Question

A block of mass m is attached to a pulley disc of equal mass m ,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 5 m/s.its velocity when the string becomes taut will be

Answer 1

initial velocity of block, $v_1=5m/s$
Let final velocity of block , $v_2$

from Newton's 2nd law,
$F=\frac{dP}{dt}=\frac{m(v_2-v_1)}{dt}$

we know, torque is the rate of change of angular momentum.
e.g., $\tau=\frac{dL}{dt}$

or, $-F.r=\frac{dL}{dt} here negative sign shows that force on the pulley is opposing to the force on the mass of the table. or, [tex]-\frac{m(v_2-v_1)}{dt}.r=\frac{I(\omega_2-\omega_1)}{dt}$

but Initial angular velocity , $\omega_1=0$

now, $m(v_1-v_2).r=I\omega_2$

moment of inertia of disc about an axis passing through centre of it and perpendicular to its plane is given by, I = mr²/2

so, $m(v_1-v_2).r=mr^2/2\frac{v_2}{r}$

or, $2(v_1-v_2)=v_2$

or, $2v_1=3v_2$

so, $v_2=\frac{2}{3}v_1$

hence, $v_2=10/3m/s$

therefore, velocity of block when the string becomes taut will be 10/3 m/s .