**Gravitational Potential:**

The work done in bringing a body of unit mass from infinity to the point considered without acceleration is defined as the gravitational potential.

Let us consider the earth as a perfect sphere of radius R and mass M. Let A be the point at a distance r from the center of the earth, where the gravitational potential has to be calculated. Let us take a point C at a distance x from O as shown in the figure. Gravitational force of attraction on a body of unit mass at point C will be

F = GM/ x^{2}

If we displace the unit mass through small displacement dx from C to B, then small amount of work done has to be done which is given by

dW = F dx = GM/x^{2}dx

Total work done in bringing the object of unit mass from ꚙ to the point A is obtained by integrating the above equation within the limits x=ꚙ and x = r, we get

W = ^{r}ʃ_{ꚙ} GM/ x^{2} * dx = - GM/ r

The work done remains in the form of gravitational potential at A.

V_{A} = W = -GM/ r^{2} ..…… (i)

**Expression for Gravitational Potential energy**

Gravitational potential energy of an object at a point in the gravitational field is the work done to bring the object from infinity to that point without acceleration.

Let the earth be a perfect sphere of radius R and mass M. the mass if the earth can be supposed to be concentrated at its center O. Lets calculate the gravitational potential energy of the body of mass m placed at point A in gravitational field, where OA = r as shown in figure. Take two points B and C on the line of OP produced onwardssuch thar OC = x and BC = dx.

Then the gravitational force of attraction on the body at point C will be

F = GMm/ x^{2}

When the body is moved through a very small distance CB = dx, then the small amount of work has to be done which is given by,

dW = F dx = GMm/ x^{2}dx

Total work done on bringing body from infinity to point A is given by,

W = ^{r}ʃ_{ꚙ} GMm/ x^{2} * dx = - GMm/ r

Since the work done is stored in the body as it gravitational potential energy U, the gravitational potential energy of a body at point A in the gravitational field will be

U = -GMm/r

This is the expression for gravitational potential energy of a body of mass m at distance r from the mass M. Therefore, from Equation (i) and (ii) we have

U = V_{A} * m

i.e. gravitational potential energy = gravitation potential * mass of the object

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