Question

give an expression for gravitational potential energy ​

Answer 1

Answer:

Consider a source mass ‘M’ is placed at a point along the x-axis, initially, a test mass ‘m’ is at infinity. A small amount of work done in bringing it without acceleration through a very small distance (dx) is given by

dw = Fdx

Here, F is an attractive force and the displacement is towards the negative x-axis direction so F and dx are in the same direction. Then,

dw = (GMm/x2)dx

Integrating on both sides

w=∫r∞GMmx2dx w=−[GMmx]r∞

⇒ w = -GMm/r – (-GMm/∞)

⇒ w = -GMm/r

Since this work done is stored as its potential energy U, therefore gravitational potential energy at a point which is at a distance ‘r’ from the source mass is given by;

U = -GMm/r

If a test mass moves from a point inside the gravitational field to the other point inside the same gravitational field of source mass, then the change in potential energy of the test mass is given by;

ΔU = GMm (1/ri – 1/rf)

If ri > rf then ΔU is negative.

⇒ Check: Acceleration due to Gravity

Expression for Gravitational Potential Energy at Height (h) – Derive ΔU = mgh

If a body is taken from the surface of the earth to a point at a height ‘h’ above the surface of the earth, then ri = R and rf = R + h then,

ΔU = GMm [1/R – 1/(R+h)]

ΔU = GMmh/R(R + h)

When, h<<R, then, R + h = R and g = GM/R2.

On substituting this in the above equation we get,

Gravitational Potential Energy ΔU = mgh