Rotational Kinetic Energy, Work & Power

A spinning wheel stores energy in the motion of all its parts.

10 minUpdated 21 October 2026rotational kinetic energywork by a torquepower P = τωflywheel

Lesson at a glanceShow the key points

A body rotating about a fixed axis has kinetic energy K = ½Iω²; it is just Σ½mᵢvᵢ² with vᵢ = rᵢω.
Energy conservation must include the rotating parts: for a block falling a height h while unwinding a string from a wheel, mgh = ½mv² + ½I(v/r)².
Power delivered by a torque: P = τω.
Work done by a torque: dW = τ dθ, so W = ∫τ dθ (10.14).

A potter’s wheel keeps turning after the foot stops pushing; a flywheel smooths an engine’s jerks. Spinning things store energy, and we can count it.

Kinetic energy of a spinning body

A body turning about a fixed axis at angular speed ω has its i-th particle moving at $v_i = r_i\omega$ . Add up the ordinary kinetic energies:

K = \sum \tfrac12 m_i (r_i\omega)^2 = \tfrac12\Big(\sum m_i r_i^2\Big)\omega^2

Rotational kinetic energy

K = \tfrac12 I\omega^2

the twin of ½mv², with I in place of m and ω in place of v

Energy bookkeeping with wheels

When a block on a string unwinds a wheel, the string doesn’t slip, so the rim moves as fast as the block: ω = v/r. As the block falls a height h, gravity’s lost potential energy goes into both bodies:

mgh = \tfrac12 mv^2 + \tfrac12 I\Big(\frac vr\Big)^2 \quad\Rightarrow\quad v = \sqrt{\frac{2mgh}{m + I/r^2}}

Interactive

Where the falling block's energy goes

A block hangs from a string wrapped round a wheel of radius 20 cm. As it falls 1.5 m, the potential energy it loses is shared between its own motion (½mv²) and the wheel's spin (½Iω², with ω = v/r). A heavier wheel steals more and the block lands more slowly.

0.0

PE lost

0.0

KE block

0.0

KE wheel

a = \dfrac{mg}{m + I/r^2}

2.80m/s²

v

0.00m/s

\omega = v/r

0.0rad/s

speed at the bottom2.90m/s

Block mass

m

2.0kg

Wheel's moment of inertia

I

0.20kg·m²

Worked example

A block unwinding a wheel

A 2.0 kg block hangs from a light string wrapped round a wheel of radius 20 cm and moment of inertia 0.20 kg·m², which turns freely. Released from rest, how fast is the block moving after falling 1.5 m?

Work and power from a torque

A torque that speeds a body up raises its kinetic energy at a rate

P = \frac{dK}{dt} = \frac{d}{dt}\Big(\tfrac12 I\omega^2\Big) = I\omega\frac{d\omega}{dt} = I\alpha\,\omega = \tau\omega

Power and work of a torque(10.14)

P = \tau\omega, \qquad dW = \tau\,d\theta, \qquad W = \int_{\theta_1}^{\theta_2}\tau\,d\theta

compare P = Fv and W = ∫F dx

Worked example

An electric motor

A motor delivers a steady torque of 20 N·m to a shaft turning at 3000 rpm. (a) What power does it supply? (b) How much work does it do in 10 revolutions?

What to remember

1 $K_{\text{rot}} = \tfrac12 I\omega^2$ , the sum of ½mv² over the body.
2Include rotating parts in energy conservation; a non-slipping string links v = ωr.
3Power $P = \tau\omega$ ; work $W = \int\tau\,d\theta$ .

Check yourself

Two wheels have the same I. One spins twice as fast. Its kinetic energy is…

Increasing the wheel's I in the falling-block set-up makes the block reach the floor…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchRotational kinetic energyKhan Academy ReadMoment of inertia and rotational kinetic energyOpenStax University Physics §10.4 ReadWork and power for rotational motionOpenStax University Physics §10.8 ReadRotational kinetic energyHyperPhysics ReadRotational energyWikipedia

PreviousAngular Momentum & Its Conservation Next Calculating Moment of Inertia

Rotational Kinetic Energy, Work & Power

A spinning wheel stores energy in the motion of all its parts.

10 minUpdated 21 October 2026rotational kinetic energywork by a torquepower P = τωflywheel

Lesson at a glanceShow the key points

A body rotating about a fixed axis has kinetic energy K = ½Iω²; it is just Σ½mᵢvᵢ² with vᵢ = rᵢω.
Energy conservation must include the rotating parts: for a block falling a height h while unwinding a string from a wheel, mgh = ½mv² + ½I(v/r)².
Power delivered by a torque: P = τω.
Work done by a torque: dW = τ dθ, so W = ∫τ dθ (10.14).

A potter’s wheel keeps turning after the foot stops pushing; a flywheel smooths an engine’s jerks. Spinning things store energy, and we can count it.

Kinetic energy of a spinning body

A body turning about a fixed axis at angular speed ω has its i-th particle moving at $v_i = r_i\omega$ . Add up the ordinary kinetic energies:

K = \sum \tfrac12 m_i (r_i\omega)^2 = \tfrac12\Big(\sum m_i r_i^2\Big)\omega^2

Rotational kinetic energy

K = \tfrac12 I\omega^2

the twin of ½mv², with I in place of m and ω in place of v

Energy bookkeeping with wheels

mgh = \tfrac12 mv^2 + \tfrac12 I\Big(\frac vr\Big)^2 \quad\Rightarrow\quad v = \sqrt{\frac{2mgh}{m + I/r^2}}

Interactive

Where the falling block's energy goes

0.0

PE lost

0.0

KE block

0.0

KE wheel

a = \dfrac{mg}{m + I/r^2}

2.80m/s²

v

0.00m/s

\omega = v/r

0.0rad/s

speed at the bottom2.90m/s

Block mass

m

2.0kg

Wheel's moment of inertia

I

0.20kg·m²

Worked example

A block unwinding a wheel

Work and power from a torque

A torque that speeds a body up raises its kinetic energy at a rate

P = \frac{dK}{dt} = \frac{d}{dt}\Big(\tfrac12 I\omega^2\Big) = I\omega\frac{d\omega}{dt} = I\alpha\,\omega = \tau\omega

Power and work of a torque(10.14)

P = \tau\omega, \qquad dW = \tau\,d\theta, \qquad W = \int_{\theta_1}^{\theta_2}\tau\,d\theta

compare P = Fv and W = ∫F dx

Worked example

An electric motor

A motor delivers a steady torque of 20 N·m to a shaft turning at 3000 rpm. (a) What power does it supply? (b) How much work does it do in 10 revolutions?

What to remember

1 $K_{\text{rot}} = \tfrac12 I\omega^2$ , the sum of ½mv² over the body.
2Include rotating parts in energy conservation; a non-slipping string links v = ωr.
3Power $P = \tau\omega$ ; work $W = \int\tau\,d\theta$ .

Check yourself

Two wheels have the same I. One spins twice as fast. Its kinetic energy is…

Increasing the wheel's I in the falling-block set-up makes the block reach the floor…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousAngular Momentum & Its Conservation Next Calculating Moment of Inertia

Kinetic energy of a spinning body#

Energy bookkeeping with wheels#

Work and power from a torque#

Check yourself

Keep exploring

Kinetic energy of a spinning body#

Energy bookkeeping with wheels#

Work and power from a torque#

Check yourself

Keep exploring

Kinetic energy of a spinning body

Energy bookkeeping with wheels

Work and power from a torque

Kinetic energy of a spinning body

Energy bookkeeping with wheels

Work and power from a torque