Question

A uniform circular turntable (mass $M$, radius $r$, centre $O$) is at rest in the XY-plane and is mounted on a frictionless axle, which lies along the vertical Z axis. A lump of putty (mass $m$) is thrown with speed $v$ towards the edge of the turntable so it approaches along a line that passes within a distance $b$ of $O$. When the putty hits the turntable, it sticks to the edge and the two rotate together with angular velocity equal to:

• A. $\frac{m}{m+\frac{M}{2}}\times \frac{vb}{r^2}$
• B. $\frac{M}{m+\frac{M}{2}}\times \frac{vb}{r^2}$
• C. $\frac{m}{m+\frac{M}{2}}\times \frac{v^{2}b}{r^2}$
• D. $\frac{vb}{r^2}$

Answer 1

The correct option is A $\frac{m}{m+\frac{M}{2}}\times \frac{vb}{r^2}$

Initial angular momentum of the system $L_i=0+mvb=mvb$

Let the final angular velocity of the system be $w$.

Final angular momentum $L_f=I_{system}w=(\frac{1}{2}M{r}^2+m{r}^2)w$

Conservation of angular momentum : $L_i=L_f$

$\therefore$ $mvb=(\frac{1}{2}M+m)w{r}^2$

$⟹w=\frac{m}{\frac{M}{2}+m}\times \frac{vb}{r^2}$