Question
Define gravitational potential energy of a body. Derive an expression for the gravitational potential energy of a body of mass 'm' located at a distance 'r' from the centre of the earth.
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Answer
Gravitational potential energy. The work done in carrying a mass 'm' from infinity to a point at distance r is called gravitational potential energy. $\text{G.P.E.}=-\frac{\text{GMm}}{\text{r}}$i.e., G.P.E. = Mass × Gravitational potential
It is a scalar quantity measured in joule. Negative sign means that the mass is bound to M. The gravitational force of attraction between M and m when x is the distance between their centres is given by, $\text{F}=\frac{\text{GMm}}{\text{x}^2}$ Suppose the body is moved through a distance dx, therefore, work done is given by, $\text{dW}=\text{Fdx}=\frac{\text{GMm}}{\text{x}^2}\text{dx}$ When the body is brought from infinity to some distance r, We write, $\int\text{dW}=\int^\limits{\text{x}=\text{r}}_\limits{\text{x}=\infty}\frac{\text{GMm}}{\text{x}^2}\text{dx}$ $\text{or }\text{W}=\text{GMm}\Big[\frac{-1}{\text{x}}\Big]^{\text{r}}_{\infty}$ $=-\text{GMm}\Big[\frac{1}{\text{r}}-\frac{1}{\infty}\Big]=\frac{-\text{GMm}}{\text{r}}$ This amount of work done is the change in the potential energy of the body. $\therefore\text{P.E.}\text{ U }=\frac{-\text{GMm}}{\text{r}}$
It is a scalar quantity measured in joule. Negative sign means that the mass is bound to M. The gravitational force of attraction between M and m when x is the distance between their centres is given by, $\text{F}=\frac{\text{GMm}}{\text{x}^2}$ Suppose the body is moved through a distance dx, therefore, work done is given by, $\text{dW}=\text{Fdx}=\frac{\text{GMm}}{\text{x}^2}\text{dx}$ When the body is brought from infinity to some distance r, We write, $\int\text{dW}=\int^\limits{\text{x}=\text{r}}_\limits{\text{x}=\infty}\frac{\text{GMm}}{\text{x}^2}\text{dx}$ $\text{or }\text{W}=\text{GMm}\Big[\frac{-1}{\text{x}}\Big]^{\text{r}}_{\infty}$ $=-\text{GMm}\Big[\frac{1}{\text{r}}-\frac{1}{\infty}\Big]=\frac{-\text{GMm}}{\text{r}}$ This amount of work done is the change in the potential energy of the body. $\therefore\text{P.E.}\text{ U }=\frac{-\text{GMm}}{\text{r}}$
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