The inclined surface is rough `mu = (1)/(2)`. For different values of m and M system slides down or up the plane or remains stationary. Match the appropriate entries of column-1 with those of column-2

`{:(,"Column-1",,"Column-2"),("(A)","Minimum value of " (m)/(M) "so theta m slides down","(P)",(5)/(3)),("(B)","Minimum valie of " (M)/(m)"so that m slides up","(Q)",1),("(C)","Value of" (m)/(M) "so that friction force on m is zero","(R)",(3)/(5)),("(D)",underset("and acceleration of M")"Ratio of vertical component of acceleration of m","(S)",5):}`

Answer

A, B, C, D

Explanation

`implies T = mg`
`T + N mu = mg sin theta`
`N = m g cos theta`
`implies Mg + mg cos theta mu = m g sin theta`
`implies (m)/(M) = (1)/(sin theta - mu cos theta) = 5`
(B) `implies M = m (sin theta + mu cos theta)`
`implies M = m (sin theta + mu cos theta)`
`implies (M)/(m) = (3)/(5) + ((1)/(2)) (4)/(5) = 1`
(C ) `T = m g sin theta = Mg implies m//M = (1)/(sin theta) = (5)/(3)` `implies` vertical component of acceleration of m
(D ) `= a sin theta` And vertical acc of M = a
`implies` ratio `= sin theta = (3)/(5)`