question_answer

A big size balloon of mass M is held stationary in air with the help of a small block of mass \[\mathbf{M}/\mathbf{2}\]tied to it by a light string such that both float in mid air.  Describe the motion of the balloon and the block when the string is cut. Support your answer with calculations.                               

Answer:

                The situation is shown in Fig. When the balloon is held stationary in air, the forces acting on it get balanced. Up thrust \[=\] Weight of balloon \[+\] Tension in string \[U=Mg+T\]                                           ...(1) For the small block of mass M / 2 floating stationary in air, \[T=\frac{M}{2}g\]                                                          ? (2) From (1) and (2), \[U=Mg+\frac{M}{2}g=\frac{3}{2}Mg\] When the string is cut, \[T=\text{ }0\]. The small block will begin to fall freely. The balloon will rise up with an acceleration \[a\] such that \[U-Mg=Ma\] or                \[\frac{3}{2}Mg-Mg=Ma\] or                \[a=\frac{g}{2},\]in the upward direction.