question_answer

Two masses m and 2m are attached with each other by a rope passing over a frictionless and massless pulley. If the pulley is accelerated upwards with an acceleration a, what is the value of T ?

A)  \[\frac{g+a}{3}\]             

B)         \[\frac{g-a}{3}\]

C) \[\frac{4m(g+a)}{3}\]                      

D)  \[\frac{m(g-a)}{3}\]

Correct Answer: C

Solution :

 Let mass \[2m\] come downwards with acceleration \[x\] relative to pulley. The mass m rises up with acceleration \[x\] with respect to pulley. The entire system is lifted upwards with acceleration\[a\]. So, the mass m moves up with acceleration\[a+x\] and mass 2m moves up with acceleration\[a-x\]. Now, free body diagram of each mass is as shown: So,       \[T-mg=m(a+x)\] and       \[T-2mg=2m(a-x)\] or         \[T-mg-ma=mx\]                        ? (i) and       \[T-2mg-2ma=-2mx\]                  ? (ii) Dividing equations (i) by fu^ we have             \[\frac{T-2mg-2ma}{T-mg-ma}=-2\] or         \[T-2mg-2ma=-2T+2mg+2ma\] or         \[T=4\frac{m}{3}(g+a)\]