A balloon of mass \(M\) descends with an acceleration \(a( < g)\). What mass need to be removed from the…

A balloon of mass \(M\) descends with an acceleration \(a( < g)\). What mass need to be removed from the balloon, so that it starts ascending with acceleration, \(a\) ?
  1. \(\frac{2 M}{(a+g)}\)
  2. \(\frac{2 M a}{(a+g)}\)
  3. \(\frac{2 M a}{(a-g)}\)
  4. \(\frac{2 M a}{(g-a)}\)

Solution

The given situation is shown in the figure
In above figures, \(M\) is the mass of balloon and \(M^{\prime}\) is the mass remove from the ballon. According to Newton's law of motion in first case, \(M g-F=M a \Rightarrow F=M g-M a\) ...(i) According to Newton's second law of motion, for second case, \(\begin{aligned} & F-\left(M-M^{\prime}\right) g=\left(M-M^{\prime}\right) a \\ & \Rightarrow F-M g+M^{\prime} g=M a-M^{\prime} a \\ & \Rightarrow M g-M a-M g+M^{\prime} g=M a-M^{\prime} a \text { [from Eq. (i)] } \\ & \Rightarrow M^{\prime} g+M^{\prime} a=M a+M a \\ & \Rightarrow M^{\prime}(g+a)=2 M a \\ & \Rightarrow M^{\prime}=\frac{2 M a}{g+a} \\ \end{aligned}\)

Asked in: AP EAMCET 2020 (18 Sep Shift 2)

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