In the system shown in figure all pulleys are mass less and the string is inextensible and light. After the system is released, if $m_{1} = 1 kg, m_{2} = 2 kg$ and $m_{3} = 3 kg$ then what must be value of mass $m_{4}$ so that it accelerates downwards ?

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TestHub · Accepted Answer

Take downward as positive; let T be the tension in each segment.

$a_{4} = a_{1} + a_{2} + a_{3} (rope length)$

$m_{i} a_{i} = m_{i} g - 2 T (i = 1, 2, 3)$

$m_{4} a_{4} = m_{4} g - 2 T$

$a_{1} + a_{2} + a_{3} = 3 g - 2 T (1 + \frac{1}{2} + \frac{1}{3}) = 3 g - \frac{11 T}{3}$

$g - \frac{2 T}{m _{4}} = 3 g - \frac{11 T}{3}$

$T = \frac{6 g m _{4}}{11 m _{4} - 6}$

$a_{4} = g - \frac{2 T}{m _{4}} = \frac{g ( 11 m _{4} - 18 )}{11 m _{4} - 6}$

$a_{4} > 0 \Rightarrow m_{4} > \frac{18}{11} kg (C)$

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