The Gravitational Force

The Gravitational Force - All points on the small hoop...

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The Gravitational Force Gravity is the weakest force we know, but it is the force of gravity that controls the evolution of the universe. Every body in the universe attracts every other body. Newton proposed that the magnitude of this force is given by where m 1 and m 2 are the masses of the particles, r is the distance between them and G is a universal constant whose value is G = 6.67 x 10 -11 N m 2 /kg 2 The gravitational forces between two particles act along the line joining them, and form an action-reaction pair (see Figure 14.1). Figure 14.1. The gravitational force. In real life we are not dealing with point particles; instead we are dealing with extended objects. To evaluate the gravitational force between extended objects, the shell theorem can be used: "A uniform shell of matter attracts an external particle as if all the shell's mass were concentrated at its center" Proof : Figure 14.2 shows a shell located a distance r from a particle with mass m. The radius of the shell is R and its mass is M. The mass density of the shell is given by
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Unformatted text preview: All points on the small hoop indicated in Figure 14.2 have the same distance to the particle m. The magnitude of the gravitational attraction between any of these points and the mass m is therefore the same. The net force between the hoop and mass m acts along the axis connecting the center of the shell and mass m. The area of the hoop is given by Figure 14.2. Shell theorem. and its mass m is equal to The net force is equal to The angles [theta] and a can be eliminated by using the following relations: and Differentiating the first of these two equations with respect to [theta] we obtain or Further more we see that The total force acting on mass m can now be obtained easily The shell theorem immediately shows that a sphere of uniform density (and mass M) attracts an external particle as if all the mass of the sphere is concentrated in its center. In a similar fashion we can proof that a uniform shell of matter exerts no gravitational force on a particle located inside it ....
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