A pulley fixed to the ceiling of an elevator car carries a thread whose ends are attached to the masses `m_(1) and m_(2)`. The car starts going up with an acceleration `a_(0)`. Assuming the masses of the pulley and the thread as well as the friction to be negligible, find:

(a) the acceleration of the load `m_(1)` relative to the elevator shaft and relative to the car
(b) the force exerted by the pulley on the ceiling of the car. Given `m_(1) gt m_(2)`.

Explanation

(a) The elevator is an accelerated frame which is non-inertial, pseudo force `m_(1)a_(0) lt m_(2)a_(0)` have been taken into consideration for `m_(1) and m_(2)`, respectively.
`m_(1)g+m_(1)a_(0)-T=m_(1)a_(1)" "...(i)`
`T-m_(2)g-m_(2)a_(0)=m_(2)a_(2)" " ...(ii)`
`a_(1)=a_(2)" "...(iii)`
Solving the above equations, we get,
`a_(1)=((m_(1)-m_(2)).(a_(0)+g))/((m_(1)+m_(2)))`
This is the acceleration of `m_(1)` w.r.t., car.
(b) Acceleration of the mass `m_(1)` w.r.t. the elevator shaft `=a_(0)-a_(1)`
`=(2a_(0)m_(2)-g(m_(1)-m_(2)))/((m_(1)+m_(2)))`
Force exerted by the pulley on the ceiling of the car
`=2.T=(m_(1)a_(0) +m g-m_(1)a_(1))xx2=(4m_(1)m_(2)(a_(0)+g))/((m_(1)+m_(2)))`