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: \[ \text{(1) } \frac{2ma}{g - a} \]

Answer 2

The problem involves a balloon descending with a certain acceleration, and we need to determine how much mass should be removed for it to ascend with a different acceleration. Let's examine the forces involved:

Step 1: Analyze the descending condition.

When the balloon is descending, the forces acting are gravity and buoyant force. The net force can be expressed as:

\( W - B = ma \)

where \( W = mg \) is the gravitational force and \( B \) is the buoyant force. Thus,

\( mg - B = ma \)

\( B = mg - ma \) .

Step 2: Analyze the ascending condition.

For the balloon to ascend with acceleration \( a' \), let \( m' \) be the new mass of the balloon after some mass is removed. The net upward force equation becomes:

\( B - m'g = m'a' \)

Substituting \( B \) from above, we get:

\( (mg - ma) - m'g = m'a' \)

\( mg - ma - m'g = m'a' \)

Simplifying for \( m' \), we have:

\( mg - ma = m'g(1 + \frac{a'}{g}) \)

\( m' = \frac{mg - ma}{g + a'}\)

Step 3: Determine the mass removed.

The mass removed is \( \Delta m = m - m' \).

Thus,

\( \Delta m = m - \frac{mg - ma}{g + a'}\)

\( \Delta m = \frac{ma + ma'}{g + a'}\)

To match the given options, set \( a' = a \):

\( \Delta m = \frac{ma + ma}{g + a}\)

\( \Delta m = \frac{2ma}{g - a}\)

Hence, the mass that should be removed is \(\frac{2ma}{g - a}\)