Solved: (a) In 77, remove sphere A and calculate the gravitational potential energy of

In Fig. 13-21, a central particle of mass M is surrounded by a square array of other particles, separated by either distance d or distance d/2 along the perimeter of the square. What are the magnitude and direction of the net gravitational force on the central particle due to the other particles?

Figure 13-22 shows three arrangements of the same identical particles, with three of them placed on a circle of radius 0.20 m and the fourth one placed at the center of the circle. (a) Rank the arrangements according to the magnitude of the net gravitational force on the central particle due to the other three particles, greatest first. (b) Rank them according to the gravitational potential energy of the four-particle system, least negative first.

In Fig. 13-23, a central particle is surrounded by two circular rings of particles, at radii r and R, with R > r. All the particles have mass m. What are the magnitude and direction of the net gravitational force on the central particle due to the particles in the rings?

Figure 13-25 shows three situations involving a point particle P with mass m and a spherical shell with a uniformly distributed mass M. The radii of the shells are given. Rank the situations according to the magnitude of the gravitational force on particle P due to the shell, greatest first.

In Fig. 13-26, three particles are fixed in place. The mass of B is greater than the mass of C. Can a fourth particle (particle D) be placed somewhere so that the net gravitational force on particle A from particles B, C, and D is zero? If so, in which quadrant should it be placed and which axis should it be near?

Figure 13-27 gives the gravitational acceleration ag for four planets as a function of the radial distance r from the center of the planet, starting at the surface of the planet (at radius\(R_{1}, R_{2}, R_{3} \text {, or } R_{4}\)). Plots 1 and 2 coincide for\(r \geq R_{2}\); plots 3 and 4 coincide for \(r \geq R_{2}\). Rank the four planets according to (a) mass and (b) mass per unit volume, greatest first. Text Transcription: R_1,R_2,R_3,R_4 r ? R2 r ? R2

Figure 13-28 shows three particles initially fixed in place, with B and C identical and positioned symmetrically about the y axis, at distance d from A. (a) In what direction is the net gravitational force \(\vec{F}_{\text {net }}\)t on A? (b) If we move C directly away from the origin, does \(\vec{F}_{\text {net }}\)change in direction? If so, how and what is the limit of the change? Text Transcription: arrow right over F_net

Figure 13-29 shows six paths by which a rocket orbiting a moon might move from point a to point b. Rank the paths according to (a) the corresponding change in the gravitational potential energy of the rocket–moon system and (b) the net work done on the rocket by the gravitational force from the moon, greatest first.

Figure 13-30 shows three uniform spherical planets that are identical in size and mass. The periods of rotation T for the planets are given, and six lettered points are indicated—three points are on the equators of the planets and three points are on the north poles. Rank the points according to the value of the free-fall acceleration g at them, greatest first.

In Fig. 13-31, a particle of mass m (which is not shown) is to be moved from an infinite distance to one of the three possible locations a, b, and c. Two other particles, of masses m and 2m, are already fixed in place on the axis, as shown. Rank the three possible locations according to the work done by the net gravitational force on the moving particle due to the fixed particles, greatest first.

The radius \(R_{h}\) of a black hole is the radius of a mathematical sphere, called the event horizon, that is centered on the black hole. Information from events inside the event horizon cannot reach the outside world. According to Einstein’s general theory of relativity,\(R_{h}=2 G M / c^{2}\) , where M is the mass of the black hole and c is the speed of light. Suppose that you wish to study a black hole near it, at a radial distance of 50\(\(R_{h}\)\). However, you do not want the difference in gravitational acceleration between your feet and your head to exceed 10 \(\mathrm{m} / \mathrm{s}^{2}\)when you are feet down (or head down) toward the black hole. (a) As a multiple of our Sun’s mass\(M_{s}\), approximately what is the limit to the mass of the black hole you can tolerate at the given radial distance? (You need to estimate your height.) (b) Is the limit an upper limit (you can tolerate smaller masses) or a lower limit (you can tolerate larger masses)? Text Transcription: R_h R_h = 2GM/c^2 M_s m/s^2

Several planets (Jupiter, Saturn, Uranus) are encircled by rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ring-like structures. Consider a homogeneous thin ring of mass M and outer radius R (Fig. 13-52). (a) What gravitational attraction does it exert on a particle of mass m located on the ring’s central axis a distance x from the ring center? (b) Suppose the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring?

Figure 13-53 is a graph of the kinetic energy K of an asteroid versus its distance r from Earth’s center, as the asteroid falls directly in toward that center. (a) What is the (approximate) mass of the asteroid? (b) What is its speed at r = 1.945 × \(10^{7}\)m? Text Transcription: 10^7

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

Four uniform spheres, with masses \(m_{A}\)= 40 kg, \(m_{B}\)= 35 kg, \(m_{C}\)= 200 kg, and \(m_{B}\)= 50 kg, have (x, y) coordinates of (0, 50 cm), (0, 0), (?80 cm, 0), and (40 cm, 0), respectively. In unit-vector notation, what is the net gravitational force on sphere B due to the other spheres? Text Transcription: M_A M_B M_C M_D

A certain triple-star system consists of two stars, each of mass m, revolving in the same circular orbit of radius r around a central star of mass M (Fig. 13-54). The two orbiting stars are always at opposite ends of a diameter of the orbit. Derive an expression for the period of revolution of the stars.

The fastest possible rate of rotation of a planet is that for which the gravitational force on material at the equator just barely provides the centripetal force needed for the rotation. (Why?) (a) Show that the corresponding shortest period of rotation is \(T=\sqrt{\frac{3 \pi}{G \rho}}\) where \(\rho\)is the uniform density (mass per unit volume) of the spherical planet. (b) Calculate the rotation period assuming a density of 3.0 \(\mathrm{g} / \mathrm{cm}^{3}\), typical of many planets, satellites, and asteroids. No astronomical object has ever been found to be spinning with a period shorter than that determined by this analysis. Text Transcription: T=sqrt 3 pi / G_p g/cm^3 rho

In a double-star system, two stars of mass 3.0 × \(10^{30}\)kg each rotate about the system’s center of mass at radius 1.0 × \(10^{11}\)m. (a) What is their common angular speed? (b) If a meteoroid passes through the system’s center of mass perpendicular to their orbital plane, what minimum speed must it have at the center of mass if it is to escape to “infinity” from the two-star system? Text Transcription: 10^30 10^11

A satellite is in elliptical orbit with a period of 8.00 × \(10^{4}\) s about a planet of mass 7.00 × \(10^{24}\)kg. At aphelion, at radius 4.5 × \(10^{7}\) m, the satellite’s angular speed is 7.158 × \(10^{-5}\)rad/s. What is its angular speed at perihelion? Text Transcription: 10^24 10^4 10^7 10^-5

In a shuttle craft of mass m = 3000 kg, Captain Janeway orbits a planet of mass M = 9.50 × \(10^{25}\)kg, in a circular orbit of radius r = 4.20 × \(10^{7}\)m. What are (a) the period of the orbit and (b) the speed of the shuttle craft? Janeway briefly fires a forwardpointing thruster, reducing her speed by 2.00%. Just then, what are (c) the speed, (d) the kinetic energy, (e) the gravitational potential energy, and (f) the mechanical energy of the shuttle craft? (g) What is the semimajor axis of the elliptical orbit now taken by the craft? (h) What is the difference between the period of the original circular orbit and that of the new elliptical orbit? (i) Which orbit has the smaller period? Text Transcription: 10^25 10^7

Consider a pulsar, a collapsed star of extremely high density, with a mass M equal to that of the Sun (1.98 × \(10^{30}\)kg), a radius R of only 12 km, and a rotational period T of 0.041 s. By what percentage does the free-fall acceleration g differ from the gravitational acceleration \(a_{g}\)at the equator of this spherical star? Text Transcription: 10^30 A_g

An object lying on Earth’s equator is accelerated (a) toward the center of Earth because Earth rotates, (b) toward the Sun because Earth revolves around the Sun in an almost circular orbit, and (c) toward the center of our galaxy because the Sun moves around the galactic center. For the latter, the period is 2.5 × \(10^{8}\)y and the radius is 2.2 × \(10^{20}\)m. Calculate these three accelerations as multiples of g = 9.8 \(\mathrm{m} / \mathrm{s}^{2}\). Text Transcription: 10^8 10^20 m/s^2

The orbit of Earth around the Sun is almost circular: The closest and farthest distances are 1.47 × \(10^{8}\) km and 1.52 × \(10^{8}\)km respectively. Determine the corresponding variations in (a) total energy, (b) gravitational potential energy, (c) kinetic energy, and (d) orbital speed. (Hint: Use conservation of energy and conservation of angular momentum.) Text Transcription: 10^8

We watch two identical astronomical bodies A and B, each of mass m, fall toward each other from rest because of the gravitational force on each from the other. Their initial center-to-center separation is \(\boldsymbol{R}_{i}\). Assume that we are in an inertial reference frame that is stationary with respect to the center of mass of this twobody system. Use the principle of conservation of mechanical energy (\(K_{f}+U_{f}=K_{i}+U_{i}\) ) to find the following when the centerto-center separation is 0.5\(\boldsymbol{R}_{i}\): (a) the total kinetic energy of the system, (b) the kinetic energy of each body, (c) the speed of each body relative to us, and (d) the speed of body B relative to body A. Next assume that we are in a reference frame attached to body A (we ride on the body). Now we see body B fall from rest toward us. From this reference frame, again use \(K_{f}+U_{f}=K_{i}+U_{i}\)+ Ui to find the following when the center-to-center separation is 0.5Ri: (e) the kinetic energy of body B and (f) the speed of body B relative to body A. (g) Why are the answers to (d) and (f) different? Which answer is correct? Text Transcription: Kf + Uf = Ki + U R_i

Planet Roton, with a mass of 7.0 × \(10^{24}\)kg and a radius of 1600 km, gravitationally attracts a meteorite that is initially at rest relative to the planet, at a distance great enough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, find the speed of the meteorite when it reaches the planet’s surface. Text Transcription: 10^24

An object of mass m is initially held in place at radial distance r = \(3 R_{E})from the center of Earth, where \(R_{E}\)is the radius of Earth. Let ME be the mass of Earth. A force is applied to the object to move it to a radial distance r =\(4 R_{E}\), where it again is held in place. Calculate the work done by the applied force during the move by integrating the force magnitude. Text Transcription: 3R_E 4R_E R_E

To alleviate the traffic congestion between two cities such as Boston and Washington, D.C., engineers have proposed building a rail tunnel along a chord line connecting the cities (Fig. 13-55). A train, unpropelled by any engine and starting from rest, would fall through the first half of the tunnel and then move up the second half. Assuming Earth is a uniform sphere and ignoring air drag and friction, find the city-to-city travel time.

A thin rod with mass M = 5.00 kg is bent in a semicircle of radius R = 0.650 m (Fig. 13-56). (a) What is its gravitational force (both magnitude and direction) on a particle with mass m = 3.0 × \(10^{-3}\)kg at P, the center of curvature? (b) What would be the force on the particle if the rod were a complete circle? Text Transcription: 10^-3

In Fig. 13-57, identical blocks with identical masses m = 2.00 kg hang from strings of different lengths on a balance at Earth’s surface. The strings have negligible mass and differ in length by h = 5.00 cm. Assume Earth is spherical with a uniform density \(\rho\)= 5.50 \(\mathrm{g} / \mathrm{cm}^{3}\). What is the difference in the weight of the blocks due to one being closer to Earth than the other? Text Transcription: g/cm^3 rho

In Fig. 13-24, two particles, of masses m and 2m, are fixed in place on an axis. (a) Where on the axis can a third particle of mass 3m be placed (other than at infinity) so that the net gravitational force on it from the first two particles is zero: to the left of the first two particles, to their right, between them but closer to the more massive particle, or between them but closer to the less massive particle? (b) Does the answer change if the third particle has, instead, a mass of 16m? (c) Is there a point off the axis (other than infinity) at which the net force on the third particle would be zero?

Rank the four systems of equalmass particles shown in Checkpoint 2 according to the absolute value of the gravitational potential energy of the system, greatest first.

A typical neutron star may have a mass equal to that of the Sun but a radius of only 10 km. (a) What is the gravitational acceleration at the surface of such a star? (b) How fast would an object be moving if it fell from rest through a distance of 1.0 m on such a star? (Assume the star does not rotate.)

The mysterious visitor that appears in the enchanting story The Little Prince was said to come from a planet that “was scarcely any larger than a house!” Assume that the mass per unit volume of the planet is about that of Earth and that the planet does not appreciably spin. Approximate (a) the free-fall acceleration on the planet’s surface and (b) the escape speed from the planet

A very early, simple satellite consisted of an inflated spherical aluminum balloon 30 m in diameter and of mass 20 kg. Suppose a meteor having a mass of 7.0 kg passes within 3.0 m of the surface of the satellite. What is the magnitude of the gravitational force on the meteor from the satellite at the closest approach?

(a) In Problem 77, remove sphere A and calculate the gravitational potential energy of the remaining three-particle system. (b) If A is then put back in place, is the potential energy of the four-particle system more or less than that of the system in (a)? (c) In (a), is the work done by you to remove A positive or negative? (d) In (b), is the work done by you to replace A positive or negative?

A projectile is fired vertically from Earth’s surface with an initial speed of 10 km/s. Neglecting air drag, how far above the surface of Earth will it go?

(a) If the legendary apple of Newton could be released from rest at a height of 2 m from the surface of a neutron star with a mass 1.5 times that of our Sun and a radius of 20 km, what would be the apple’s speed when it reached the surface of the star? (b) If the apple could rest on the surface of the star, what would be the approximate difference between the gravitational acceleration at the top and at the bottom of the apple? (Choose a reasonable size for an apple; the answer indicates that an apple would never survive near a neutron star.)

With what speed would mail pass through the center of Earth if falling in a tunnel through the center?

A 50 kg satellite circles planet Cruton every 6.0 h. The magnitude of the gravitational force exerted on the satellite by Cruton is 80 N. (a) What is the radius of the orbit? (b) What is the kinetic energy of the satellite? (c) What is the mass of planet Cruton?

A 150.0 kg rocket moving radially outward from Earth has a speed of 3.70 km/s when its engine shuts off 200 km above Earth’s surface. (a) Assuming negligible air drag acts on the rocket, find the rocket’s kinetic energy when the rocket is 1000 km above Earth’s surface. (b) What maximum height above the surface is reached by the rocket?

Two 20 kg spheres are fixed in place on a y axis, one at y = 0.40 m and the other at y = ?0.40 m. A 10 kg ball is then released from rest at a point on the x axis that is at a great distance (effectively infinite) from the spheres. If the only forces acting on the ball are the gravitational forces from the spheres, then when the ball reaches the (x, y) point (0.30 m, 0), what are (a) its kinetic energy and (b) the net force on it from the spheres, in unit-vector notation?

Sphere A with mass 80 kg is located at the origin of an xy coordinate system; sphere B with mass 60 kg is located at coordinates (0.25 m, 0); sphere C with mass 0.20 kg is located in the first quadrant 0.20 m from A and 0.15 m from B. In unit-vector notation, what is the gravitational force on C due to A and B?

In his 1865 science fiction novel From the Earth to the Moon, Jules Verne described how three astronauts are shot to the Moon by means of a huge gun. According to Verne, the aluminum capsule containing the astronauts is accelerated by ignition of nitrocellulose to a speed of 11 km/s along the gun barrel’s length of 220 m. (a) In g units, what is the average acceleration of the capsule and astronauts in the gun barrel? (b) Is that acceleration tolerable or deadly to the astronauts? A modern version of such gun-launched spacecraft (although without passengers) has been proposed. In this modern version, called the SHARP (Super High Altitude Research Project) gun, ignition of methane and air shoves a piston along the gun’s tube, compressing hydrogen gas that then launches a rocket. During this launch, the rocket moves 3.5 km and reaches a speed of 7.0 km/s. Once launched, the rocket can be fired to gain additional speed. (c) In g units, what would be the average acceleration of the rocket within the launcher? (d) How much additional speed is needed (via the rocket engine) if the rocket is to orbit Earth at an altitude of 700 km?

A spaceship is on a straight-line path between Earth and the Moon. At what distance from Earth is the net gravitational force on the spaceship zero?