A straight, smooth tunnel is dug through a uniform

True or false: (a) For Keplers law of equal areas to be valid, the force of gravity must vary inversely with the square of the distance between a given planet and the Sun. (b) The planet closest to the Sun has the shortest orbital period. (c) Venuss orbital speed is larger than the orbital speed of Earth. (d) The orbital period of a planet allows accurate determination of that planets mass.

If the mass of a small Earth-orbiting satellite is doubled, the radius of its orbit can remain constant if the speed of the satellite (a) increases by a factor of 8, (b) increases by a factor of 2, (c) does not change, (d) is reduced by a factor of 8, (e) is reduced by a factor of 2.

During what season in the northern hemisphere does Earth attain its maximum orbital speed about the Sun? In what season does it attain its minimum orbital speed? Hint: Earth is at perihelion in early January.

Haleys comet is in a highly elliptical orbit about the Sun with a period of about 76 y. Its last closest approach to the Sun occurred in 1987. In what years of the twentieth century was it traveling at its fastest or slowest orbital speed about the Sun?

Venus has no natural satellites. However, artificial satellites have been placed in orbit around it. To use one of their orbits to determine the mass of Venus, what orbital parameters would you have to measure? How would you then use these parameters to do the mass calculation?

A majority of the asteroids are in approximately circular orbits in a belt between Mars and Jupiter. Do they all have the same orbital period about the Sun? Explain.

At the moons surface, the acceleration due to the gravity of the moon is a. At a distance from the moons center equal to four times the radius of the moon, the acceleration due to the gravity of the moon is (a) 16a, (b) (c) (d) (e) None of the above.

At a depth equal to half the radius of Earth, the acceleration due to gravity is about (a) g (b) 2g (c) (d) (e) ( ) You cannot determine the answer based on the data given.

Two stars orbit their common center of mass as a binary star system. If each of their masses were doubled, what would have to happen to the distance between them in order to maintain the same gravitational force? The distance would have to (a) remain the same, (b) double, (c) quadruple, (d) be reduced by a factor of 2, (e) You cannot determine the answer based on the data given.

CONTEXT-RICH If you had been working for NASA in the 1960s and planning the trip to the moon, you would have determined that a unique location exists somewhere between Earth and the moon, where a spaceship is, for an instant, truly weightless. (Consider only the moon, Earth, and the Apollo spaceship, and neglect other gravitational forces.) Explain this phenomenon and explain whether this location is closer to the moon, midway on the trip, or closer to Earth.

Suppose the escape speed from a planet is only slightly larger than the escape speed from Earth, yet the planet is considerably larger than Earth. How would the planets (average) density compare to Earths (average) density? (a) It must be denser, (b) It must be less dense. (c) It must be the same density. (d) You cannot determine the answer based on the data given. SSM

Suppose that, using a telescope in your backyard, you discovered a distant object approaching the Sun, and were able to determine both its distance from the Sun and its speed. How would you be able to predict whether the object will remain bound to the solar system, or if it is an interstellar interloper that would come in, turn around, and escape, never to return?

CONTEXT-RICH, ENGINEERING APPLICATION Near the end of their useful lives, several large Earth-orbiting satellites have been maneuvered so they burn up as they enter Earths atmosphere. These maneuvers have to be done carefully so large fragments do not impact populated land areas. You are in charge of such a project. Assuming a satellite of interest has on-board propulsion, in what direction would you fire the rockets for a short burn time to start this downward spiral? What would happen to the kinetic energy, gravitational potential energy, and total mechanical energy following the burn as the satellite came closer and closer to Earth?

ENGINEERING APPLICATION During a trip back from the moon, the Apollo spacecraft fires its rockets to leave its lunar orbit. Then, it coasts back to Earth where it enters the atmosphere at high speed, survives a blazing reentry, and parachutes safely into the ocean. In what direction do you fire the rockets to initiate this return trip? Explain the changes in kinetic energy, gravitational potential, and total mechanical energy that occur to the spacecraft from the beginning to the end of this journey.

Explain why the gravitational field inside a solid sphere of uniform mass is directly proportional to r rather than inversely proportional to r.

In the movie 2001: A Space Odyssey, a spaceship containing two astronauts is on a long-term mission to Jupiter. Amodel of their ship could be a uniform pencil-like rod (containing the propulsion systems) with a uniform sphere (the crew habitat and flight deck) attached to one end (Figure 11-23). The design is such that the radius of the sphere is much smaller than the length of the rod. At a location a few meters away from the ship, at point P on the perpendicular bisector of the rodlike section, what would be the direction of the gravitational field due to the ship alone (that is, assuming all other gravitational fields are negligible)? Explain your answer. At a large distance from the ship, what would be the dependence of the ships gravitational field on the distance from the ship?

Estimate the mass of our galaxy (the Milky Way) if the Sun orbits the center of the galaxy with a period of 250 million years at a mean distance of Express the mass in terms of multiples of the solar mass (Neglect the mass farther from the center than the Sun, and assume that the mass closer to the center than the Sun exerts the same force on the Sun, as would a point particle of the same mass located at the center of the galaxy.)

Besides studying samples of the lunar surface, the Apollo astronauts had several ways of determining that the moon is not made of green cheese. Among these ways are measurements of the gravitational acceleration at the lunar surface. Estimate the gravitational acceleration at the lunar surface if the moon were, in fact, a solid block of green cheese and compare your answer to the known value of the gravitational acceleration at the lunar surface.

CONTEXT-RICH, ENGINEERING APPLICATION You are in charge of the first manned exploration of an asteroid. You are concerned that, due to the weak gravitation field and resulting low escape speed, tethers might be required to bind the explorers to the surface of the asteroid. Therefore, if you do not wish to use tethers, you have to be careful about which asteroids to choose to explore. Estimate the largest radius the asteroid can have that would still allow you to escape its surface by jumping. Assume spherical geometry and reasonable rock density.

One of the great discoveries in astronomy in the past decade is the detection of planets outside the solar system. Since 1996, more than 100 planets have been detected orbiting stars other than the Sun. While the planets themselves cannot be seen directly, telescopes can detect the small periodic motion of the star as the star and planet orbit around their common center of mass. (This is measured using the Doppler effect, which is discussed in Chapter 15.) Both the period of this motion and the variation in the speed of the star over the course of time can be determined observationally. The mass of the star is found from its observed luminance and from the theory of stellar structure. Iota Draconis is the eighth brightest star in the constellation Draco. Observations show that a planet, with an orbital period of is orbiting this star. The mass of Iota Draconis is (a) Estimate the size (in AU) of the semimajor axis of this planets orbit. (b) The radial speed of the star is observed to vary by Use conservation of momentum to find the mass of the planet. Assume the orbit is circular, we are observing the orbit edge-on, and no other planets orbit Iota Draconis. Express the mass as a multiple of the mass of Jupiter.

One of the biggest unresolved problems in the theory of the formation of the solar system is that, while the mass of the Sun is 99.9 percent of the total mass of the solar system, it carries only about 2 percent of the total angular momentum. The most widely accepted theory of solar system formation has as its central hypothesis the collapse of a cloud of dust and gas under the force of gravity, with most of the mass forming the Sun. However, because the net angular momentum of this cloud is conserved, a simple theory would indicate that the Sun should be rotating much more rapidly than it currently is. In this problem, you are to show why it is important that most of the angular momentum was somehow transferred to the planets. (a) The Sun is a cloud of gas held together by the force of gravity. If the Sun were rotating too rapidly, gravity could not hold it together. Using the known mass of the Sun and its radius estimate the maximum angular speed that the Sun can have if it is to stay intact. What is the period of rotation corresponding to this rotation rate? (b) Calculate the orbital angular momentum of Jupiter and of Saturn from their masses (318 and 95.1 Earth masses, respectively), mean distances from the Sun (778 and 1430 million km, respectively), and orbital periods (11.9 and respectively). Compare them to the experimentally measured value of the Suns angular momentum of (c) If we were to somehow transfer all of Jupiters and Saturns angular momentum to the Sun, what would be the Suns new rotational period? The Sun is not a uniform sphere of gas, and its moment of inertia is given by the formula Compare this to the maximum rotational period of Part (a).

The new comet Alex-Casey has a very elliptical orbit with a period of If the closest approach of Alex-Casey to the Sun is 0.1 AU, what is its greatest distance from the Sun?

The radius of Earths orbit is and that of Uranus is What is the orbital period of Uranus?

The asteroid Hektor, discovered in 1907, is in a nearly circular orbit of radius 5.16 AU about the Sun. Determine the period of this asteroid.

One of the so-called Kirkwood gaps in the asteroid belt occurs at an orbital radius at which the period of the orbit is half that of Jupiters. The reason there is a gap for orbits of this radius is because of the periodic pulling (by Jupiter) that an asteroid experiences at the same location with every other orbit around the Sun. Repeated tugs from Jupiter of this kind would eventually change the orbit of such an asteroid. Therefore, all asteroids that would otherwise have orbited at this radius have presumably been cleared away from the area due to this resonance phenomenon. How far from the Sun is this particular 2:1 resonance Kirkwood gap?

The tiny Saturnian moon, Atlas, is locked into what is known as an orbital resonance with another moon, Mimas, whose orbit lies outside that of Atlas. The ratio between periods of these orbits is 3:2, that is, for every 3 orbits of Atlas, Mimas completes 2 orbits. Thus, Atlas, Mimas and Saturn are aligned at intervals equal to two orbital periods of Atlas. If Mimas orbits Saturn at a radius of 186,000 km, what is the radius of Atlass orbit?

The asteroid Icarus, discovered in 1949, was so named because its highly eccentric elliptical orbit brings it close to the Sun at perihelion. The eccentricity e of an ellipse is defined by the relation where is the perihelion distance and a is the semimajor axis. Icarus has an eccentricity of 0.83 and a period of (a) Determine the semimajor axis of the orbit of Icarus. (b) Determine the perihelion and aphelion distances of the orbit of Icarus.

CONTEXT-RICH, ENGINEERING APPLICATION, BIOLOGICAL APPLICATION A manned mission to Mars and its attendant problems due to the extremely long time the astronauts would spend weightless and without supplies space have been extensively discussed. To examine this issue in a simple way, consider one possible trajectory for the spacecraft: the Hohmann transfer orbit. This orbit consists of an elliptical orbit tangent to the orbit of Earth at its perihelion and tangent to the orbit of Mars at its aphelion. Given that Mars has a mean distance from the Sun of 1.52 times the mean SunEarth distance, calculate the time spent by the astronauts during the outbound part of the trip to Mars. Many adverse biological effects (such as muscle atrophy and decreased bone density) have been observed in astronauts returning from near-Earth orbit after only a few months in space. As the flight doctor, are there any health concerns that you should be aware of?

ESTIMATION Kepler determined distances in the solar system from his data. For example, he found the relative distance from the Sun to Venus (as compared to the distance from the Sun to Earth) as follows. Because Venuss orbit is closer to the Sun than is Earths orbit, Venus is a morning or evening starits position in the sky is never very far from the Sun (Figure 11-24). If we suppose the orbit of Venus is a perfect circle, then consider the relative orientation of Venus, Earth, and the Sun at maximum extension, that is, when Venus is farthest from the Sun in the sky. (a) Under this condition, show that angle b in Figure 11-24 is (b) If the maximum elongation angle a between Venus and the Sun is what is the distance between Venus and the Sun in AU? (c) Use this result to estimate the length of a Venusian year.

At apogee, the center of the moon is 406,395 km from the center of Earth and at perigee, the moon is 357,643 km from the center of Earth. What is the orbital speed of the moon at perigee and at apogee? The mass of Earth is

Jupiters satellite Europa orbits Jupiter with a period of 3.55 d at an average orbital radius of (a) Assuming that the orbit is circular, determine the mass of Jupiter from the data given. (b) Another satellite of Jupiter, Callisto, orbits at an average radius of with an orbital period of Show that these data are consistent with an inverse-square force law for gravity (Note: Do NOT use the value of G anywhere in Part (b)).

BIOLOGICAL APPLICATION Some people think that shuttle astronauts are weightless because they are beyond the pull of Earths gravity. In fact, this is completely untrue. (a) What is the magnitude of the gravitational field in the vicinity of a shuttle orbit? A shuttle orbit is about 400 km above the ground. (b) Given the answer in Part (a), explain why shuttle astronauts suffer from adverse biological affects such as muscle atrophy even though they are not actually weightless?

The mass of Saturn is (a) Find the period of its moon Mimas, whose mean orbital radius is (b) Find the mean orbital radius of its moon Titan, whose period is

Calculate the mass of Earth from the period of the moon, its mean orbital radius, and the known value of G.

Suppose you leave the solar system and arrive at a planet that has the same mass-to-volume ratio as Earth but has 10 times Earths radius. What would you weigh on this planet compared with what you weigh on Earth?

Suppose that Earth retained its present mass but was somehow compressed to half its present radius. What would be the value of g at the surface of this new, compact planet?

A planet orbits a massive star. When the planet is at perihelion, it has a speed of and is from the star. The orbital radius increases to at aphelion. What is the planets speed at aphelion?

What is the magnitude of the gravitational field at the surface of a neutron star whose mass is 1.60 times the mass of the Sun and whose radius is 10.5 km?

The speed of an asteroid is at perihelion and at aphelion. (a) Determine the ratio of the aphelion to perihelion distances. (b) Is this asteroid farther from the Sun or closer to the Sun than Earth, on average? Explain.

A satellite that has a mass of moves in a circular orbit above Earths surface. (a) What is the gravitational force on the satellite? (b) What is the speed of the satellite? (c) What is the period of the satellite?

A superconducting gravity meter can measure changes in gravity on the order (a) You are hiding behind a tree holding the meter, and your 80-kg friend approaches the tree from the other side. How close to you can your friend come before the meter detects a change in g due to his presence? (b) You are in a hot air balloon and are using the gravity meter to determine the rate of ascent (assume the balloon has constant acceleration). What is the smallest change in altitude that results in a detectable change in the gravitational field of Earth?

Suppose that the attractive interaction between a star of mass M and a planet of mass is of the form where K is the gravitational constant. What would be the relation between the radius of the planets circular orbit and its period?

Earths radius is 6370 km and the moons radius is 1738 km. The acceleration of gravity at the surface of the moon is What is the ratio of the average density of the moon to that of Earth?

The weight of a standard object defined as having a mass of exactly is measured to be 9.81 N. In the same laboratory, a second object weighs 56.6 N. (a) What is the mass of the second object? (b) Is the mass you determined in Part (a) gravitational or inertial mass?

ESTIMATION The Principle of Equivalence states that the free-fall acceleration of any object in a gravitational field is independent of the mass of the object. This can be deduced from the law of universal gravitation, but how well does it hold up experimentally? The RollKrotkovDicke experiment performed in the 1960s indicates that the free-fall acceleration is independent of mass to at least 1 part in Suppose two objects are simultaneously released from rest in a uniform gravitational field. Also, suppose one of the objects falls with a constant acceleration of exactly while the other falls with a constant acceleration that is greater than by one part in How far will the first object have fallen when the second object has fallen 1.00 mm farther than the first object has? Note that this estimate provides only an upper bound on the difference in the accelerations; most physicists believe that there is no difference in the accelerations.

(a) If we take the potential energy of a 100-kg object and Earth are zero when the two are separated by an infinite distance, what is the potential energy when the object is at the surface of Earth? (b) Find the potential energy of the same object at a height above Earths surface equal to Earths radius. (c) Find the escape speed for a body projected from this height.

Knowing that the acceleration of gravity on the moon is 0.166 times that on Earth and that the moons radius is find the escape speed for a projectile leaving the surface of the moon.

What initial speed would a particle need to be given at the surface of Earth if it is to have a final speed that is equal to its escape speed when it is very far from Earth? Neglect any effects due to air resistance.

CONTEXT-RICH, ENGINEERING APPLICATION While preparing its budget for the next fiscal year, NASA wants to report to the nation a rough estimate of the cost (per kilogram) of launching a modern satellite into near-Earth orbit. You are chosen for this task, because you know both physics and accounting. (a) Determine the energy, in , necessary to place a 1.0-kg object in low-Earth orbit. In low-Earth orbit, the height of the object above the surface of Earth is much smaller than Earths radius. Take the orbital height to be 300 km. (b) If this energy can be obtained at a typical electrical energy rate of what is the minimum cost of launching a 400-kg satellite into low-Earth orbit? Neglect any effects due to air resistance.

The science fiction writer Robert Heinlein once said, If you can get into orbit, then youre halfway to anywhere. Justify this statement by comparing the minimum energy needed to place a satellite into low Earth orbit to that needed to set it completely free from the bonds of Earths gravity. Neglect any effects due to air resistance.

An object is dropped from rest from a height of above the surface of Earth. If there is no air resistance, what is its speed when it strikes Earth?

An object is projected straight upward from the surface of Earth with an initial speed of What is the maximum height it reaches?

A particle is projected from the surface of Earth with a speed twice the escape speed. When it is very far from Earth, what is its speed?

When we calculate escape speeds, we usually do so with the assumption that the object from which we are calculating escape speed is isolated. This is, of course, generally not true in the solar system. Show that the escape speed at a point near a system that consists of two stationary massive spherical objects is equal to the square root of the sum of the squares of the escape speeds from each of the two objects considered individually.

Calculate the minimum necessary speed, relative to Earth, for a projectile launched from the surface of Earth to escape the solar system. The answer will depend on the direction of launch. Explain the choice of direction you would make for the direction of the launch in order to minimize the necessary launch speed relative to Earth. Neglect Earths rotational motion and effects due to air resistance.

An object is projected vertically from the surface of Earth at less than the escape speed. Show that the maximum height reached by the object is where is the height that it would reach if the gravitational field were constant. Neglect any effects due to air resistance.

A 100-kg spacecraft is in a circular orbit about Earth at a height (a) What is the orbital period of the spacecraft? (b) What is the spacecrafts kinetic energy? (c) Express the angular momentum L of the spacecraft about the center of Earth in terms of the kinetic energy K and find the numerical value of L.

ESTIMATION The orbital period of the moon is 27.3 d, the average center-to-center distance between the moon and Earth is the length of an Earth year 365.25 d, and the average center-to-center distance between Earth and the Sun is Use this data to estimate the ratio of the mass of the Sun to the mass of Earth. Compare this estimation to the measured ratio of List some neglected factors that might account for any discrepancy.

Many satellites orbit Earth at maximum altitudes above Earths surface of or less. Geosynchronous satellites, however, orbit at an altitude of 35 790 km above Earths surface. How much more energy is required to launch a 500-kg satellite into a geosynchronous orbit than into an orbit above the surface of Earth?

CONTEXT-RICH, ENGINEERING APPLICATION The idea of a spaceport orbiting Earth is an attractive proposition for launching probes and/or manned missions to the outer planets of the solar system. Suppose such a platform has been constructed, and orbits Earth at a distance of above Earths surface. Your research team is launching a lunar probe into an orbit that has its perigee at the spaceports orbital radius, and its apogee at the moons orbital radius. (a) To launch the probe successfully, first determine the orbital speed for the platform. (b) Next, determine the necessary speed relative to the platform that is necessary to launch the probe so that it attains the desired orbit. Assume that any effects due to the gravitational pull of the moon on the probe are negligible. In addition, assume that the launch takes place in a negligible amount of time. (c) You have the probe designed to radio back when it has reached apogee. How long after launch should you expect to receive this signal from the probe (neglect the second or so delay for the transit time of the signal back to the platform)?

A 3.0-kg space probe experiences a gravitational force of as it passes through point P. What is the gravitational field at point P?

The gravitational field at some point is given by What is the gravitational force on a 0.0040 kg object located at that point?

Apoint particle of mass m is on the x axis at and an identical point particle is on the y axis at (a) What is the direction of the gravitational field at the origin? (b) What is the magnitude of this field?

Five objects, each of mass M, are equally spaced on the arc of a semicircle of radius R, as in Figure 11-25. An object of mass m is located at the center of curvature of the arc. (a) If M is m is and R is what is the gravitational force on the particle of mass m due to the five objects? (b) If the object whose mass is m is removed, what is the gravitational field at the center of curvature of the arc?

Apoint particle of mass is at the origin and a second point particle of mass is on the x axis at Find the gravitational field at (a) and (b) (c) Find the point on the x axis for which

Show that on the x axis, the maximum value of g for the field of Example 11-7 occurs at points x _ _a>12.

A nonuniform thin rod of length L lies on the x axis. One end of the rod is at the origin, and the other end is at The rods mass per unit length varies as where C is a constant. (Thus, an element of the rod has mass ) (a) Determine the total mass of the rod. (b) Determine the gravitational field due to the rod on the x axis at where

Auniform thin rod of mass M and length L lies on the positive x axis with one end at the origin. Consider an element of the rod of length dx, and mass dm, at point x, where (a) Show that this element produces a gravitational field at a point on the x axis in the region is given by (b) Integrate this result over the length of the rod to find the total gravitational field at the point due to the rod. (c) Find the gravitational force on a point particle of mass at (d) Show that for the field of the rod approximates the field of a point particle of mass M at

Auniform thin spherical shell has a radius of 2.0 m and a mass of What is the gravitational field at the following distances from the center of the shell: (a) (b) (c)

A uniform thin spherical shell has a radius of 2.00 m and a mass of 300 kg and its center is located at the origin of a coordinate system. Another uniform thin spherical shell with a radius of 1.00 m and a mass of 150 kg is inside the larger shell, with its center at on the x axis. What is the gravitational force of attraction between the two shells?

Two widely separated solid spheres, and each have radius R and mass M. Sphere is uniform, whereas the density of is given by where r is the distance from its center. If the gravitational field strength at the surface of is what is the gravitational field strength at the surface of

Two widely separated uniform solid spheres, and have equal masses, but different radii, and If the gravitational field strength on the surface of is what is the gravitational field strength on the surface of

Two concentric uniform thin spherical shells have masses and and radii a and 2a, as in Figure 11-26. What is the magnitude of the gravitational force on a point particle of mass m (not shown) located (a) a distance 3a from the center of the shells? (b) a distance 1.9a from the center of the shells? (c) a distance 0.9a from the center of the shells?

The inner spherical shell in Problem 73 is shifted so that its center is now on the x axis at What is the magnitude of the gravitational force on a particle of point mass m located on the x axis at (a) (b) (c)

Suppose you are standing on a spring scale in an elevator that is descending at costant speed in a mine shaft located on the equator. Model Earth as a homogeneous sphere. (a) Show that the force on you due to Earths gravity alone is proportional to your distance from the center of the planet. (b) Assume that the mine shaft is located on the equator and is vertical. Do not neglect Earths rotational motion. Show that the reading on the spring scale is proportional to your distance from the center of the planet.

CONTEXT-RICH Suppose Earth were a nonrotating uniform sphere. As a reward for earning the highest lab grade, your physics professor chooses your laboratory team to participate in a gravitational experiment at a deep mine on the equator. This mine has an elevator shaft going into Earth. Before making the measurement, you are asked to predict the decrease in the weight of a team member, who weighs at the surface of Earth, when she is at the bottom of the shaft. The density of Earths crust actually increases with depth. Is your answer higher or lower than the actual experimental result?

A solid sphere of radius R has its center at the origin. It has a uniform mass density except that the sphere has a spherical cavity in it of radius centered at as in Figure 11-27. Find the gravitational field at points on the x axis for Hint: The cavity may be thought of as a sphere of mass m _ (4>3)pr3r0 plus a sphere of negative mass m. SSM

For the sphere with the cavity in Problem 77, show that the gravitational field is uniform throughout the cavity, and find its magnitude and direction there.

A straight, smooth tunnel is dug through a uniform spherical planet of mass density The tunnel passes through the center of the planet and is perpendicular to the planets axis of rotation, which is fixed in space. The planet rotates with a constant angular speed so objects in the tunnel have no apparent weight. Find the required angular speed of the planet

The density of a sphere is given by The sphere has a radius of 5.0 m and a mass of (a) Determine the constant C. (b) Obtain expressions for the gravitational field for the regions (1) and (2)

A small-diameter hole is drilled into the sphere of Problem 80 toward the center of the sphere to a depth of below the spheres surface. A small mass is dropped from the surface into the hole. Determine the speed of the small mass when it strikes the bottom of the hole. SSM

CONTEXT-RICH, ENGINEERING APPLICATION As a geologist for a mining company, you are working on a method for determining possible locations of underground ore deposits. Assume that where the company owns land the crust of Earth is thick and has a density of about Suppose a spherical deposit of heavy metals with a density of and radius of is centered below the surface. You propose to detect it by determining its effect on the local surface value of g. Find at the surface directly above this deposit, where g is the increase in the gravitational field due to the deposit.

Two identical spherical cavities are made in a lead sphere of radius R. The cavities each have a radius They touch the outside surface of the sphere and its center as in Figure 11-28. The mass of a solid uniform lead sphere of radius R is M. Find the force of attraction on a point particle of mass m located a distance d from the center of the lead sphere as shown. SSM

A globular cluster is a roughly spherical collection of up to millions of stars bound together by the force of gravity. Astronomers can measure the velocities of stars in the cluster to study its composition and to get an idea of the mass distribution within the cluster. Assuming that all of the stars have approximately the same mass and are distributed uniformly within the cluster, show that the mean speed of a star in a circular orbit around the center of the cluster should increase linearly with its distance from the center.

The mean distance of Pluto from the Sun is 39.5 AU. Calculate the period of Plutos orbital motion.

Calculate the mass of Earth using the known values of G, g, and

The force exerted by Earth on a particle of mass m a distance r from the center of Earth has the magnitude where (a) Calculate the work you must do to move the particle from distance to distance (b) Show that when and the result can be written as (c) Show that when the work is given approximately by

The average density of the moon is Find the minimum possible period T of a spacecraft orbiting the moon.

A neutron star is a highly condensed remnant of a massive star in the last phase of its evolution. It is composed of neutrons (hence the name), because the stars gravitational force causes electrons and protons to coalesce into the neutrons. Suppose at the end of its current phase, the Sun collapsed into a neutron star (it cannot actually do this because it does not have enough mass) of radius without losing any mass in the process. (a) Calculate the ratio of the gravitational acceleration at the surface of the Sun following the collapse compared to the value at the surface of the Sun today. (b) Calculate the ratio of the escape speed from the surface of the neutron-Sun to the Suns value today.

CONTEXT-RICH, ENGINEERING APPLICATION Suppose the Sun could collapse into a neutron star of radius as in Problem 89. Your research team is in charge of sending a probe from Earth to study the transformed Sun, and the probe needs to end up in a circular orbit from the neutron- Suns center. (a) Calculate the orbital speed of the probe. (b) Later on, plans call for construction of a permanent spaceport in that same orbit. To transport equipment and supplies, scientists on Earth need you to determine the escape speed for rockets launched from the spaceport (relative to the spaceport) in the direction of the spaceports orbital velocity at takeoff time. What is that speed, and how does it compare to the escape speed at the surface of Earth?

A satellite is circling the moon (radius ) close to the surface at a speed A projectile is launched vertically up from the moons surface at the same initial speed How high will the projectile rise?

Black holes are objects whose gravitational field is so strong that not even light can escape. One way of thinking about this is to consider a spherical object whose density is so large that the escape speed at its surface is greater than the speed of light, c. If a stars radius is smaller than a value called the Schwarzschild radius then the star will be a black hole, that is, light originating from its surface cannot escape. (a) For a nonrotating black hole, the Schwarzschild radius depends only upon the mass of the black hole. Show that it is related to that mass M by (b) Calculate the value of the Schwarzschild radius for a black hole whose mass is ten solar masses.

In a binary star system, two stars follow circular orbits about their common center of mass. If the stars have masses and and are separated by a distance r, show that the period of rotation is related to r by

Two particles of masses and are released from rest at a large separation distance. Find their speeds and when their separation distance is r. The initial separation distance is given as large, but large is a relative term. Relative to what distance is it large?

Uranus, the seventh planet in the solar system, was first observed in 1781 by William Herschel. Its orbit was then analyzed in terms of Keplers laws. By the 1840s, observations of Uranus clearly indicated that its true orbit was different from the Keplerian calculation by an amount that could not be accounted for by observational uncertainty. The conclusion was that there must be another influence other than the Sun and the known planets lying inside Uranuss orbit. This influence was hypothesized to be due to an eighth planet, whose predicted orbit was described independently in 1845 by two astronomers: John Adams (no relation to the former president of the United States) and Urbain LeVerrier. In September of 1846, John Galle, searching in the sky at the place predicted by Adams and LeVerrier, made the first observation of Neptune. Uranus and Neptune are in orbit about the Sun with periods of 84.0 and 164.8 years, respectively. To see the effect that Neptune had on Uranus, determine the ratio of the gravitational force between Neptune and Uranus to that between Uranus and the Sun, when Neptune and Uranus are at their closest approach to one another (i.e., when aligned with the Sun). The masses of the Sun, Uranus, and Neptune are 333,000, 14.5, and 17.1 times that of Earth, respectively.

It is believed that there is a supermassive black hole at the center of our galaxy. One datum that leads to this conclusion is the important recent observation of stellar motion in the vicinity of the galactic center. If one such star moves in an elliptical orbit with a period of 15.2 years and has a semimajor axis of 5.5 light-days (the distance light travels in 5.5 days), what is the mass around which the star moves in its Keplerian orbit?

Four identical planets are arranged in a square, as shown in Figure 11-29. If the mass of each planet is M and the edge length of the square is a, what must their speed be if they are to orbit their common center under the influence of their mutual attraction? SSM SSM

A hole is drilled from the surface of Earth to its center, as in Figure 11-30. Ignore Earths rotation and any effects due to air resistance, and model Earth as a uniform sphere. (a) How much work is required to lift a particle of mass m from the center of Earth to Earths surface? (b) If the particle is dropped from rest at the surface of Earth, what is its speed when it reaches the center of Earth? (c) What is the escape speed for a particle projected from the center of Earth? Express your answers in terms of

A thick spherical shell of mass M and uniform density has an inner radius and an outer radius Find the gravitational field as a function of r for Sketch a graph of versus r.

(a) Athin uniform ring of mass Mand radius R lies in the plane and is cenered at the origin. Sketch a plot of the gravitational field versus x for all points on the x axis. (b) At what point, or points, on the axis is the magnitude of a maximum?

Find the magnitude of the gravitational field that is at a distance r from an infinitely long uniform thin rod whose mass per unit length is

One question in early planetary science was whether each of the rings of Saturn were solid or were, instead, composed of individual chunks, each in its own orbit. The issue could be resolved by an observation in which astronomers would measure the speed of the inner and outer portions of the ring. If the inner portion of the ring moved more slowly than the outer portion, then the ring was solid; if the opposite was true, then it was actually composed of separate chunks. Let us see how this results from a theoretical viewpoint. Let the radial width of a given ring (there are many) be the average distance of that ring from the center of Saturn be represented by R, and the average speed of that ring be (a) If the ring is solid, show that the difference in speed between its outermost and innermost portions, is given by the expression Here, is the speed of the outermost portion of the ring, and is the speed of the innermost portion. (b) If, however, the ring is composed of many small chunks, show that (Assume that )

Find the gravitational potential energy of the thin rod in Example 11-8 and a point particle of mass that is on the x axis at where (a) Show that the potential energy shared by an element of the rod of mass dm (shown in Figure 11-14) and the point particle of mass is given by where at (b) Integrate your result for Part (a) over the length of the rod to find the total potential energy for the system. Generalize your function to any place on the x axis in the region by replacing with a general coordinate x and write it as (c) Compute the force on at a general point x using and compare your result with where g is the field at calculated in Example 11-8.

A uniform sphere of mass M is located near a thin, uniform rod of mass m and length L, as in Figure 11-31. Find the gravitational force of attraction exerted by the sphere on the rod.

A thin uniform 20-kg rod with a length equal to 5.0 m is bent into a semicircle. What is the gravitational force exerted by the rod on a 0.10-kg point mass located at the center of curvature of the circular arc?

Both the Sun and the moon exert gravitational forces on the oceans of Earth, causing tides. (a) Show that the ratio of the force exerted on a point particle on the surface of Earth by the Sun to that exerted by the moon is Here and represent the masses of the Sun and moon, and and are the distances of the particle from Earth to the Sun and Earth to the moon, respectively. Evaluate this ratio numerically. (b) Even though the Sun exerts a much greater force on the oceans than does the moon, the moon has a greater effect on the tides because it is the difference in the force from one side of Earth to the other that is important. Differentiate the expression to calculate the change in F due to a small change in r. Show that (c) The oceanic tidal bulge (that is, the elongation of the liquid water of the oceans causing two opposite high and two opposite low spots) is caused by the difference in gravitational force on the oceans from one side of Earth to the other. Show that for a small difference in distance compared to the average distance, the ratio of the differential gravitational force exerted by the Sun to the differential gravitational force exerted by the moon on Earths oceans is given by Calculate this ratio. What is your conclusion? Which object, the moon or the Sun, is the main cause of the tidal stretching of the oceans on Earth?

CONTEXT-RICH, ENGINEERING APPLICATION United Federation Starship Excelsior drops two small robotic probes toward the surface of a neutron star for exploration. The mass of the star is the same as that of the Sun, but the stars diameter is only 10 km. The robotic probes are linked together by a 1.0-mlong steel cord (which includes communication lines between the two probes), and are dropped vertically (that is, one always above the other). The ship hovers at rest above the stars surface. As the Chief of Materials Engineering on the ship, you are concerned that the communication between the two probes, a crucial aspect of the mission, will not survive. (a) Outline your briefing session to the mission commander and explain the existence of a stretching force that will try to pull the robots apart as they fall toward the planet. (See Problem 106 for hints.) (b) Assume that the cord in use has a breaking tension of and that the robots each have a mass of How close will the robots be to the surface of the star before the cord breaks?