Question

A cylindrical shell of mass 2.0 kg and diameter 15 cm starts to rotate from rest around its central axis with a constant angular acceleration. It takes 2 minutes for the cylindrical shell to make 30 revolutions.

A) Find the magnitude of the net torque exerted on the cylindrical shell when the cylindrical shell made 30 revolutions from rest?

B) Find the magnitude of the total linear acceleration when the cylindrical shell made 20 revolutions from rest.

C) Find the angular momentum when the cylindrical shell made 10 revolutions from rest.

Answer 1

Mass of the cylindrical shell, m = 2 kg

Diameter of the cylindrical shell, d = 15 cm

Radius if the cylindrical shell, r = d/2 = 15 /2 = 7.5 cm = 0.075 m

Initial angular velocity, ω1 = 0 rad/s

Time taken to complete 30 revolutions. T = 2 minutes = 120 s

Frequency of revolution, f = 30/120 = 0.25 rps

Moment of inertia of the cylindrical shell, I = mr² = 2 x 0.075 x 0.075 = 0.01125 kgm²

Angular acceleration, α = (ω2 – ω1)/t = 2πf/t = 2 x 3.14 x 0.25/120 = 0.013 rad/s²

A)

Torque, Ꞇ = Iα = 0.01125 x 0.013 = 1.46 x 10^-4 Nm

Torque, Ꞇ = 1.46 x 10^-4 Nm

B)

Linear acceleration, a = rα = 0.075 x 0.013 = 9.75 x 10^-4 m/s²

C )

Angular momentum, L = Iω

Angular velocity when it completes 10 revolutions, ω2² = ω1² + 2αθ = 0 + 2 x 0.013 x 2 x 3.14 x 10 = 1.63

Angular velocity when it completes 10 revolutions, ω2= 1.28 rad/s

Angular momentum, L = Iω2 = 0.01125 x 1.28 = 0.0144 Js

Angular momentum, L = 0.0144 Js