First slide

Connected bodies

Question

A large cubical shaped block of mass M rests on a fixed horizontal surface. Two blocks of mass $m_{1}$ aod $m_{2}$ are connected by a light inextensible string passing over a light pulley as shown. Neglect friction everyvhere. Then the constant horizontal force of magnitude F that should be applied to M so that $m_{1} and m_{2}$ do not move relative to M is:

Difficult

A

$F = \frac{m_{2}}{m_{1}} (m_{1} + m_{2} + M) g$
B

$F = \frac{m_{1}}{m_{2}} (m_{1} + m_{2} + M) g$
C

$F = \frac{m_{1}}{m_{2}} (m_{1} + M) g$
D

$F = \frac{m_{2}}{m_{1}} (m_{1} + M) g$

Solution

Let a be the acceleration of cubical block of mass M from ground frame. Then from the frame of block of mass M, forces on $m_{1} and m_{2}$ are as shown. Hence for $m_{1} and m_{2}$ to remain at rest in frame of M
$m_{2} a = m_{1} g or a = \frac{m_{1}}{m_{2} g}$
$∴ F = \frac{m_{1}}{m_{2}} (m_{1} + m_{2} + M) g$

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