AP Physics 1 Rotational Energy: How Spinning Systems Store and Transfer Energy

Rotational kinetic energy

The kinetic energy an object has due to its rotation, arising because its mass elements move in circles around an axis (not because the center of mass is translating).

Translational kinetic energy

The kinetic energy associated with the motion of an object’s center of mass, given by K = (1/2)mv^2.

Rigid body (rotation model)

An object treated as many small mass elements at distances r from an axis, all rotating with the same angular speed ω but having different tangential speeds.

Angular speed (ω)

A measure of how fast an object rotates (in rad/s); determines tangential speeds via v = rω.

Tangential speed (v)

The linear speed of a point at distance r from the rotation axis; for rigid rotation v = rω.

Moment of inertia (I) / rotational inertia

A measure of how mass is distributed relative to a rotation axis; for discrete masses I = Σ(mᵢ rᵢ²).

Rotational kinetic energy formula

K_rot = (1/2)Iω².

Interpretation of moment of inertia

The rotational analog of mass that depends strongly on how far mass lies from the axis; larger I means harder to spin up and more rotational energy at a given ω.

Point mass moment of inertia

For a point mass m at radius r: I = mr² (not generally valid for extended objects).

Thin hoop (ring) moment of inertia (about central axis)

I = mR².

Solid disk / solid cylinder moment of inertia (about central axis)

I = (1/2)mR².

Thin rod moment of inertia (about center, axis ⟂ rod)

I = (1/12)mL².

Thin rod moment of inertia (about one end, axis ⟂ rod)

I = (1/3)mL².

Axis dependence of moment of inertia

The same object can have different I values depending on where the rotation axis is located and how it is oriented.

Hoop vs. disk comparison (same m and R)

A hoop has more mass at the rim, so it has larger I and thus larger K_rot than a solid disk for the same angular speed ω.

Total kinetic energy for rolling motion

For an object that both translates and rotates: Ktotal = (1/2)mvcm² + (1/2)I_cm ω².

Rolling without slipping condition

Constraint linking translation and rotation: v_cm = ωR.

Static friction in rolling without slipping

Friction can be present but does not necessarily do mechanical work on the rolling object when there is no slipping because the contact point is instantaneously at rest relative to the ground.

Energy conservation for rolling down a height

For rolling without slipping from rest: mgh becomes translational plus rotational kinetic energy, mgh = (1/2)mv² + (1/2)Iω² (with v = ωR).

Effect of rotation on rolling speed

Because some energy goes into rotation, a rolling object’s translational speed at the bottom is less than if it slid without friction (which would give v = √(2gh)).

Torque (τ)

A measure of how effectively a force causes rotation about an axis; magnitude τ = rF sinθ.

Lever arm (r)

The distance from the rotation axis to the point where the force is applied (used in τ = rF sinθ).

Work done by a constant torque

Rotational work for constant torque: W = τΔθ, where Δθ must be in radians and sign depends on rotational direction.

Rotational work-energy theorem

The net work done by torques equals the change in rotational kinetic energy: Wnet = ΔKrot.

Rotational power

The rate of doing rotational work: P = τω (analogous to linear P = Fv).