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# Solutions for a number of questions regarding gravitational force, circular orbits, and harmonic motion

Dear OTA,

I just need explanations, can you do it please have it in detail.

#### Solution Preview

The answers can be found below and also attached. Please see the attachment for figures and equations.

5) Field Inside a Spherical Shell (see attachment for figure)

This turns out to be surprisingly simple! We imagine the shell to be very thin, with a mass density kg per square meter of surface. Begin by drawing a two-way cone radiating out from the point P, so that it includes two small areas of the shell on opposite sides: these two areas will exert gravitational attraction on a mass at P in opposite directions. It turns out that they exactly cancel.

This is because the ratio of the areas A1 and A2 at distances r1 and r2 are given by A1/A2 = r1^2/r2^2 : since the cones have the same angle, if one cone has twice the height of the other, its base will have twice the diameter, and therefore four times the area. Since the masses of the bits of the shell are proportional to the areas, the ratio of the masses of the cone bases is also r1^2/r2^2. But the gravitational attraction at P from these masses goes as Gm/r^2 , and that r2 term cancels the one in the areas, so the two opposite areas have equal and opposite gravitational forces at P.

In fact, the gravitational pull from every small part of the shell is balanced by a part on the opposite side?you just have to construct a lot of cones going through P to see this. (There is one slightly tricky point?the line from P to the sphere's surface will in general cut the surface at an angle. However, it will cut the opposite bit of sphere at the same angle, because any line passing through a sphere hits the two surfaces at the same angle, so the effects balance, and the base areas of the two opposite small cones are still in the ratio of the squares of the distances r1, r2.)

6) This is a straight problem, you can solve it also by guass law. Simply take the mass of shell as m and ...

#### Solution Summary

Answer and explanation is given for a series of problems relating to gravitational force, circular orbits, and harmonic motion.

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