Question
Derive a relation for work done in a gravitational field. Using it, (i) find potential difference between a pair of points. (ii) express whether gravitational force is conservative or non-conservative.
✓
Answer
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 be 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. $=-\text{GMm}\Big[\frac{1}{\text{r}}-\frac{1}{\infty}\Big]=\frac{-\text{GMm}}{\text{r}}$ Gravitational potential $\text{V}=\frac{\text{U}}{\text{m}}=\frac{-\text{GM}}{\text{r}}$ The general expression for gravitational potential due to the earth (mass M) at
- Distance r is, V $=\frac{-\text{GM}}{\text{r}}$
Potential at a point A (ra) $=-\frac{\text{GM}}{\text{r}_{\text{a}}}$
Potential at a point B (rb) $=-\frac{\text{GM}}{\text{r}_{\text{b}}}$
$\therefore$ Difference in Potential between the points
$=-\text{GM}\Big(\frac{1}{\text{r}_{\text{a}}}-\frac{1}{\text{r}_{\text{b}}}\Big)$
- Since work done against gravitational force is (a) independent of path and dependent only on the initial and final points and (b) the work done in a closed path is zero, it is a conservative force.
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