QS Study

Gravitational potential due to a point mass

We know, the amount of work done in bringing a unit mass from infinity to a point in the gravitational field is called the gravitational potential at that point. Now, we need to find the equation of gravitational potential due to a point mass.

Let an object of mass M be placed at point O [Figure]. P is a point at a distance r from O. The gravitational potential at point P is to be found out.

At point P the attractive force on unit mass in the direction of O i.e., gravitational intensity = GM/r2. Now, the amount of work done i.e., potential in taking the unit mass by a very small distance dr is,

dV = force x displacement = intensity displacement = (Gm/r2) dr

So, Total work done in bringing the unit mass from infinity to point P is,

V = ∫ dV = r=rr=∞ (Gm/r2) dr

or, V = GM r=rr=∞ (1/r2.dr) = GM [- 1/r]r

or, V = – GM/r

Here negative sign indicates that the work has not been done by any external agent; the gravitational force itself has done the work.

Special cases:

(i) By considering all the masses of a uniform solid sphere or uniform sphere at their centre potential at outside point of those spheres can be found out.

(ii) Potential at all points inside a solid sphere remains constant. This potential is equal to the potential at the surface of the sphere. If M is the mass of the sphere and r is the radius then potential at any point inside sphere is V = – GM/r.