First slide

Pseudo force

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 ?

Moderate

A

$\frac{2 m a}{g + a}$
B

$\frac{2 m a}{g - a}$
C

$\frac{m a}{g + a}$
D

$\frac{m a}{g - a}$

Solution

Let F be the upthrust of the air. As the balloon is descending down with an acceleration a.
$∴ m g - F = m a . . . . (i)$
Let mass $m_{0}$ be removed from the balloon so that it starts moving up with an acceleration a. Then,
$\begin{array}{l} F - (m - m_{0}) g = (m - m_{0}) a \\ F - m g + m_{0} g = m a - m_{0} a \end{array}$ . . . .(ii)
$Adding eqn. (i) and eqn. (ii), we get$
$\begin{array}{l} m_{0} g = 2 m a - m_{0} a \\ m_{0} g + m_{0} a = 2 m a \\ m_{0} (g + a) = 2 m a \\ m_{0} = \frac{2 m a}{g + a} \end{array}$

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