Three masses $m_{1}$ $m_{2}$ and $m_{3}$ are attached to a string as shown in the figure. All three masses are held at rest and then released. To keep $m_{3}$ at rest the condition is

\frac{4}{m_{1}} + \frac{1}{m_{2}} = \frac{3}{m_{3}}

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m_{1} + m_{2} = m_{3}

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\frac{4}{m_{3}} = \frac{1}{m_{1}} + \frac{1}{m_{2}}

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\frac{1}{m_{1}} + \frac{2}{m_{2}} = \frac{3}{m_{3}}

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Solution

The correct option is D $\frac{4}{m_{1}} + \frac{1}{m_{2}} = \frac{3}{m_{3}}$
$T = m_{3} g$
$2 T - m_{1} g = m_{1} a_{1}$ __(1)
$m_{2} g - 2 T = 2 m_{2} a_{1}$
$\Rightarrow 2 m_{2} g - 25 = 4 m_{2} a_{1}$ __(2)
from (1)
$2 m_{3} g - m_{1} g = m_{1} a_{1}$
from (2)
$2 m_{2} g - 2 m_{2} g = 4 m_{2} a_{1}$
$\begin{matrix} (1) \times 4 m_{2} 8 m_{2} m_{3} g - 4 m_{1} m_{2} g = 4 m_{1} m_{2} a_{1} (2) \times m_{1} 2 m_{1} m_{2} g - 2 m_{1} m_{3} g = 4 m_{1} m_{2} a_{1} 8 m_{2} m_{3} g - 4 m_{1} m_{2} g = 2 m_{1} m_{2} g - 2 m_{1} m_{3} g 8 m_{2} m_{3} + 2 m_{1} m_{3} = 6 m_{1} m_{2} \Rightarrow \frac{8}{m_{1}} + \frac{2}{m_{2}} = \frac{6}{m_{3}} \Rightarrow \frac{4}{m_{1}} + \frac{1}{m_{2}} = \frac{3}{m_{3}} \end{matrix}$
$∴$ No. $A$ is correct.

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Three masses m1 m2 and m3 are attached to a string as shown in the figure. All three masses are held at rest and then released. To keep m3 at rest the condition is

Three masses $m_{1}$ $m_{2}$ and $m_{3}$ are attached to a string as shown in the figure. All three masses are held at rest and then released. To keep $m_{3}$ at rest the condition is