Question

A Rope ladder with a length $l$ meters carrying a man with a mass $m$ at its end is attached to the basket of a balloon with a mass $M$. The entire system is in equilibrium in the air. As the man climbs up the ladder into a balloon, the balloon descends by a height $h$. In this Case, the potential Energy of the balloon:
(A) Decreases by $mgh$
(B) Increases by $mgh$
(C) Increases by $mg(l - h)$
(D) Increases by $mgl$

Answer 1

Hint: When the man climbs up the rope there is application of force. This force results in the displacement of the balloon. Hence, there is a certain work done in changing the height of the balloon. This work done will be equal to the change in potential energy.
Formula Used:-
Potential Energy, $P.E = mgh$
$ \to m$is the mass
$ \to g$ is the acceleration due to gravity
$ \to h$ is the height

Complete step by step solution:
As we know the formula for potential energy due to gravity is given by,
Potential Energy, $P.E = mgh$
Hence, the total work done by the man is given as $mgl$
That is, work done, $dW = mgl$
Where, $l$ is the length of the ladder. (As mentioned in Question)
     $m$ is the mass of the man
     $g$ is the acceleration due to gravity
As the man moves up through the rope, the balloon descends a height $h$
So, the gain in potential energy of the man is given by $(mgl - mgh)$
Taking $dE$ is the change in potential energy of the balloon.
From the law of conservation of energy, we get,
Total work done of the system= Change in potential energy of the system
So, we have
$mgl = mgl - mgh + dE$…………………………………………………………………………………………………….. (1)
So,
By further simplification of equation we get,
$dE = mgh$
So, the potential Energy of the balloon increases by $mgh$.

So the final answer is option (B) Increases by $mgh$

Note: The gravitational potential energy of an object near Earth's surface is due to its position in the mass-Earth system. It is the energy stored in an object as the result of its height or its vertical position. The energy stored within the system is because of the gravitational attraction of the Earth for the system.