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

Short answer

The correct answer is (A) 20 dyne.

Step-by-step solution

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.

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 \(\frac{10 \; g}{2.5 \;\text{s}} = 4\; \frac{g}{s}\).

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 = \(4\; \frac{g}{s} \times 5\;\frac{cm}{s}\) = \(20\;\frac{g\cdot cm}{s^2}\).

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.