Gravitational Potential

“The gravitational potential at a point in a gravitational field is defined as the work done against the field in bringing a unit mass from infinity to the point considered”. It is a scalar quantity & is denoted by ‘V’. \(V=\frac{W}{m}\),Work done against the field in bringing the mass ‘m’ from infinity to the point in question. But \(dw=-F.dx\) $$ \therefore V = \frac{W}{m} = \frac{1}{m}\int^{r}_{\infty}dw = \frac{1}{m}\int^{r}_{\infty}-F.dx $$ $$ = -\frac{1}{m}\int^{r}_{\infty}\left(-\frac{GMm}{x^2}\right).dx $$ $$ = GM\int^{r}_{\infty}\frac{dx}{x^2} $$ $$ = GM \left\{-\frac{1}{x}\bigg]^{r}_{\infty}\right\} $$ $$ =GM\left(-\frac{1}{r}\right) $$ $$ \therefore V = -\frac{GM}{r} $$ Here, ‘V’ is the gravitational potential due to the body having mass ‘M’ at a point which is located at a distance ‘r’ from it.

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