Two blocks of masses `m_(1) amd m_(2)` connected by a string and placed on an rough inclined plane having coefficient of friction `mu` is shown in fig 7.89 Find the retio of masses `m_(1) //m_(2)` so that

a. the block `m_(1)` start moving downward
b. the block `m_(1)` start moving upwnward

c. no motion takes plane

Explanation

a. If the block `m_(1)` start moving downward the block `m_(2)` will have tendency in move up .Hence friction will act in downward direction for this case
`T gt m_(2) g sin theta + mu m_(2) g cos theta`…(i)
But `T = m_(2) g`….(ii)
From Eq (i) and (ii) we have `(m_(1))/(m_(2)) gt sin theta + mu cos theta `
b. If the block `m_(1)` starts moving upnward the block `m_(2)` will have tendency in move down .Hence friction will act in upward direction for this case
`m_(2) g sin theta gt T + mu m_(2) g cos theta` ....(iii)
But `T = m_(2) g`….(iv)
From Eq (iii) and (iv) we have `(m_(1))/(m_(2)) gt sin theta - mu cos theta `

c. If no motion take place `T = mu m_(2) g sin theta`....(v)
But `T = m_(2) g`….(vi)
From Eq (v) and (vi) we have `(m_(1))/(m_(2)) = sin theta`