Problem 401 - JEE Physics

Question

A balloon has a mass of \(10 \mathrm{~g}\) in air, The air escapes from the balloon at a uniform rate with a velocity of \(5 \mathrm{~cm} / \mathrm{s}\) and the balloon shrinks completely in \(2.5 \mathrm{sec}\). calculate the average force acting on the balloon. (A) 20 dyne (B) 5 dyne (C) 0 dyne (D) 10 dyne

Step-by-Step Solution

Answer

The correct answer is (A) 20 dyne.

Step 1: Write down the initial and final mass of the balloon

Initially, the balloon has a mass of \(10 \;g\). Since the balloon shrinks completely in \(2.5 \;\text{seconds}\) and the air escapes from it, we can consider the final mass of the balloon to be zero.

Step 2: Calculate the air's mass loss rate

We are given that the air escapes from the balloon at a uniform rate with a velocity of \(5 \;\frac{cm}{s}\). To calculate the mass loss rate, we need to calculate the mass of air that comes out of the balloon per second. The time it takes for the balloon to shrink completely is \(2.5 \;\text{seconds}\), and it loses all its mass in this time. So, the mass loss rate would be \(\frac{10 \; g}{2.5 \;\text{s}} = 4\; \frac{g}{s}\).

Step 3: Calculate the momentum change per second

Now, we have the mass loss rate and the velocity of the air escaping the balloon. Using these, we can calculate the rate at which the momentum of the air changes. The momentum change per second = mass loss rate × velocity = \(4\; \frac{g}{s} \times 5\;\frac{cm}{s}\) = \(20\;\frac{g\cdot cm}{s^2}\).

Step 4: Calculate the average force acting on the balloon

The rate at which the momentum changes is given by the force acting on the system. So, the average force acting on the balloon equals the momentum change per second which we calculated in step 3. Hence, the average force acting on the balloon is \(20\;\frac{g\cdot cm}{s^2}\) = \(20 \;\text{dyne}\). So, the correct answer is (A) 20 dyne.

Key Concepts

Momentum Change

Momentum change refers to the alteration in momentum that a body experiences over time. In this scenario, as air escapes the balloon, the air exerts a force causing a change in momentum. Momentum, which is the product of mass and velocity, has the formula:

• Momentum = Mass × Velocity

The momentum change occurs when there is either a change in mass, velocity, or both—typically from forces acting over a time span.
The formula for the rate of momentum change is:

• Rate of Momentum Change = Mass Loss Rate × Velocity of Air Escape

This was calculated in step 3 to be 20 dynes, showing the average force acting on the balloon due to the escaping air.

Mass Loss Rate

Understanding the mass loss rate is crucial in calculating forces and momentum changes. This rate signifies how much mass leaves the system over time. Here, it describes how fast the air mass escapes the balloon during the shrinkage process.
To compute the mass loss rate, we apply the formula:

• Mass Loss Rate = Initial Mass / Time

For the balloon, the rate was calculated as:

• 4 grams per second (g/s), indicating how fast the mass diminishes each second

Knowing this value helps find the rate of change of momentum, further leading us to calculate the average force.

Velocity of Air Escape

The velocity of air escape indicates the speed at which air exits the balloon. It plays a significant role in determining the force exerted by the escaping air. Velocity, when combined with mass changes, directly affects momentum.
Given in the problem, the velocity of air escape from the balloon is 5 centimeters per second (cm/s). This uniform rate enables us to uniformly calculate the mass loss and momentum change over the 2.5 second period.

• Using velocity alongside the mass loss rate provides the link to find momentum change and hence calculating average force.

Dyne Unit

The dyne is a unit of force in the centimeter-gram-second (CGS) system of units. It is often used in physics and mechanics to measure small forces and is especially handy in problems involving mass and acceleration in grams and centimeters.
One dyne equates to the force necessary to accelerate a 1 gram mass by 1 centimeter per second squared (cm/s²). Hence, in this context, when we calculated the average force acting on the balloon as 20 dyne, it implies that the escaping air provided enough force to achieve a momentum change of 20 gram-centimeters per second squared per unit of time, ultimately showing the average force acting on the balloon during air escape.