Question

A balloon with mass $ m $ is descending down with an acceleration $ a $ (where, $ a<g $). How much mass should be removed from it so that it starts moving up with an acceleration $ a' $?

• A. \( \frac{2ma}{g - a} \)
• B. \( \frac{ma}{g - a} \)
• C. \( \frac{ma}{g + a} \)
• D. \( \frac{ma}{g} \)

Hint:

To make the balloon ascend, reduce the mass to make the upward force greater than the downward force caused by gravity.

Answer 1

The Correct Option is A

Let the mass of the balloon be \( m \), and the acceleration due to gravity be \( g \). The balloon is descending with an acceleration \( a \), so the net force on the balloon is: \[ F_{\text{net}} = mg - T = ma \] where \( T \) is the tension in the string. Rearranging this, we get: \[ T = mg - ma \] Now, we need to remove mass from the balloon so that it starts moving upward with acceleration \( a' \). The tension in the string should now be equal to the force required to accelerate the balloon upwards: \[ T = m' g + m' a' \] where \( m' \) is the new mass of the balloon after mass is removed. Equating the tensions: \[ mg - ma = m' g + m' a' \] Since we need the balloon to move upwards with acceleration \( a' \), and \( a' = a \), we can solve for \( m' \). Substituting the values: \[ m' = \frac{2ma}{g - a} \]
Thus, the mass that should be removed from the balloon is \( m - m' \), which gives: \[ m - \frac{2ma}{g - a} \]
Thus, the correct answer is: \[ \text{(1) } \frac{2ma}{g - a} \]