Question

A small body is initially at a distance \[r\]from the center of earth. \[r\] is greater than the radius of the earth. If it takes \[W\]joule of work to move the body from this position to another position at a distance $2r$ measured from the center of the earth, then how many joules would be required to move it from this position to a new position at a distance of $3r$ from the center of the earth.
\[A.\dfrac{W}{5}\;\]
\[B.\dfrac{W}{3}\]
\[C.\dfrac{W}{2}\]
\[D.\dfrac{W}{6}\]

Answer 1

Hint: To solve the problem the potential energy has to be found and after this, the work done by the body can be calculated using the relation between potential energy and the work done. Note that, the work done has to be calculated for two cases given in the problem and then compared to each other. Gravitational potential energy is the energy possessed or acquired by an object due to a change in its position when it is present in a gravitational field. In simple terms, it can be said that gravitational potential energy is an energy that is related to gravitational force or gravity.

Formula used:
Potential energy for the distance ${R_1}$ from the center of the earth, ${U_1} = - \dfrac{{GMm}}{{{R_1}}}$
Potential energy for the distance ${R_2}$ from the center of the earth ${U_2} = - \dfrac{{GMm}}{{{R_2}}}$
And,
Work done $W = {U_2} - {U_1}$
\[G = \] gravitational constant
$M = $ Mass of earth
$m = $ mass of planet

Complete answer:
Work done by an object in the field of gravitation is defined by the change in potential energy of the object.
Now, for the distance ${R_1}$ from the center of the earth, the Potential energy will be ${U_1} = - \dfrac{{GMm}}{{{R_1}}}$
And, that for the distance ${R_2}$ from the center of the earth will be ${U_2} = - \dfrac{{GMm}}{{{R_2}}}$
\[G = \] gravitational constant
$M = $ Mass of earth
$m = $ mass of planet
Hence, by definition the Work done $W = {U_2} - {U_1}$
For the 1st case,
Given that, ${R_1} = r$and, ${R_2} = 2r$ and work done = $W$
So, \[W = - \dfrac{{GMm}}{{2r}} - \left( { - \dfrac{{GMm}}{r}} \right)\]
$ \Rightarrow W = \dfrac{{GMm}}{{2r}}$
Now in the 2nd case, the body moves from a distance ${R_1} = 2r$to a distance ${R_2} = 3r$
If work done is $W'$
then, \[W' = - \dfrac{{GMm}}{{3r}} - \left( { - \dfrac{{GMm}}{{2r}}} \right)\]
$ \Rightarrow W' = \dfrac{1}{3}\dfrac{{GMm}}{{2r}}$
$ \Rightarrow W' = \dfrac{1}{3}W$ [since, $ \Rightarrow W = \dfrac{{GMm}}{{2r}}$]

Option \[B. \Rightarrow \dfrac{W}{3}\] is correct.

Note:
The force of gravity can be considered as a conservation force since the work done does not depend upon the path followed. The potential energy that arises due to this force is called gravitational potential energy. Also, Newton’s law of universal gravitation states that the gravitational force attraction between any two particles of different masses separated by distance will have magnitude F. This is gravity force and here we get a new constant called gravitational constant.