Question

A circular hoop of mass $m$, and radius $R$ rests flat on a horizontal frictionless surface. A bullet, also of mass $m$, and moving with a velocity $v$, strikes the hoop and gets embedded in it. The thickness of the hoop is much smaller than $R$. The angular velocity with which the system rotates after the bullet strikes the hoop is

• A. $\frac{ V }{4 R }$
• B. $\frac{ V }{3 R }$
• C. $\frac{2 V }{3 R }$
• D. $\frac{3 V }{4 R }$

Answer 1

Answer: (B)

$mv \frac{ R }{2}= I \omega$
$I =\left[ mR ^{2}+\frac{ mR ^{2}}{4}\right]+\frac{ mR ^{2}}{4}=\frac{3 mR ^{2}}{4}$
$mv \frac{ R }{2}=\frac{3 mR ^{2}}{2} \omega$
$\omega=\frac{ v }{3 R }$