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Gravitation — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsGravitation5 Marks
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.
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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

  1. 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

$=-\text{GM}\Big(\frac{1}{\text{r}_{\text{a}}}-\frac{1}{\text{r}_{\text{b}}}\Big)$

  1. 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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