A balloon of mass M with a light rope and monkey of mass m are at rest in mid air. If the monkey climbs up the rope and reaches the top of the rope, the distance by which the balloon descends will be-?

To solve the problem of how far the balloon descends when the monkey climbs to the top of the rope, we can apply the principle of conservation of momentum and consider the system as a whole. Let’s break it down step by step.

Understanding the System

We have a balloon of mass M and a monkey of mass m, both initially at rest in mid-air. When the monkey climbs up the rope, it exerts a force on the rope, which in turn affects the balloon. Since the system is isolated, we can analyze the movements of both the monkey and the balloon.

Key Concepts

• Conservation of Momentum: In an isolated system, the total momentum before an event must equal the total momentum after the event.
• Relative Motion: The movement of one object can affect the position of another object in a connected system.

Analyzing the Movement

When the monkey climbs up the rope, it moves a distance d upwards. According to Newton's third law, the balloon will move downwards in response to the monkey's upward movement. We need to find out how far the balloon descends when the monkey reaches the top of the rope.

Setting Up the Equation

Let’s denote the distance the balloon descends as D. When the monkey climbs a distance d, the rope effectively shortens by d, causing the balloon to descend. However, we must consider the total movement of the system:

When the monkey climbs a distance d, the balloon will descend a distance D. The key relationship here is that the total length of the rope remains constant. Therefore, we can express this relationship mathematically:

When the monkey climbs up a distance d, the balloon descends a distance D, and the total change in height must equal the length of the rope:

d + D = L

Where L is the initial length of the rope. Since the monkey climbs d, the balloon must descend D such that:

D = d * (m / M)

Final Calculation

From the equation above, we can see that the distance D the balloon descends is proportional to the distance d the monkey climbs, scaled by the ratio of the masses. This means that if the monkey is significantly lighter than the balloon, the balloon will descend a smaller distance compared to the distance the monkey climbs.

Example

For instance, if the monkey has a mass of 10 kg and the balloon has a mass of 90 kg, and the monkey climbs 1 meter (d = 1 m), we can calculate D:

D = 1 m * (10 kg / 90 kg) = 1/9 m ≈ 0.11 m

This means the balloon would descend approximately 0.11 meters when the monkey climbs 1 meter. This relationship illustrates how the masses of the objects influence their movements in a connected system.

In summary, the distance the balloon descends is determined by the ratio of the masses of the monkey and the balloon, scaled by the distance the monkey climbs. This principle of conservation of momentum helps us understand the dynamics of the system effectively.