Question

A balloon of mass M is descending with a constant acceleration g/3. When a mass m is released from the balloon it starts rising with the same acceleration g/3. The value of m is (Assuming that its volume does not change)

Answer 1

In any observable way, mass does not impact acceleration due to gravity and both the quantities are independent of one another. Only when forces other than gravity are also at work, Light things accelerate more gently than heavy things.

Complete step by step answer:
Let M be the mass when the balloon descends with a constant acceleration of g/3 and m be the mass when it starts to rise with the same acceleration ie., g/3.
The force acting on the balloon while descending is given by,

$$R = M(g - a) \\\ \Rightarrow a = \dfrac{g}{3} \\\ \Rightarrow R = M(g - \dfrac{g}{3}) \\\ \Rightarrow R = M\dfrac{{2g}}{3} \\\$$

The force acting on the balloon while rising is given by,
$R = (M - m)(g + a)$
Substitute the value of R and a in the above equation
$\left( {\dfrac{{2Mg}}{3}} \right) = (M - m)\left( {g + \dfrac{g}{3}} \right) \\\ \Rightarrow \left( {\dfrac{{2Mg}}{3}} \right) = (M - m)\left( {\dfrac{{4g}}{3}} \right) \\\ \Rightarrow M = 2(M - m) \\\ \Rightarrow M = 2M - 2m \\\ \Rightarrow 2m = 2M - M \\\ \Rightarrow 2m = M \\\ \therefore m = \dfrac{M}{2} \\\$
Therefore, the value of the mass is $m = \dfrac{M}{2}$

Note: Often there is a confusion between mass and weight. Mass is the amount of matter that an object contains. Whereas weight is the force exerted by the gravity on an object or the force that is necessary to support the object.