Question

A block of mass $m$ is attached to a pulley disc of equal mass $m$ and radius $r$ by means of a slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of $5m/s$. Its velocity when the string becomes taut will be

• A. $3m/s$
• B. $2.5m/s$
• C. $\frac{5}{3}m/s$
• D. $\frac{10}{3}m/s$

Answer 1

Answer: (D)

$\frac{10}{3}m/s$

We have momentum variation for final and initial velocities of the block as
$P=m(v_2-v_1)$
Similarly for pulley
$L=I({\omega }_2-{\omega }_1)=I{\omega }_2$ (as initial velocity of pulley is zero)
equating the moment of the linear momentum of the block to the angular momentum of the pulley we get
$-m(v_2-v_1)r=I{\omega }_2$(here the minus sign shows the force on the pulley is opposing to the force on the block)
or
$m(v_1-v_2)r=\frac{m{r}^2}{2}\times \frac{v_2}{r}$
or
$2(v_1-v_2)=v_2$
or
$v_2=\frac{2}{3}{v}_1$
or
$v_2=\frac{2}{3}\left(5\right)=\frac{10}{3}m/s$