@Ganesh, I got the question paper of GATE 2005 & read that question. Still I took much time to understand the question right. May be because of I am not aware of how exactly the elevator works. I always wondered that why that counterweight(CW) is there when we have electric motor in place. Then I googled it finally understands the use of CW is to save the energy.
Now coming back to the question;
Here, we can see the energy in the cage is in the form of K.E. which equals to, 0.5*m*v*v
Also K.E. in counterweight(CW) is, 0.5*m*v*v
Even the cage & CW are moving in opposite direction, they are pulling the chain in same direction (clockwise).
So combining the two K.E.
(0.5*m*v*v) + (0.5*m*v*v) = m*v*v
Now the motor should have minimum energy which could be able to oppose this combined energy. In other words, the motor should do “WORK” to oppose the K.E. which is equals to, m*v*v.
So, minimum work the motor must generate = m*v*v
Now as we know, Power = Work done / time
we can say,
Minimum power the motor must have = (m*v*v)/t

@Ganesh, I got the question paper of GATE 2005 & read that question. Still I took much time to understand the question right. May be because of I am not aware of how exactly the elevator works. I always wondered that why that counterweight(CW) is there when we have electric motor in place. Then I googled it finally understands the use of CW is to save the energy.

Now coming back to the question;

Here, we can see the energy in the cage is in the form of K.E. which equals to, 0.5*m*v*v

Also K.E. in counterweight(CW) is, 0.5*m*v*v

Even the cage & CW are moving in opposite direction, they are pulling the chain in same direction (clockwise).

So combining the two K.E.

(0.5*m*v*v) + (0.5*m*v*v) = m*v*v

Now the motor should have minimum energy which could be able to oppose this combined energy. In other words, the motor should do “WORK” to oppose the K.E. which is equals to, m*v*v.

So, minimum work the motor must generate = m*v*v

Now as we know, Power = Work done / time

we can say,

Minimum power the motor must have = (m*v*v)/t

check out the answer.