11.1.1 Concept of moment of inertia

The moment of inertia (often denoted $( I )$ is a measure of an object's resistance to changes in its rotational motion, similar to how mass is a measure of resistance to linear acceleration. It represents how difficult it is to accelerate or decelerate an object in rotation around a particular axis.

Definition of Moment of Inertia:

• For a single point mass $m$ at a distance $r$ from the axis of rotation, the moment of inertia is given by:

$$I = mr^2$$

Where:

• $m$ = mass of the object
• $r$ = distance from the mass to the axis of rotation
• For an extended object (one made up of multiple point masses), the moment of inertia can be calculated by summing the individual moments of inertia of each point mass:

$$I = \sum mr^2$$

Factors Affecting Moment of Inertia:

1. Mass of the Object: The larger the mass, the greater the moment of inertia.
2. Distribution of Mass Relative to the Axis of Rotation: The further the mass is distributed from the axis, the larger the moment of inertia. For instance, a figure skater can increase or decrease their rotation speed by extending or retracting their arms and legs. Bringing limbs closer to the body reduces the distance from the axis, decreasing the moment of inertia and increasing rotation speed.

Calculating Moment of Inertia for a System:

To find the moment of inertia of a system, add the moments of inertia for each component.

Adding Mass to a System:

If a new mass $m$ is added at distance $r$ from the axis, the new moment of inertia becomes:

$$I_{\text{new}} = I + mr^2$$

where $I$ is the initial moment of inertia.