A.4.2 Moment of inertia (HL only)

Moment of Inertia

Imagine trying to spin a bicycle wheel.
If the mass is concentrated near the rim, it feels harder to start or stop the rotation compared to when the mass is closer to the center.

This resistance to rotational motion is described by the moment of inertia ( $I$ ), which is the rotational equivalent of mass.

Definition of Moment of Inertia

Definition

Inertia

Inertia is the tendency of an object to resist changes in its state of motion.

In other words, it quantifies how difficult it is to change the rotational motion of an object.

It depends on two factors:

Mass of the object
Distribution of mass relative to the axis of rotation

Mathematically, the moment of inertia for a system of point masses is given by:

$I = \sum m_{i} r_{i}^{2}$

where:

$m_{i}$ is the mass of each point
$r_{i}$ is the distance of the mass from the axis of rotation

Example

Single Particle: A particle of mass $m$ at a distance $R$ from the axis has a moment of inertia $I = m R^{2}$ .

Multiple Particles: Two particles, each of mass $m$ , at distances $R_{1}$ and $R_{2}$ from the axis have a total moment of inertia $I = m R_{1}^{2} + m R_{2}^{2}$ .

Moment of Inertia as the Rotational Equivalent of Mass

In linear motion, mass measures an object's resistance to changes in velocity.
Similarly, in rotational motion, the moment of inertia measures an object's resistance to changes in angular velocity.

Note

The moment of inertia depends on the axis of rotation.
The same object can have different moments of inertia if rotated about different axes.

Rotational Kinetic Energy

Just as a moving object has kinetic energy, a rotating object has rotational kinetic energy.

Definition

Rigid body

A rigid body is an idealized object in which the relative positions of all particles remain fixed, regardless of external forces or torques.

Formula for Rotational Kinetic Energy

Definition

Rotational kinetic energy

Rotational kinetic energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy.

The rotational kinetic energy of a rigid body is given by:

$E_{k} = \frac{1}{2} I ω^{2}$

where:

$I$ is the moment of inertia
$ω$ is the angular velocity

Example question

Rotational kinetic energy

A ring of mass $0.45 kg$ and radius $0.20 m$ rotates with an angular velocity of $5.2 rad/s$ . Calculate its rotational kinetic energy.

Solution

Calculate the moment of inertia: $I = M R^{2} = 0.45 \times {0.20}^{2} = 0.018 {kg m}^{2}$
Calculate the kinetic energy: $E_{k} = \frac{1}{2} I ω^{2} = \frac{1}{2} \times 0.018 \times {5.2}^{2} = 0.24 J$

Relationship to Linear Kinetic Energy

In linear motion, kinetic energy is given by $E_{k} = \frac{1}{2} m v^{2}$ .
In rotational motion, mass is replaced by the moment of inertia ( $I$ ), and linear velocity ( $v$ ) is replaced by angular velocity ( $ω$ ).

Analogy

Think of rotational kinetic energy as the energy of spinning.
Just as a faster-moving car has more kinetic energy, a faster-spinning wheel has more rotational kinetic energy.

Factors Affecting Moment of Inertia

The moment of inertia is influenced by:

Distribution of Mass: The farther the mass is from the axis, the greater the moment of inertia.
Axis of Rotation: Changing the axis changes the distances ( $r_{i}$ ) in the formula, affecting the moment of inertia.

Hint

You don't have to memorize the formulas for the moment of inertia of common bodies like spheres and discs: they will be given in the task if needed.

Note

If you have two attached bodies of different shapes, the moment of inertia is added together.

Examples of Mass Distribution

Solid Sphere: Mass is evenly distributed, resulting in a smaller moment of inertia compared to a hollow sphere of the same mass and radius.
Ring: All mass is concentrated at the same distance from the axis, leading to a larger moment of inertia.

Common Mistake

A common mistake is to assume that the moment of inertia depends only on the mass.

In reality, the distribution of mass is equally important.

Axis of Rotation

The moment of inertia changes with the axis of rotation.
For example, a rod rotating about its center has a different moment of inertia than when it rotates about one end.

Tip

To minimize the moment of inertia, keep the mass as close to the axis of rotation as possible.
This is why figure skaters pull their arms in to spin faster.

Reflection

Self review

What is the formula for the moment of inertia of a point mass?
How does the moment of inertia affect rotational kinetic energy?
Why does a hollow sphere have a larger moment of inertia than a solid sphere of the same mass and radius?

Theory of Knowledge

How do the principles of rotational motion apply to the design of modern technologies, such as wind turbines or gyroscopes?
What ethical considerations arise when using these technologies to address global challenges?

Understanding the moment of inertia and rotational kinetic energy helps explain why some objects are easier to spin than others.

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A.4.2 Moment of inertia (HL only)

Moment of Inertia

Definition of Moment of Inertia

Moment of Inertia as the Rotational Equivalent of Mass

Rotational Kinetic Energy

Formula for Rotational Kinetic Energy

Rotational kinetic energy

Relationship to Linear Kinetic Energy

Factors Affecting Moment of Inertia

Examples of Mass Distribution

Axis of Rotation

Reflection

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Introduction to Moment of Inertia