Example 13.1 Billiards, Anyone? Three 0.300-kg billiard balls are placed on a table at the corners of a right triangle. The sides of the triangle are of lengths a = 0.400 m, b = 0.300 m, and c = 0.500 m. Calculate the gravitational force vector on the cue ball (designated m1) resulting from the other two balls as well as the magnitude and direction of this force.

Example 13.2 The Density of the Earth Using the known radius of the Earth and that g = 9.80 m/s² at the Earth’s surface, find the average density of the Earth.

Example 13.3 The Weight of the Space Station The International Space Station operates at an altitude of 350 km. Plans for the final construction show that material of weight 4.22 x 106 N, measured at the Earth’s surface, will have been lifted off the surface by various spacecraft during the construction process. What is the weight of the space station when in orbit?

Example 13.4 The Mass of the Sun Calculate the mass of the Sun, noting that the period of the Earth’s orbit around the Sun is 3.156 = 107 s and its distance from the Sun is 1.496 = 1011 m.

Example 13.5 A Geosynchronous Satellite Consider a satellite of mass m moving in a circular orbit around the Earth at a constant speed v and at an altitude h above the Earth’s surface.

(A) Determine the speed of satellite in terms of G, h, RE (the radius of the Earth), and ME (the mass of the Earth).

(B) If the satellite is to be geosynchronous (that is, appearing to remain over a fixed position on the Earth), how fast is it moving through space?

Example 13.6 The Change in Potential Energy A particle of mass m is displaced through a small vertical distance Δy near the Earth’s surface. Show that in this situation the general expression for the change in gravitational potential energy given by Equation 13.13 reduces to the familiar relationship ΔU = mg Δy.

Example 13.7 Changing the Orbit of a Satellite A space transportation vehicle releases a 470-kg communications satellite while in an orbit 280 km above the surface of the Earth. A rocket engine on the satellite boosts it into a geosynchronous orbit. How much energy does the engine have to provide?

Example 13.8 Escape Speed of a Rocket Calculate the escape speed from the Earth for a 5000-kg spacecraft and determine the kinetic energy it must have at the Earth’s surface to move infinitely far away from the Earth.