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Time period of particle in a hole along diameter of earth.

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1. A hole is drilled straight through the center of the earth and a particle is dropped into the hole. Neglect rotational effects:

a) Show that the particle's motion is simple harmonic motion.

b) Calculate the Period of oscillation.

2. If a field vector is independent of the radial distance within a sphere, find the function describing the density , as a function of r within the sphere.

Please give a very clear and complete explanation...this prob has me very confused

The Field is radial and we want that its magnitude remains constant with the distance r from the center. For this how the density of the sphere should change with the distance r from the center.

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Solution Preview

Please see the attached files.

gravitation/potentials
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If a field vector is independent of the radial distance within a sphere, find the function describing the density , as a function of r within the sphere.

Please give a very clear and complete explanation...this prob has me very confused

The Field is radial and we want that its magnitude remains constant with the distance r from the center. For this how the density of the sphere should change with the distance r from the center.

The gravitational field at any point inside a uniform spherical shell is zero. Corresponding to every part of the shell within a small solid angle at that point there is other part on the opposite side such that the field due to both equal and opposite and hence the resultant field will be zero. similarly we can prove that field due to whole shell at a point inside it is zero.

To calculate field at a point inside a sphere of radius R, at a distance r from the center we have to consider the whole mass within the sphere of radius r only and that too at its center O. The mass outside the sphere of radius r will have resultant field zero.

Now the magnitude of the field strength at the surface of a sphere of mass M and radius R is given by

g = GM/R2.

This gives GM = gR2
Here G is the universal gravitation constant.

Consider the sphere of radius r and total mass m, then the magnitude of the gravitational field g at the surface is given by the relation

Gm = gr2 ----------------------- (1)

Now consider an infinitesimally thin layer of thickness dr makes the radius of the sphere equal to r + dr. If the density at this distance r is r then mass of this thin layer will be
dm = area*thickness*density = 4r2*dr*

Hence ...

Solution Summary

The force on the particle inside the earth is calculated and the motion of the particle is discussed. It is Simple Harmonic and the time period of this motion is derived.

The second problem is to derive density as a function of distance in a sphere for the field to remain constant.

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