Question

A gramophone record of mass M and radius R is rotating at an angular velocity ω. A win of mass M is gently placed on the record at a distance R/2. from its centre. The new angular velocity of the system is 

(A) [(2ωM) (2M + m)] 

(B) [(2ωM) (M + 2m)] 

(C) ω 

(D) (ωm / M)

Answer 1

Correct option: (A) [(2ωM) (2M + m)]

Explanation:

initial angular momentum of rotating record = L = Iω = [(MR²) / 2] ∙ ω -- (1)

----------as I = [(MR²) / 2]

when coin of mass m is placed on it, let velocity be ω¹.

Hence new angular momentum = L¹ = Iω¹ + mr²ω¹            (2)

No external torque acts on the system hence momentum is conserved

L = L¹ 

from (1) & (2),

[(MR²ω) / 2] = ω¹(I + mr²)

ω¹ = [{(MR²ω) / 2} / {{(MR²) / 2} + m(R²/4)}]----since r = (R/2)

= [{(Mω) / 2} / {(M/2) + (R/4)}]

= [{(Mω) / 2} / {{(2M + m) / 4}}]

= [(2Mω) / (2M + m)]