# Angular momentum – problems and solutions

1. An object with the moment of inertia of 2 kg m^{2} rotates at 1 rad/s. What is the angular momentum of the object?

__Known :__

Moment of inertia (I) = 2 kg m^{2}

Angular speed (ω) = 1 rad/s

__Wanted :__ Angular momentum (L)

__Solution :__

Formula of angular momentum :

L = I ω

*L = **angular momentum **(kg m*^{2}*/s), I = **moment of inertia **(kg m*^{2}*), **ω **= **angular speed **(rad/s)*

The angular momentum :

L = I ω = (2)(1) = 2 kg m^{2}/s

2. A 2-kg cylinder pulley with radius of 0.1 m rotates at a constant angular speed of 2 rad/s. What is the angular momentum of the pulley ?

__Known :__

Mass of pulley (m) = 2 kg

Radius of pulley (r) = 0.1 m

Angular speed (ω) = 2 rad/s

__Wanted :__ Angular momentum

__Solution :__

Formula of moment of inertia for solid cylinder :

I = 1/2 m r^{2 }

*I = **moment of inertia **(kg m*^{2}*), m = mass (kg), **r = **radius **(m)*

Moment of inertia :

I = 1/2 (2)(0.1)^{2} = (1)(0.01) = 0.01 kg m^{2}

The angular speed :

L = I ω = (0.01)(2) = 0.02 kg m^{2}/s

3. A 2-kg uniform sphere with radius of 0.2 m rotates at 4 rad/s. What is the angular momentum of the ball.

__Known :__

Mass of ball (m) = 2 kg

Radius of ball (r) = 0.2 m

Angular speed (ω) = 4 rad/s

__Wanted :__ Angular momentum

__Solution :__

Formula of moment of inertia for uniform sphere :

I = (2/5) m r^{2 }

*I = momen**t of inertia **(kg m*^{2}*), m = mass (kg), **r = **radius **(m)*

The moment of inertia for uniform sphere :

I = (2/5)(2)(0.2)^{2} = (4/5)(0.04) = 0.032 kg m^{2}

The angular momentum of sphere :

L = I ω = (0.032)(4) = 0.128 kg m^{2}/s

4. A 1-kg particle rotates at a constant angular speed of 2 rad/s. What is the angular speed if the radius of circle is 10 cm.

__Known :__

Mass of object (m) = 1 kg

The radius of circle (r) = 10 cm = 10/100 = 0.1 m

The angular speed (ω) = 2 rad/s

__Wanted :__ Angular momentum

__Solution :__

Formula of moment of inertia for particle :

I = m r^{2 }= (1)(0.1)^{2} = (1)(0.01) = 0.01 kg m^{2}

Angular momentum :

L = I ω = (0.01)(2) = 0.02 kg m^{2}/s