In the arrangement shown below, pulleys are mass-less and friction-less and threads are in-extensible. Block of mass `m_(1)` will remain at rest if :

Options

A. `(1)/(m_(1)) = (1)/(m_(2)) + (1)/(m_(3))`
B. `m_(1) = m_(2) + m_(3)`
C. `(4)/(m_(1)) = (1)/(m_(2)) + (1)/(m_(3))`
D. `(6)/(m_(1)) = (1)/(m_(2)) + (2)/(m_(3))`

Answer

C

Explanation

`m_(1)` will be at rest if `2T = m_(1) g implies T = (m_(1) g)/(2)`
As shown in figure `a = ((m_(2) - m_(2))g)/((m_(2) + m_(3)))`
For `m_(2) : m_(2) g - T = m_(2) a implies m_(2) g - (m_(1) g)/(2) = m_(2) [((m_(2) - m_(3))g)/((m_(2) + m_(3)))]`
`implies m_(2) [1- ((m_(2) - m_(3))/(m_(2) + m_(3)))] = (m_(1))/(2)`
`implies (m_(2) (2m_(3)))/(m_(2) + m_(3)) = (m_(1))/(2) implies (4)/(m_(1)) = (1)/(m_(2)) + (1)/(m_(3))`