Question

3) A circular disc of mass M and radius R can rotate freely about an axis passing through its centre and
perpendicular to its plane. A bullet of mass m travelling with speed v hits the disc as shown and gets
stuck to it. The angular velocity of the system if the disc was initially at rest. (M = 2m)​

Answer 1

Answer:

A) v/4R

Points to be known:-

MoI of disc and ring

Angular momentum and its conservation

Please refer the attachment

Hope this answer helped you

Answer 2

Given:

A disc with radius R and mass M, initially at rest

A bullet of mass m hitting the disc with velocity v at an angle α with the radius

M= 2m

To Find:

The final angular momentum of the system

Solution:

Since there is no external torque, the angular momentum is conserved.

⇒ Initial Angular Momentum (Li) = Final Angular Momentum (Lf)

Li = Ldisc + Lbullet      

(L initial of dic = 0 because it is at rest)

= 0 + mvR sin α = mvR sin 30°

=mvR/2

Lf = Ldisc + Lbullet

=MR²ω/2 + mR²ω

= (M/2 + m) R²ω

Equating Li and Lf.

(M/2 + m) R²ω = mvR/2

Substituting, M/2 = m

2m R ω = mv/2

or ω = v / 4R

Hence, (1) v / 4R is the angular velocity of the disc bullet system.