Question

Derive the formula for gravitational potential energy of a body located at depth d below the surface of a sphere of radius r​

Answer 1

Answer:

Gravitational Potential of a Uniform Solid Sphere

Consider a thin uniform solid sphere of the radius (R) and mass (M) situated in space. Now,

Case 1: If point ‘P’ lies Inside the uniform solid sphere (r < R):

Inside the uniform solid sphere, E = -GMr/R3.

Using the relation V=−∫E⃗.dr→V=-\mathop{\int }\vec{E}.\overrightarrow{dr}V=−∫E

.dr

over a limit of (0 to r).

The value of gravitational potential is given by,

V = -GM [(3R2 – r2)/2R2]

Case 2: If point ‘P’ lies On the surface of the uniform solid sphere ( r = R ):

On the surface of a uniform solid sphere, E = -GM/R2. Using the relation V=−∫E⃗.dr→V=-\mathop{\int }\vec{E}.\overrightarrow{dr}V=−∫E

.dr

over a limit of (0 to R) we get,

V = -GM/R.

Case 3: If point ‘P’ lies Outside the uniform solid sphere ( r> R):

Using the relation over a limit of (0 to r) we get, V = -GM/R.

Case 4: Gravitational potential at the centre of the solid sphere is given by V = -3/2 × (GM/R).