Gravitational Potential

Physics

definition

Gravitational Potential (V)

The gravitational potential (V) is the gravitational potential energy (U) per unit mass: where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity.

definition

Gravitational Potential due to a point mass

Gravitational Potential due to a point mass is given by:

example

Gravitational Potential due to a uniform ring

Example: Two rings having masses and , respectively, having the same radius are placed coaxially as shown in the figure. If the mass distribution on both the rings is non-uniform, then what is the gravitational potential at point ?

Solution:
Gravitational potential due to ring () at an axial point which is x unit away from the center,    Thus gravitational potential at P,     



example

Gravitational Potential inside and outside of a thin spherical shell

Gravitational field inside the shell:


Gravitational field outside the shell:

example

Gravitational Potential due to a solid sphere

Example The earth does not have a uniform density; it is most dense at its centre and least dense at its surface. An approximation of its density is  , where and    is the distance from the centre of earth. Use m for the radius of earth approximated as a sphere, Imagine dividing the earth into concentric, elementary spherical shells. Each shell has radius ,  thickness , volume and mass . By integrating    from zero to  the mass of earth can be found. Knowing the fact that a uniform spherical shell gives no contribution to acceleration due to gravity inside it, we can also find    as a function of . If B 0, then find gravitational potential at the centre?

Solution:
We have,


     
Now,


Thus,

We know,



Integrating from 0 to R,


                       

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