Question

Consider a cylindrical turntable whose mass is M and radius is R, turning with an initial angular speed ω1.

(a) A parakeet of mass m, after hovering in flight above the outer edge of the turntable, gently lands on it and stays in one place on it, as shown below. What is the angular speed of the turntable after the parakeet lands? (Use any variable or symbol stated above as necessary.) ωf =

(b) Becoming dizzy, the parakeet jumps off (not flies off) with a velocity vector v relative to the turntable. The direction of vector v is tangent to the edge of the turntable and in the direction of its rotation. What will be the angular speed of the turntable afterwards? Express your answer in terms of the two masses m and M, the radius R, the parakeet speed and the initial angular speed ω1. (Use any variable or symbol stated above along with the following as necessary: v for |vector v|.) ωf =

Answer 1

a)

I_t = moment of inertia of cylinderical turntable = (0.5) MR²

W_i = initial angular velocity = W₁

Using conservation of angular momentum

I_t W_i = (I_t + mR²) W_f

(0.5) MR² W₁ = ((0.5) MR² + mR²) W_f

W_f = (0.5) MR² W₁ / ((0.5) MR² + mR²)

b)

W_pt = angular velocity of parakeet relative to turntable

W_pg = angular velocity of parakeet relative to ground

W_tg = angular velocity of turntable relative to ground

W_pt = W_pg - W_tg

V/R + W_f = W_pg   

V_pg = RW_pg                

V_pg = V + RW_f                   eq-1

Using conservation of angular momentum

((0.5) MR² + mR²) W_f = m V_pg R + I_t W_f'

(0.5) MR² W₁ = m (V + RW_f ) R + (0.5) MR² W_f'

2 [(0.5) MR² W₁ - m (V + RW_f ) R ] / (MR² ) = W_f'