JEE Main & Advanced Physics Gravitation / गुरुत्वाकर्षण Gravitational Potential Energy

The gravitational potential energy of a body at a point is defined as the amount of work done in bringing the body from infinity to that point against the gravitational force.       

\[W=\int_{\infty }^{r}{\frac{GMm}{{{x}^{2}}}dx=-GMm\,\left[ \frac{1}{x} \right]_{\infty }^{r}}\]            

\[W=-\frac{GMm}{r}\]                          

This work done is stored inside the body as its gravitational potential energy            

\[\therefore U=-\frac{GMm}{r}\]

(i) Potential energy is a scalar quantity.

(ii) Unit : Joule

(iii) Dimension : \[[M{{L}^{2}}{{T}^{-2}}]\]

(iv) Gravitational potential energy is always negative in the gravitational field because the force is always attractive in nature.

(v) As the distance \[r\] increases, the gravitational potential energy becomes less negative i.e., it increases.

(vi) If \[r=\infty \] then it becomes zero (maximum)

(vii) In case of discrete distribution of masses

Gravitational potential energy

\[U=\sum {{u}_{i}}=-\left[ \frac{G{{m}_{1}}{{m}_{2}}}{{{r}_{12}}}+\frac{G{{m}_{2}}{{m}_{3}}}{{{r}_{23}}}+........ \right]\]

(viii) If the body of mass \[m\] is moved from a point at a distance \[{{r}_{1}}\] to a point at distance \[{{r}_{2}}({{r}_{1}}>{{r}_{2}})\] then change in potential energy \[\Delta U=\int_{{{r}_{1}}}^{{{r}_{2}}}{\frac{GMm}{{{x}^{2}}}}dx=-GMm\,\left[ \frac{1}{{{r}_{2}}}-\frac{1}{{{r}_{1}}} \right]\]

or \[\Delta U=GMm\,\left[ \frac{1}{{{r}_{1}}}-\frac{1}{{{r}_{2}}} \right]\]

As \[{{r}_{1}}\] is greater than \[T=2\pi \,\,\sqrt{\frac{{{\left( R+h \right)}^{3}}}{g\,{{R}^{2}}}}\], the change in potential energy of the body will be negative. It means that if a body is brought closer to earth it's potential energy decreases.

(ix) Relation between gravitational potential energy and potential \[U=-\frac{GMm}{r}\]\[=m\left[ \frac{-GM}{r} \right]\] \ \[=2\,\pi \,\,\sqrt{\frac{R}{g}}{{\left( 1+\frac{h}{R} \right)}^{3/2}}\]

(x) Gravitational potential energy at the centre of earth relative to infinity.

\[{{U}_{centre}}=m\,{{V}_{centre}}\]\[=m\left( -\frac{3}{2}\frac{GM}{R} \right)\] \[=-\frac{3}{2}\frac{GMm}{R}\]

(xi) Gravitational potential energy of a body at height h from the earth surface is given by \[GM=g{{R}^{2}}\]