Question

Calculate the angular momentum of the following systems. Refer to Worksheet 19 for the

moment of inertia of highly-symmetric objects. For all items, you need to convert from

rpm to rad/s first.

1. Blu-ray disc (inner diameter 0.015 m, outer diameter 0.120 m, mass 0.016 kg)

spinning at a rate of 10,000 rpm

2. Revolving door with four wings, each with mass 20.0 kg, rotating at a rate of 8

rpm. Each wing can be considered to be a single door with height 3.00 m and

width 1.75 m.

3. Bicycle wheel with mass 2.0 kg and diameter 0.63 m rotating at a rate of 2.00

rpm. Assume that the bicycle wheel is a thin-walled hollow cylinder.

Answer 1

1. The moment of inertia:

$$I = m\dfrac{r_1^2 + r_2^2}{2}\\ I = 0.016\cdot \dfrac{0.015^2 + 0.120^2}{2} = 1.17\times 10^{-4}kg\cdot m^2$$

The angular frequency:

$$\omega = 2\pi\dfrac{10000}{60} \approx 1047.2\space rad/s$$

The angular momentum:

$$L = I\omega\\ L = 1.17\times 10^{-4}\cdot 1047.2 \approx 0.12\space kg\cdot m^2/s$$

2. The moment of inertia:

$$I = 4\times m\dfrac{w^2}{3}\\ I = 4\times 20\cdot \dfrac{1.75^2}{3} \approx 81.67\space kg\cdot m^2$$

The angular frequency:

$$\omega = 2\pi\dfrac{8}{60} \approx 0.84\space rad/s$$

The angular momentum:

$$L = I\omega\\ L \approx 68.4\space kg\cdot m^2/s$$

3. The moment of inertia:

$$I = mr^2\\ I = 2\cdot (0.63/2)^2 =0.19845 \space kg\cdot m^2$$

The angular frequency:

$$\omega = 2\pi\dfrac{2}{60} \approx 0.21\space rad/s$$

The angular momentum:

$$L = I\omega\\ L \approx 0.042\space kg\cdot m^2/s$$

Answer. 1) 0.12 kg*m^2/s, 2) 68.4 kg*m^2/s, 3) 0.042 kg*m^2/s.