Question

All contact surfaces shown are smooth and distance between the block
and the pulley is at t=0 time taken by pulley : ???

Answer 1

Given:

Distance between the block and the pulley is at $t\;=\;0$

To find:

The time taken by the pulley

Explanation:

For the given bigger block of mass $M$,

$2F \;=\;Ma_{2}$

$a_{2} \;=\;\frac{2F}{M}$

For the smaller block of mass $m$,

$F\;=\;ma_{1}$

$a_{1} \;=\;\frac{F}{m}$

We know that, $a_{rel} \;=\;a_{1} \;+\;a_{2}$

$a_{rel}\;=\;\frac{F}{m} \;+\;\frac{2F}{M}$

⇒ $a_{rel} \;=\;F(\frac{1}{m} \;+\;\frac{2}{M} )$

According to the second equation of motion, $S\;=\;ut\;+\;\frac12at^2$

⇒ $l\;=\;0\;+\;\frac12\lbrack F(\frac1m\;+\;\frac2M)\rbrack t^2$

⇒ $t^2=\;\frac{2lMm}{F(M+2m)}$

⇒ $t\;=\;\sqrt{\frac{2lMm}{F(M+2m)}}$

Answer:

The time taken by the pulley is Option (B) $t\;=\;\sqrt{\frac{2lMm}{F(M+2m)}}$