(87) Topic 5.4 - Rotational Inertia

Rotational Inertia

Definition

Rotational inertia, also known as moment of inertia, is a fundamental property of a rigid body that quantifies its resistance to angular acceleration when a torque is applied. It is analogous to mass in linear motion, which determines how much an object resists changes to its velocity. Simply put, the greater the rotational inertia, the more force is required to change the object's rotational speed.

Factors Affecting Rotational Inertia

The rotational inertia of a system is influenced by two primary factors:

1. The mass of the system (m): Heavier objects have a greater inertia and require more torque to initiate or change rotation.

2. The distribution of that mass: Objects with mass distributed farther from the axis of rotation have a higher inertia compared to those with the same mass concentrated closer to the axis. For example, a solid sphere has less rotational inertia compared to a hollow sphere of the same mass and radius due to the distribution of mass.

Equation for Rotational Inertia

1. For a single object rotating at a distance R:[ I = mR^2 ] where ( I ) is the rotational inertia, ( m ) is the mass, and ( R ) is the distance from the rotational point.

2. For a collection of objects:[ I = \sum (mR^2) ] This requires knowing the individual masses and their respective distances from the axis of rotation to calculate the total rotational inertia of the system.

Parallel Axis Theorem

Description

The Parallel Axis Theorem provides a method to calculate the rotational inertia of an object about any axis that is parallel to an axis through its center of mass. This theorem is particularly useful in cases where the rotation axis does not pass through the center of mass, as it allows for the determination of the rotational inertia about a new axis using known quantities.

Equation

[ I = I_{com} + mR^2 ]where ( I ) is the rotational inertia about the new axis, ( I_{com} ) is the rotational inertia about the center of mass, ( m ) is the mass of the object, and ( R ) is the distance between the two axes.

Types of Objects and Their Rotational Inertia

Various shapes and structural configurations have distinct formulas for calculating their rotational inertia:

• Hoop (Thin Ring): ( I = mR^2 )A thin ring has its entire mass located at a distance R from the axis, resulting in maximum inertia.

• Solid Sphere: ( I = \frac{2}{5}mR^2 )The mass is more evenly distributed, yielding a lower inertia than a hoop of the same mass.

• Spherical Shell: ( I = \frac{2}{3}mR^2 )A hollow sphere has a higher inertia compared to a solid sphere due to mass being farther from the axis.

• Solid Disc:The inertia varies based on thickness and how mass is distributed across the radius, typically given by ( I = \frac{1}{2}mR^2 ) for discs rotating about an axis through the center.

Key Takeaway

The object with the smallest rotational inertia will accelerate quicker and reach the bottom of an inclined plane first, assuming all other factors like mass and friction are constant. This highlights the relationship between rotational inertia and angular acceleration, paralleling Newton's second law of motion.

Example Problem: Using Parallel Axis Theorem

Problem: Demonstrate that the rotational inertia about one end of a rod is ( \frac{1}{3} mR^2 ).

1. Determine I(center of mass): ( I = \frac{1}{12}mR^2 )

2. Apply Parallel Axis Theorem: ( I ) about end = ( I_{com} + m(R/2)^2 )where ( R/2 ) is the distance from the center of mass to one end.

3. Simplifying yields: ( I = \frac{1}{12}mR^2 + m\frac{R^2}{4} )

4. Combine terms to find: ( I = \frac{1}{3}mR^2 )

AP Exam Note

Students may be asked to show steps derivatively applying the parallel axis theorem, so understanding each component of the theorem is essential for problem-solving in exams.