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

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) ( V /4 R ) (B) ( V /3 R ) (C) (2 V /3 R ) (D) (3 V /4 R )

Tardigrade Admin · Accepted Answer

mv ( R /2)= I ω I =[ mR 2+( mR 2/4)]+( mR 2/4)=(3 mR 2/4) mv ( R /2)=(3 mR 2/2) ω ω=( v /3 R )

$\frac{V}{4 R}$

$\frac{V}{3 R}$

$\frac{2 V}{3 R}$

$\frac{3 V}{4 R}$

$m v \frac{R}{2} = I ω$
$I = [m R^{2} + \frac{m R ^{2}}{4}] + \frac{m R ^{2}}{4} = \frac{3 m R ^{2}}{4}$
$m v \frac{R}{2} = \frac{3 m R ^{2}}{2} ω$
$ω = \frac{v}{3 R}$

4RV​

3RV​

3R2V​

4R3V​

mv2R​=Iω I=[mR2+4mR2​]+4mR2​=43mR2​ mv2R​=23mR2​ω ω=3Rv​

$\frac{V}{4 R}$

$\frac{V}{3 R}$

$\frac{2 V}{3 R}$

$\frac{3 V}{4 R}$

$m v \frac{R}{2} = I ω$
$I = [m R^{2} + \frac{m R ^{2}}{4}] + \frac{m R ^{2}}{4} = \frac{3 m R ^{2}}{4}$
$m v \frac{R}{2} = \frac{3 m R ^{2}}{2} ω$
$ω = \frac{v}{3 R}$