# Gravity: Magnitude of force, shortest period of rotation, altitude of satellite orbit, maximum height reached by a rocket

See the attached file for the diagrams.

1. In the picture two spheres of mass m and a third sphere of mass M form an equilateral triangle and a fourth sphere of mass m 4 is at the center of the triangle. The net gravitational force on that central sphere from the three other spheres is zero. (a) What is M in terms of m ? (b) if we double the value of m 4 , what then is the magnitude of the net gravitational force on the central sphere ?

2. 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

3. Two concentric shells of uniform density having masses M 1 and M2 are situated as shown in the picture. Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at (a) point A at distance r = a from the center (b) point B at r = b and (c) point C at r = c. The distance r is measured from the center of the shells.

4. A satellite hovers over a certain spot on the equator of (rotating ) earth. What is the altitude of its orbit (called a geosynchronous orbit )

5. A 150.00 kg rocket moving radially outward from Earth has a speed of 3.70 km/s when its engine shuts off 200 km above earths surface. (a) assuming negligible air drag, find the rockets kinetic energy when the rocket is 1000 km above Earths surface. (b) what maximum height above the surface is reached by the rocket ?

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#### Solution Preview

See attached file for full solutions.

1. In the picture two spheres of mass m and a third sphere of mass M form an equilateral triangle and a fourth sphere of mass m 4 is at the center of the triangle. The net gravitational force on that central sphere from the three other spheres is zero. (a) What is M in terms of m ? (b) if we double the value of m 4 , what then is the magnitude of the net gravitational force on the central sphere ?

By symmetry, it should be obvious that mass M has the same mass as the two mass m's. That is because the geometry is so symmetric - with three evenly distributed masses, the only way the force will go to zero at the centre is if all the masses are the same.

From these same symmetry arguments, you should see that the net force at the centre of the triangle is zero no matter what mass m4 you place there.

2. 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

The answer to the (why?) should be obvious - if the planet spins any ...

#### Solution Summary

The magnitude of force and the shortest period of rotation for gravity are analyzed. The detailed solutions, graphs and formulas solve the problems.