A ring of mass M and radius r is moving with angular velocity ω. If another two bodies each of mass m is placed on its diameter, the resultant angular velocity will be :

The correct answer is option 3) i.e. \(\frac{ω M}{M+2m}\).

CONCEPT:

• Angular momentum: The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity.
  • Angular momentum also obeys the law of conservation of momentum i.e. angular momentum before and after is conserved.
  • Angular momentum = I × ω = mr²ω 

Where I is the moment of inertia, ω is the angular velocity, m is the mass of the object, and r is the radius of curvature.

CALCULATION:

Let ω be the initial angular velocity of the ring and ω' be the new angular velocity after placement of two bodies on it.

From the conservation of angular momentum:

Momentum before = Momentum after

The momentum of the ring before = Momentum of the ring after placing the bodies + Momentum of the bodies

Mr2ω = Mr²ω' + 2 × (mr²ω')

Mω = ω' (M+2m)

⇒ ω' = \(\frac{\omega M}{M+2m}\)