Rotation of a Rigid Body

Every point goes round in a circle, and all of them share one angle.

11 minUpdated 21 October 2026rigid bodyaxis of rotationangular velocityangular accelerationrotational kinematics

Lesson at a glanceShow the key points

In rotation about a fixed axis every particle moves in a circle centred on the axis, in a plane perpendicular to it; all particles turn through the same angle in the same time.
Angular velocity ω = dθ/dt and angular acceleration α = dω/dt describe the whole body; ω points along the axis by the right-hand rule.
For constant α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ (10.1–10.3).
A particle at distance r from the axis has speed v = rω and tangential acceleration aₜ = rα; particles on the axis do not move.
1 revolution = 2π rad; rpm × 2π/60 converts revolutions per minute to rad/s.

A bucket drops into a well and the pulley above it whirls faster and faster. The pulley’s centre never moves, so the net force on it is zero, yet clearly something is changing. Welcome to rotation.

What “rotating about an axis” means

Pinch a book between two fingertips and twist it. Whatever you do, the two pinched points stay put and every other bit of the book swings round in a circle. The circles all lie in parallel planes, each centred on the line joining your fingertips, and every particle completes its circle in the same time. Particles on that line don’t move at all; those far from it sweep out big circles.

Definition · Rotation about a fixed axis

Motion of a rigid body in which every particle moves on a circle whose centre lies on one fixed straight line (the axis of rotation), with the plane of each circle perpendicular to that line.

Hinged doors, ceiling fans and the knob on a gas stove all rotate this way. The axis need not even pass through the body: a fly sitting near the rim of a spinning turntable rotates about the turntable’s axis.

One angle describes the whole body

Because the body is rigid, when one particle’s radius turns through an angle θ, every particle’s radius turns through the same θ. So a single number, the angular position θ of the body, tells us where everything is. From it we build the same ladder of quantities we used for a particle on a line:

\omega = \frac{d\theta}{dt}, \qquad \alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}

Angular velocity is measured in rad/s (often quoted in rev/s or rpm; 1 rev = 2π rad). It is given a direction along the axis by the right-hand rule: curl your fingers the way the body turns and your thumb points along $\vec\omega$ . Choose one sense of rotation as positive and stick with it; θ, ω and α can then be positive or negative, just like x, v and a.

Rotation with constant angular acceleration(10.1–10.3)

\omega = \omega_0 + \alpha t, \qquad \theta = \omega_0 t + \tfrac12\alpha t^2, \qquad \omega^2 = \omega_0^2 + 2\alpha\theta

Interactive

Switch it off and count the turns

A grinder spins at a set speed and is switched off. With a constant angular deceleration, ω falls in a straight line and the angle turned is the shaded area under the ω–t graph. Both marked points share ω, but the outer one moves faster: v = rω.

Point A is 10 cm from the axis, B is 22 cm (drawn to scale).

t

0.00s

\omega

12.57 rad/s (120 rpm)

\alpha

-0.628rad/s²

\theta

0.0 rad = 0.00 rev

v_B = r_B\,\omega

2.76m/s

Starting speed120rpm

Time to stop20s

Worked example

How many turns before it stops?

A bench grinder spinning at 120 rpm is switched off and comes to rest in 20 s with constant deceleration. How many revolutions does it make while stopping?

Worked example

Half speed takes half the time

A fan starting from rest reaches 600 rpm in 4.0 s with constant angular acceleration. How long does it take to reach 150 rpm?

From the body back to its particles

A particle at perpendicular distance r from the axis moves along an arc of length $s = r\theta$ . Differentiate:

Linking rotation to particle motion(10.4, 10.5)

v = r\omega, \qquad a_t = r\alpha

ω and α are shared by the whole body; v and aₜ grow with distance from the axis

Every particle also has its centripetal acceleration $\omega^2 r$ toward the axis, as in circular motion.

Worked example

Lowering a bucket

A rope runs over a fixed pulley of radius 15 cm without slipping. At one instant the bucket on the rope descends at 60 cm/s and speeds up at 3.0 m/s². Find the pulley’s angular velocity and angular acceleration.

What to remember

1In fixed-axis rotation every particle circles the axis, and all share the same θ, ω and α.
2 $\omega = d\theta/dt$ , $\alpha = d\omega/dt$ ; ω points along the axis by the right-hand rule.
3Constant α: the three kinematic equations with x, v, a replaced by θ, ω, α.
4 $v = r\omega$ and $a_t = r\alpha$ for a particle at distance r.

Check yourself

On a spinning merry-go-round, a child at the rim and a child halfway in have…

A wheel starts from rest with α = 2 rad/s². How many radians does it turn in the first 5 s?

300 rpm is how many rad/s?

Keep exploring

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

WatchRotational MotionCrash Course Physics #11 WatchAngular motion variablesKhan Academy ReadRotational variablesOpenStax University Physics §10.1 ReadRotation with constant angular accelerationOpenStax University Physics §10.2 ReadRelating angular and translational quantitiesOpenStax University Physics §10.3 ReadRotational quantitiesHyperPhysics ReadRotation in two dimensionsFeynman Lectures, Vol. I, Ch. 18

Next Torque

Rotation of a Rigid Body

Every point goes round in a circle, and all of them share one angle.

11 minUpdated 21 October 2026rigid bodyaxis of rotationangular velocityangular accelerationrotational kinematics

Lesson at a glanceShow the key points

In rotation about a fixed axis every particle moves in a circle centred on the axis, in a plane perpendicular to it; all particles turn through the same angle in the same time.
Angular velocity ω = dθ/dt and angular acceleration α = dω/dt describe the whole body; ω points along the axis by the right-hand rule.
For constant α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ (10.1–10.3).
A particle at distance r from the axis has speed v = rω and tangential acceleration aₜ = rα; particles on the axis do not move.
1 revolution = 2π rad; rpm × 2π/60 converts revolutions per minute to rad/s.

A bucket drops into a well and the pulley above it whirls faster and faster. The pulley’s centre never moves, so the net force on it is zero, yet clearly something is changing. Welcome to rotation.

What “rotating about an axis” means

Definition · Rotation about a fixed axis

Motion of a rigid body in which every particle moves on a circle whose centre lies on one fixed straight line (the axis of rotation), with the plane of each circle perpendicular to that line.

One angle describes the whole body

\omega = \frac{d\theta}{dt}, \qquad \alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}

Rotation with constant angular acceleration(10.1–10.3)

\omega = \omega_0 + \alpha t, \qquad \theta = \omega_0 t + \tfrac12\alpha t^2, \qquad \omega^2 = \omega_0^2 + 2\alpha\theta

Interactive

Switch it off and count the turns

Point A is 10 cm from the axis, B is 22 cm (drawn to scale).

t

0.00s

\omega

12.57 rad/s (120 rpm)

\alpha

-0.628rad/s²

\theta

0.0 rad = 0.00 rev

v_B = r_B\,\omega

2.76m/s

Starting speed120rpm

Time to stop20s

Worked example

How many turns before it stops?

A bench grinder spinning at 120 rpm is switched off and comes to rest in 20 s with constant deceleration. How many revolutions does it make while stopping?

Worked example

Half speed takes half the time

A fan starting from rest reaches 600 rpm in 4.0 s with constant angular acceleration. How long does it take to reach 150 rpm?

From the body back to its particles

A particle at perpendicular distance r from the axis moves along an arc of length $s = r\theta$ . Differentiate:

Linking rotation to particle motion(10.4, 10.5)

v = r\omega, \qquad a_t = r\alpha

ω and α are shared by the whole body; v and aₜ grow with distance from the axis

Every particle also has its centripetal acceleration $\omega^2 r$ toward the axis, as in circular motion.

Worked example

Lowering a bucket

What to remember

1In fixed-axis rotation every particle circles the axis, and all share the same θ, ω and α.
2 $\omega = d\theta/dt$ , $\alpha = d\omega/dt$ ; ω points along the axis by the right-hand rule.
3Constant α: the three kinematic equations with x, v, a replaced by θ, ω, α.
4 $v = r\omega$ and $a_t = r\alpha$ for a particle at distance r.

Check yourself

On a spinning merry-go-round, a child at the rim and a child halfway in have…

A wheel starts from rest with α = 2 rad/s². How many radians does it turn in the first 5 s?

300 rpm is how many rad/s?

Keep exploring

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

Next Torque

What “rotating about an axis” means#

One angle describes the whole body#

From the body back to its particles#

Check yourself

Keep exploring

What “rotating about an axis” means#

One angle describes the whole body#

From the body back to its particles#

Check yourself

Keep exploring

What “rotating about an axis” means

One angle describes the whole body

From the body back to its particles

What “rotating about an axis” means

One angle describes the whole body

From the body back to its particles