Rotational Kinematics and Moment of Inertia

Driving question

How do angular motion and mass distribution determine the response of a rotating rigid body?

Before learning: points on one wheel

Mark one point at radius $R/2$ and another at radius $R$ on a rigid wheel. As it rotates, compare their angular displacement, angular velocity, linear distance, tangential speed, and radial acceleration.

Hook: same angular speed, different linear speed

Every point on a rigid wheel completes each revolution together, so all points share angular velocity. But a point farther from the axis travels a longer arc and moves faster linearly.

Angular variables

Angular displacement is measured in radians:

Angular velocity and acceleration are

Slope of $\theta(t)$ is $\omega$; slope of $\omega(t)$ is $\alpha$. Signed area under $\omega(t)$ is angular displacement.

Constant-angular-acceleration model

When $\alpha$ is constant,

These equations mirror constant-acceleration translation but require radians and consistent signs.

Worked example

A wheel starts at $2.0\text{ rad/s}$ and accelerates at $3.0\text{ rad/s}^2$ for $4.0\text{ s}$:

Angular-to-linear relationships

For a rigid body,

and radial acceleration is

Tangential acceleration changes speed; radial acceleration changes direction. They are perpendicular components.

Rotational inertia

Rotational inertia measures resistance to angular acceleration about a stated axis:

for point masses. It depends on total mass, mass distribution, and axis. Units are $\text{kg·m}^2$.

Worked comparison

Two $0.50\text{ kg}$ point masses are placed symmetrically at radius $0.20\text{ m}$:

Moving them to $0.40\text{ m}$ makes $I=0.160\text{ kg·m}^2$, four times larger.

Common rigid-body models

About central symmetry axes:

The model must match geometry and axis. A hoop has greater $I$ than a solid disk with the same $M$ and $R$ because more mass lies farther out.

Axis dependence and parallel-axis idea

Moving an axis away from the center of mass increases inertia:

Use this theorem only for parallel axes and a known center-of-mass-axis inertia.

Rolling constraint

For rolling without slipping,

for the tangential rolling acceleration relationship under appropriate conditions. Sliding breaks the constraint.

During learning: rotational audit

1. State axis and positive rotation direction.
2. Label radians and distinguish revolutions.
3. Connect graph slope/area before equations.
4. Distinguish shared angular variables from radius-dependent linear variables.
5. Match the inertia formula to shape and axis.

Misconception clinic

“All wheel points have the same linear speed.” They share $\omega$; $v=r\omega$.

“Moment of inertia depends only on mass.” Distribution and axis matter.

“Radians have no role because they are dimensionless.” They remain essential labels in angular interpretation.

“Tangential and radial acceleration are the same.” They change speed and direction, respectively.

“Rolling always means $v=R\omega$.” Only rolling without slipping satisfies it.

After learning: retrieval and transfer

1. What do slope of $\theta(t)$ and area under $\omega(t)$ represent?
2. Compare speeds at $R/2$ and $R$.
3. How does moving point masses to twice the radius change $I$?
4. Why does a hoop have larger $I$ than a disk of equal $M,R$?
5. What observation indicates rolling slip?

AP-style evidence routine

1. Define rigid body, axis, sign, and interval.
2. Connect angular graph representations.
3. Apply constant-$\alpha$ equations only when supported.
4. Convert angular to linear quantities with radius.
5. Build $I$ from geometry/distribution and check units, limits, and axis.

Key takeaway

Rigid-body points share angular motion but not linear motion. Rotational inertia turns mass distribution and axis choice into a measurable resistance to angular acceleration.

Further learning and alignment

• College Board, “AP Physics 1: Algebra-Based Course,” https://apcentral.collegeboard.org/courses/ap-physics-1 (retrieved 2026-08-19). Alignment only.
• OpenStax, “10.2 Rotation with Constant Angular Acceleration” and “10.5 Calculating Moments of Inertia,” https://openstax.org/books/physics/pages/10-2-rotation-with-constant-angular-acceleration and https://openstax.org/books/physics/pages/10-5-calculating-moments-of-inertia (retrieved 2026-08-19). CC BY verification; no exercises copied.