Active Figure 13.1  The gravitational force between two particles is attractive. The unit vector r12 is directed from particle 1 toward particle 2. Note that F21 = – F12. At the Active Figures link at http://www/pse6.com, you can change the separation distance between the particles to see the effect on the gravitational force. Fig. 13.1, p.391

Figure 13.2  As it revolves around the Earth, the Moon experiences a centripetal acceleration aM directed toward the Earth. An object near the Earth’s surface, such as the apple shown here, experiences an acceleration g. (Dimensions are not to scale.) Fig. 13.2, p.392

Figure 13.3  The resultant gravitational force acting on the cue ball is the vector sum F21 + F31. Fig. 13.3, p.393

Figure 13.4  Schematic diagram of the Cavendish apparatus for measuring G. As the small spheres of mass m are attracted to the large spheres of mass M, the rod between the two small spheres rotates through a small angle. A light beam reflected from a mirror on the rotating apparatus measures the angle of rotation. The dashed line represents the original position of the rod. Fig. 13.4, p.393

Astronauts F. Story Musgrave and Jeffrey A Astronauts F. Story Musgrave and Jeffrey A. Hoffman, along with the Hubble Space Telescope and the space shuttle Endeavor, are all in free fall while orbiting the Earth. p.394

Table 13.1, p.395

Active Figure 13. 5 Plot of an ellipse Active Figure 13.5  Plot of an ellipse. The semimajor axis has length a, and the semiminor axis has length b. Each focus is located at a distance c from the center on each side of the center. At the Active Figures link at http://www.pse6.com, you can move the focal points or enter values for a, b, c, and e to see the resulting elliptical shape. Fig. 13.5, p.396

Figure 13.6  (a) The shape of the orbit of Pluto, which has the highest eccentricity (e = 0.25) among the planets in the solar system. The Sun is located at the large yellow dot, which is a focus of the ellipse. There is nothing physical located at the center (the small dot) or the other focus (the blue dot). (b) The shape of the orbit of Comet Halley. Fig. 13.6, p.397

Active Figure 13.7  (a) The gravitational force acting on a planet is directed toward the Sun, along the radius vector. (b) As a planet orbits the Sun, the area swept out by the radius vector in a time interval dt is equal to one-half the area of the parallelogram formed by the vectors r and dr = v dt. At the Active Figures link at http://www.pse6.com, you can assign a value of the eccentricity and see the resulting motion of the planet around the Sun. Fig. 13.7, p.398

Active Figure 13.7  (a) The gravitational force acting on a planet is directed toward the Sun, along the radius vector. At the Active Figures link at http://www.pse6.com, you can assign a value of the eccentricity and see the resulting motion of the planet around the Sun. Fig. 13.7a, p.398

Active Figure 13.7  (b) As a planet orbits the Sun, the area swept out by the radius vector in a time interval dt is equal to one-half the area of the parallelogram formed by the vectors r and dr = v dt. At the Active Figures link at http://www.pse6.com, you can assign a value of the eccentricity and see the resulting motion of the planet around the Sun. Fig. 13.7b, p.398

Figure 13.8  A planet of mass MP moving in a circular orbit around the Sun. The orbits of all planets except Mercury and Pluto are nearly circular. Fig. 13.8, p.398

Table13. 2, p.399

Figure 13.9 A satellite of mass m moving around the Earth in a circular orbit of radius r with constant speed v. The only force acting on the satellite is the gravitational force Fg. (Not drawn to scale) Fig. 13.9, p.401

Figure 13.10  (a) The gravitational field vectors in the vicinity of a uniform spherical mass such as the Earth vary in both direction and magnitude. The vectors point in the direction of the acceleration a particle would experience if it were placed in the field. The magnitude of the field vector at any location is the magnitude of the free-fall acceleration at that location. (b) The gravitational field vectors in a small region near the Earth’s surface are uniform in both direction and magnitude. Fig. 13.10, p.402

Figure 13.10  (a) The gravitational field vectors in the vicinity of a uniform spherical mass such as the Earth vary in both direction and magnitude. The vectors point in the direction of the acceleration a particle would experience if it were placed in the field. The magnitude of the field vector at any location is the magnitude of the free-fall acceleration at that location. Fig. 13.10a, p.402

Figure 13.10  (b) The gravitational field vectors in a small region near the Earth’s surface are uniform in both direction and magnitude. Fig. 13.10b, p.402

Figure 13.11  A particle moves from A to B while acted on by a central force F, which is directed radially. The path is broken into a series of radial segments and arcs. Because the work done along the arcs is zero, the work done is independent of the path and depends only on rf and ri. Fig. 13.11, p.403

Figure 13.12  As a particle of mass m moves from A to B above the Earth’s surface, the gravitational potential energy changes according to Equation 13.11. Fig. 13.12, p.403

Figure 13.13  Graph of the gravitational potential energy U versus r for an object above the Earth’s surface. The potential energy goes to zero as r approaches infinity. Fig. 13.13, p.404

Figure 13.14  Three interacting particles. Fig. 13.14, p.404

Figure 13.15  An object of mass m moving in a circular orbit about a much larger object of mass M. Fig. 13.15, p.406

Figure 13.16  An object of mass m projected upward from the Earth’s surface with an initial speed vi reaches a maximum altitude h. Fig. 13.16, p.407

Table 13.3, p.408

Figure 13.17 A black hole. The distance RS equals the Schwarzschild radius. Any event occurring within the boundary of radius RS, called the event horizon, is invisible to an outside observer. Fig. 13.17, p.409

Figure 13.18 A binary star system consisting of an ordinary star on the left and a black hole on the right. Matter pulled from the ordinary star forms an accretion disk around the black hole, in which matter is raised to very high temperatures, resulting in the emission of x-rays. Fig. 13.18, p.409

Figure 13. 19 Hubble Space Telescope images of the galaxy M87 Figure 13.19 Hubble Space Telescope images of the galaxy M87. The inset shows the center of the galaxy. The wider view shows a jet of material moving away from the center of the galaxy toward the upper right of the figure at about one tenth the speed of light. Such jets are believed to be evidence of a supermassive black hole at the galaxy center. Fig. 13.19, p.410

Fig. P13.5, p.413

Fig. P13.13, p.413

Fig. P13.14, p.413

Fig. P13.17, p.414

Fig. P13.18, p.414

Fig. P13.23, p.414

Fig. P13.24, p.414

Fig. P13.25, p.415

Fig. P13.41, p.416

Fig. P13.51, p.417

Fig. P13.53, p.417

Fig. P13.69, p.419