Consider the situation as shown in the figure . The masses of pulley and the threads , as well as the friction in the pulley , are negligible. Assuming both bodies to be motionless at the initial moment. Find the mass ratio `m_(1)//m_(2)` at which the body `m_(2)`
(a) starts coming down
(b) starts going up
( c) is at rest

Explanation

(a) `m_(2)` is coming down :
For `m_(1) : N = m_(1) g cos alpha`
`f = mu N = mu m_(1) g cos alpha`
`m_(2)` will come down if
`m_(2) g gt (m_(1) g sin alpha + f)`
`rArr m_(2) g gt (m_(1) g sin alpha + mu m_(1) g cos alpha)`
`rArr (m_(2))/(m_(1)) gt ( sin alpha + mu cos alpha)`
(b) `m_(2)` is going up
`m_(1) g sin alpha gt (m_(2) g + f)`
`rArr m_(1) g sin alpha gt (m_(2) g + mu m_(1) g cos alpha)`
`rArr m_(1) ( sin alpha - mu cos alpha) gt m_(2)`
`rArr (m_(2))/(m_(1)) lt (sin alpha - mu cos alpha)`
( c ) `m_(2)` is at rest
`( sin alpha - mu cos alpha) le (m_(2))/(m_(1)) le ( sin alpha + mu cos alpha)`
For `m_(2)` to be at rest:
Maximum value of `m_(2) = m_(1) ( sin alpha + mu cos alpha)`
Minimum value of `m_(2) = m_(1) ( sin alpha - mu cos alpha)`