Angular Momentum

Momentum is the product of mass and the velocity of the object. Any object moving with mass possesses momentum. The only difference in angular momentum is that it deals with rotating or spinning objects. So is it the rotational equivalent of linear momentum?

What is Angular Momentum?

If you try to get on a bicycle and balance without a kickstand, you will probably fall off. But once you start pedalling, these wheels pick up angular momentum. They are going to resist change, thereby making balancing gets easier.

It is the property of a rotating body given by the product of the moment of inertia and the angular velocity of the rotating object. It is a vector quantity, which implies that the direction is also considered here along with magnitude.

Angular Momentum Formula

Angular momentum can be experienced by an object in two situations. They are:

Point object: The object accelerating around a fixed point. For example, Earth revolving around the sun. Here the angular momentum is given by:

\begin{array}{l} \vec{L} = \vec{r} \times \vec{p} \end{array}

Where,

\begin{array}{l} \vec{L} i s t h e a n g u l a r v e l o c i t y \end{array}

Extended object: The object, which is rotating about a fixed point. For example, Earth rotates about its axis. Here the angular momentum is given by:

\begin{array}{l} \vec{L} = \vec{I} \times \vec{ω} \end{array}

Example

Calculating Angular Momentum

Angular momentum is the measure of the rotational motion of an object. It depends on the object's mass, velocity, and the distance from the axis of rotation. It is a vector quantity, meaning it has both magnitude and direction. The goal of calculating angular momentum is to determine the rotational motion of an object around a specific axis.

The general approach to calculating angular momentum includes:

Identifying the mass, velocity, and radius of the object.
Knowing the moment of inertia if angular velocity is used instead of linear velocity.
Applying the formula for angular momentum to calculate the result.

Angular Momentum Formula

The general formula for angular momentum is:

\[ L = mvr \]

Where:

m is the mass of the object (in kilograms, kg).
v is the velocity of the object (in meters per second, m/s).
r is the distance from the axis of rotation (in meters, m).

Example:

If an object of mass 5 kg moves at 10 m/s at a radius of 2 m, the angular momentum is:

Step 1: Multiply the mass, velocity, and radius: \( L = 5 \times 10 \times 2 = 100 \, \text{kg} \cdot \text{m}^2/\text{s} \).

Angular Momentum with Rotational Motion

Angular momentum can also be calculated when rotational motion is involved. If the moment of inertia and angular velocity are known, the formula becomes:

\[ L = I\omega \]

Where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

Example:

If a rotating wheel has a moment of inertia of 3 kg·m² and an angular velocity of 4 rad/s, the angular momentum is:

Step 1: Multiply the moment of inertia and angular velocity: \( L = 3 \times 4 = 12 \, \text{kg} \cdot \text{m}^2/\text{s} \).

Conservation of Angular Momentum

Angular momentum is conserved in a closed system, meaning the total angular momentum before and after a change in the system must remain the same. This is often observed in situations like ice skaters spinning faster when they pull in their arms.

Example:

If an ice skater has an initial moment of inertia of 10 kg·m² and angular velocity of 2 rad/s, and reduces their moment of inertia to 5 kg·m², the new angular velocity will be:

Step 1: Use conservation of angular momentum: \( I_{\text{initial}} \omega_{\text{initial}} = I_{\text{final}} \omega_{\text{final}} \).
Step 2: Solve for the final angular velocity: \( \omega_{\text{final}} = \frac{{I_{\text{initial}} \omega_{\text{initial}}}}{{I_{\text{final}}}} = \frac{{10 \times 2}}{{5}} = 4 \, \text{rad/s} \).

Real-life Applications of Angular Momentum

Calculating angular momentum has many practical applications, such as:

Studying rotational motion in physics experiments (e.g., finding the angular momentum of a spinning wheel).
Determining the effect of pulling in mass on an object's rotational speed (e.g., in figure skating).
Calculating angular momentum in space (e.g., understanding the rotational motion of celestial bodies).

Common Units of Angular Momentum

SI Unit: The standard unit of angular momentum is kilogram meters squared per second (\( \text{kg} \cdot \text{m}^2/\text{s} \)).

Angular momentum can also be expressed in other units, but the SI unit is commonly used in most calculations.

Common Operations with Angular Momentum

Constant Angular Momentum: When angular momentum remains constant over time (e.g., a rotating disk in space).

Changing Angular Momentum: When the angular momentum changes due to external forces or changes in the system (e.g., a satellite moving in orbit).

Angular Momentum Transfer: This occurs when angular momentum is transferred between objects, such as when two colliding objects exchange rotational energy.

Calculating Angular Momentum Examples Table
Problem Type	Description	Steps to Solve	Example
Calculating Angular Momentum from Mass and Velocity	Finding the angular momentum when given the mass, velocity, and radius.	Identify the mass \( m \), velocity \( v \), and radius \( r \) of the object. Use the angular momentum formula: \( L = mvr \).	For an object of mass \( 5 \, \text{kg} \) moving at \( 10 \, \text{m/s} \) at a radius of \( 2 \, \text{m} \), the angular momentum is \( L = 5 \times 10 \times 2 = 100 \, \text{kg} \cdot \text{m}^2/\text{s} \).
Calculating Angular Momentum with Rotational Velocity	Finding the angular momentum when given the moment of inertia and angular velocity.	Identify the moment of inertia \( I \) and the angular velocity \( \omega \). Use the formula: \( L = I \omega \).	For a rotating wheel with a moment of inertia of \( 3 \, \text{kg} \cdot \text{m}^2 \) and an angular velocity of \( 4 \, \text{rad/s} \), the angular momentum is \( L = 3 \times 4 = 12 \, \text{kg} \cdot \text{m}^2/\text{s} \).
Conservation of Angular Momentum	Understanding how angular momentum is conserved in a closed system.	Identify the initial and final angular velocities and moments of inertia. Use the conservation of angular momentum formula: \( L_{\text{initial}} = L_{\text{final}} \).	If a spinning ice skater pulls in their arms to decrease the radius, their angular velocity increases. If \( I_{\text{initial}} = 10 \, \text{kg} \cdot \text{m}^2 \) and \( \omega_{\text{initial}} = 2 \, \text{rad/s} \), and the final moment of inertia is \( I_{\text{final}} = 5 \, \text{kg} \cdot \text{m}^2 \), then the final angular velocity is \( \omega_{\text{final}} = \frac{{I_{\text{initial}} \omega_{\text{initial}}}}{{I_{\text{final}}}} = \frac{{10 \times 2}}{{5}} = 4 \, \text{rad/s} \).
Real-life Applications	Applying angular momentum to solve practical problems such as spinning objects, orbits, and rotational motion.	To calculate the angular momentum of a rotating disk. To determine the effect of pulling in mass on an object's rotational speed.	If a figure skater spins faster when pulling in their arms, use the conservation of angular momentum to calculate the new rotational speed after reducing their moment of inertia.