Rotational Energy and Angular Momentum

Driving question

Which quantities remain useful or conserved in rotating systems?

Before learning: spinning-student prediction

A student rotates on a low-friction stool while holding two masses. Predict what happens to angular speed when the masses move inward. Then decide whether angular momentum, rotational kinetic energy, both, or neither must remain constant.

Hook: pull inward, spin faster—with no motor

Moving mass inward reduces rotational inertia. If external torque is negligible, angular momentum stays constant, so angular speed increases. But rotational kinetic energy can increase because the student does internal work while pulling.

Rotational kinetic energy

For a rigid body rotating about a fixed axis,

Rotational inertia $I$ depends on mass distribution and axis. Angular speed $\omega$ is shared by every point of a rigid body, but linear speed depends on radius:

Points farther from the axis move faster linearly.

Worked example: flywheel energy

A flywheel with $I=0.80\text{ kg}\cdot\text{m}^2$ rotates at $6.0\text{ rad/s}$:

Doubling angular speed would quadruple its rotational kinetic energy.

Work and power in rotation

For constant torque through angular displacement,

Instantaneous rotational power is

Torque and energy both use $\text{N}\cdot\text{m}$ dimensionally, but torque is a signed rotational effect while energy is a scalar measured in joules.

Angular momentum and angular impulse

For a rigid body rotating about a fixed symmetry axis,

The general relationship between external torque and angular momentum is

Angular impulse changes angular momentum:

Signed area under a net-external-torque-versus-time graph equals angular-momentum change.

Conservation of angular momentum

If net external torque is zero or its angular impulse is negligible over the interval,

For a rotating body whose distribution changes,

Worked example: pulling masses inward

A rotating system changes from $I_i=4.0\text{ kg}\cdot\text{m}^2$ at $\omega_i=2.0\text{ rad/s}$ to $I_f=1.6\text{ kg}\cdot\text{m}^2$. With negligible external angular impulse,

Initial energy is $8.0\text{ J}$ and final energy is $20\text{ J}$. Angular momentum is conserved; kinetic energy increases because internal biochemical energy is transferred by work.

Rolling without slipping

A rolling object's kinetic energy includes translation and rotation:

For rolling without slipping,

A rolling object does not have only translational energy. Objects with different $I/(MR^2)$ ratios partition energy differently and can have different accelerations down the same incline.

Worked example: rolling solid cylinder

For a solid cylinder, $I_{cm}=\frac12MR^2$. At speed $v$ without slipping,

One-third of the total kinetic energy is rotational and two-thirds translational.

During learning: conservation audit

1. Name the system, axis, and interval.
2. Identify external torques and estimate whether their angular impulse is negligible.
3. Distinguish angular-momentum conservation from mechanical-energy conservation.
4. For rolling, include both energy terms and justify the no-slip constraint.
5. Check whether all points share $\omega$, not $v$.

Misconception clinic

“Angular momentum is conserved in every rotation.” It changes when net external angular impulse is nonzero.

“If angular momentum is conserved, kinetic energy is conserved.” Internal work can change kinetic energy while $L$ remains constant.

“A rolling object has only translational energy.” It also rotates about its center of mass.

“All points on a rigid body have the same linear speed.” They share angular speed; $v=r\omega$ varies with radius.

“Static friction always removes mechanical energy.” In ideal rolling without slipping on a fixed surface, static friction may do no work at the contact point.

After learning: retrieval and transfer

1. How does doubling $\omega$ affect $K_{rot}$ at fixed $I$?
2. What does area under a net-torque-time graph represent?
3. Why can pulling masses inward increase rotational kinetic energy while conserving angular momentum?
4. Write the complete kinetic energy for rolling without slipping.
5. What external interaction would make angular momentum about a chosen axis fail to be conserved?

AP-style evidence routine

1. Choose system, axis, and interval.
2. Inventory energy transfers and external torques.
3. Decide which conservation statements are justified independently.
4. Write relationships symbolically before substitution.
5. Apply $v_{cm}=R\omega$ only for verified no-slip rolling.
6. Check units, signs, energy accounting, and limiting cases.

Key takeaway

Angular momentum is controlled by external angular impulse, while rotational energy is controlled by work and energy transfer. Conservation of one never automatically guarantees conservation of the other.

Further learning and alignment

• College Board, “AP Physics 1: Algebra-Based Course,” https://apcentral.collegeboard.org/courses/ap-physics-1 (retrieved 2026-08-18). Alignment reference only; no endorsement implied.
• OpenStax, “11.2 Angular Momentum” and “11.3 Conservation of Angular Momentum,” https://openstax.org/books/physics/pages/11-2-angular-momentum and https://openstax.org/books/physics/pages/11-3-conservation-of-angular-momentum (retrieved 2026-08-18). CC BY verification; no exercises copied.
• OpenStax, “10.4 Rotational Kinetic Energy,” https://openstax.org/books/physics/pages/10-4-rotational-kinetic-energy-work-and-energy-revisited (retrieved 2026-08-18). CC BY verification.