earth as would occur in Earth orbit. Find L1s location relative to Earth. (Hint: This problem calls for numerical methods or solving a higher-order polynomial equation.)

Whats the approximate orbital period of GPS satellites? a. 90 min b. 8 h c. 12 h d. 24 h e. 1 week

Whats the approximate speed of GPS satellites? a. 9.8 m/s b. 500 m/s c. 1.7 km/s d. 4 km/s e. 12 km/s

Whats the approximate escape speed at GPS orbital distance? a. 4 km/s b. 5.5 km/s c. 6.3 km/s d. 9.8 km/s e. 11 km/s

The current generation of GPS satellites has masses of 844 kg. Whats the approximate total energy of such a satellite?