A monkey of mass m clings to a rope over a fixed pulley. The opposite ends of the rope is tide to a weight of mass M lying on a horizontal plate neglecting friction, find the acceleration of both bodies relative to the plate and tension of the rope in the following cases:
i) the monkey does not move with respect to rope.
ii) the monkey moves upwards with respect to rope with acceleration `beta` .
iii) the monkey moves downwards with respect to the rope with acceleration `beta`.

Explanation

i) If the monkey is at rest on the rope, then acceleration of both mass M and monkey are same, equal to a (say)
If T is tension in the rope, then the equation of motion of monkey is
` mg-T=ma_1`..........(1)
and equation of motion of mass M is
`T = Ma_1` ........... (2)
Adding (1) and (2) `mg =(M+m)a_1`
` :. a_1 = (m)/(M+m) g ` ......... (3)
ii) When monkey moves upward with acceleration `beta`, the reaction m `beta` acts downward, therefore equations of motion now take the form (`a_2` is acceleration of mass, M relative to plate)
` mg + m beta - T = ma_2` ( for monkey ) ............. (4)
` T= Ma_2 ` ( for mass M) ............. (5)
Adding , we get , `mg + m beta = ( M+ m)a_2`
`:. a_2 =(m (g + beta))/( M + m)`
` :.` The acceleration of mass relative to plane
` a_2 = ( m ( g + beta))/( M + m)` ............ (6)
and acceleration of monkey M relative to plate
`a_2 = a_2 - beta = (m (g + beta))/( M + m) - beta = ( m(g + beta ) - beta ( M + m))/( M+ m) = (mg + m beta - M beta - m beta)/( M + m) = ( Mg - M beta)/( M + m)`

iii) When monkey moves downward with acceleration `beta`, the reaction `(R= mbeta)` acts upward, therefore the equations of motion take the form (`a_3` is acceleration of block relative to plate)

`mg - m beta -T = ma_3` .......... (8)
`T = Ma_3 ` ............ (9)
Adding we get ` mg - m beta = (M + m) a_3`
` :. a_3 =( m (g - beta))/(M + m)`
and acceleration of monkey relative to plate
` a_3^1 = a_3 + beta = (m ( g -beta))/( M +m) + beta = (mg - m beta + M beta + m beta )/( M + m)`
`:. a_3^(1) = (mg + M beta )/( M + m)`