A.4.3 Conservation of angular momentum (HL only)

Angular Momentum and Torque in Rotational Motion

When dealing with extended rigid bodies, forces can cause both linear and rotational motion.
To understand this, we need to explore angular momentum, torque, and their real-world applications.

Angular Momentum: A Measure of Rotational Motion

Definition

Angular momentum

Angular momentum is the rotational equivalent of linear momentum. More precisely, it is the product of its moment of inertia and its angular velocity.

For a rigid body rotating about a fixed axis, it is defined as:

$L = I ω$

where:

$L$ is the angular momentum.
$I$ is the moment of inertia.
$ω$ is the angular velocity.

Note

Angular momentum is a vector quantity, but in this course, we focus on its magnitude.

Conservation of Angular Momentum

Definition

Conservation of angular momentum

Angular momentum is conserved unless an external torque acts on the system.

Example

Imagine a figure skater spinning with her arms extended.
As she pulls her arms in, her moment of inertia decreases.
To conserve angular momentum, her angular velocity increases, causing her to spin faster.

Example question

Conservation of angular momentum

A figure skater is spinning with her arms extended. Her moment of inertia with arms extended is $I_{1} = 5.0 kg \cdot m^{2}$ , and her initial angular velocity is $ω_{1} = 2.0 rad / s$ . She pulls her arms in, reducing her moment of inertia to $I_{2} = 2.0 kg \cdot m^{2}$ .

What is her final angular velocity $ω_{2}$ ?

Solution

The angular momentum of the system is conserved, so:

$L_{1} = L_{2}$

Where angular momentum $L$ is:

$L = I \cdot ω$

Thus:

$I_{1} \cdot ω_{1} = I_{2} \cdot ω_{2}$

Solving for $ω_{2}$ :

$ω_{2} = \frac{I_{1} \cdot ω_{1}}{I_{2}}$

Substitute the values:

$ω_{2} = \frac{5.0 \cdot 2.0}{2.0}$

$ω_{2} = 5.0 rad / s$

Result:

Final Angular Velocity: $ω_{2} = 5.0 rad / s$

Torque and Angular Acceleration

Definition

Torque

Torque is the rotational equivalent of force. It measures the ability of a force to cause an object to rotate.

Torque and Angular Acceleration

Torque is directly related to angular acceleration through the equation:

$τ = I α$

where:

$τ$ is the torque.
$I$ is the moment of inertia.
$α$ is the angular acceleration.

This equation is the rotational analogue of Newton’s second law, $F = m a$ .

Note

The unit of torque is the N m (Newton-meter).

Analogy

Think of torque as the "twisting force" needed to rotate an object, just as force is needed to accelerate a linear object.

Example question

Torque and angular acceleration

A solid disk of radius $R = 0.5 m$ and mass $M = 4.0 kg$ is initially at rest. A force $F = 10.0 N$ is applied tangentially at the edge of the disk.

Calculate the angular acceleration $α$ of the disk.

Solution

Step 1: Moment of Inertia of the Disk

The moment of inertia $I$ for a solid disk rotating about its central axis is given by:

$I = \frac{1}{2} M R^{2}$

Substitute the values:

$I = \frac{1}{2} (4.0) (0.5)^{2}$

$= \frac{1}{2} (4.0) (0.25) = 0.5 kg \cdot m^{2}$

Step 2: Torque Acting on the Disk

The torque $τ$ is calculated as:

$τ = F \cdot R$

Substitute the values:

$τ = (10.0) (0.5) = 5.0 N \cdot m$

Step 3: Angular Acceleration

Using the rotational form of Newton's second law:

$τ = I \cdot α$

Rearranging for $α$ :

$α = \frac{τ}{I}$

Substitute the values:

$α = \frac{5.0}{0.5} = 10.0 rad / s^{2}$

Result:

Angular Acceleration: $α = 10.0 rad / s^{2}$

Angular Impulse: Changing Angular Momentum

Definition

Angular impulse

Angular impulse is a concept in rotational motion that describes the change in angular momentum of a rigid body due to the application of torque over a period of time.

It is mathematically expressed as:

$Δ L = τ Δ t$

where:

$Δ L$ is the change in angular momentum.
$τ$ is the torque applied.
$Δ t$ is the time duration over which the torque acts.

Angular impulse is the rotational analogue of linear impulse, which relates force to the change in linear momentum.
Just as a force applied for a certain time changes the linear momentum of an object, a torque applied for a certain time changes the angular momentum of a rigid body.

Note

If no external torque acts on a system, $τ = 0$ , and angular momentum remains conserved ( $Δ L = 0$ ).
However, when an external torque is applied over time, the angular impulse explains how the angular momentum changes.

Example question

Angular impulse

A wheel of moment of inertia $I = 0.2 kg \cdot m^{2}$ is initially at rest. A constant torque of $0.5 N \cdot m$ is applied for $4.0 s$ .

What is the angular momentum of the wheel after this time?

Solution

Using the angular impulse formula:

$Δ L = τ Δ t$

Substitute the values:

$Δ L = (0.5) (4.0) = 2.0 kg \cdot m^{2} / s$

The angular momentum of the wheel after $4.0 s$ is $2.0 kg \cdot m^{2} / s$ .

Reflection

Self review

How does the conservation of angular momentum explain why a figure skater spins faster when pulling her arms in?

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A.4.3 Conservation of angular momentum (HL only)

Angular Momentum and Torque in Rotational Motion

Angular Momentum: A Measure of Rotational Motion

Conservation of Angular Momentum

Conservation of angular momentum

Torque and Angular Acceleration

Torque and Angular Acceleration

Torque and angular acceleration

Step 1: Moment of Inertia of the Disk

Step 2: Torque Acting on the Disk

Step 3: Angular Acceleration

Result:

Angular Impulse: Changing Angular Momentum

Angular impulse

Reflection

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Introduction to Angular Momentum