Angles, Angular Velocity & Angular Acceleration

Describe circling motion with an angle, and everything else follows.

11 minUpdated 21 October 2026angular positionangular velocityangular accelerationv = rωtangential acceleration

Lesson at a glanceShow the key points

Angular position θ locates a particle on a circle; ω = dθ/dt is angular velocity (rad/s); α = dω/dt is angular acceleration.
For constant α: θ = ω₀t + ½αt², ω = ω₀ + αt, ω² = ω₀² + 2αθ, mirroring the linear equations.
Linear speed v = rω (with θ in radians).
The rate of change of speed is aₜ = dv/dt = rα: the tangential acceleration, not the whole acceleration.

A fan blade, a merry-go-round, a satellite: for anything going round a circle, one number, an angle, tells you where it is.

Angular position and angular velocity

Let a particle P move on a circle of radius $r$ with centre O. Put the origin at O and draw an x-axis OX. The angle $\theta$ between OP and OX pins down where P is; call it the angular position. As P moves to a nearby point in time $\Delta t$ , the angle grows by $\Delta\theta$ . The rate of change is the angular velocity, and its rate of change is the angular acceleration:

\omega = \lim_{\Delta t\to0}\frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}, \qquad \alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}

Angles are measured in radians, so ω is in rad/s and α in rad/s².

Constant angular acceleration

These definitions have exactly the same form as $v = dx/dt$ and $a = dv/dt$ . So when α is constant, integrating gives three equations that mirror the equations of motion, with θ in place of x, ω for v and α for a (θ = 0 at t = 0):

Constant angular acceleration(7.1–7.3)

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

Interactive

Angle, angular velocity, angular acceleration

A bead goes round a circle starting with angular velocity ω₀ and constant angular acceleration α. The angle swept, θ, grows like the distance in straight-line motion; v = rω links the two worlds.

t

0.00s

\theta

0.00 rad

\omega = \omega_0 + \alpha t

0.50rad/s

v = r\omega

1.00m/s

a_t = r\alpha

0.60m/s²

Initial angular velocity

\omega_0

0.50rad/s

Angular acceleration

\alpha

0.30rad/s²

Radius

r

2.0m

From angles back to metres

In time $\Delta t$ the particle moves along the arc a distance $\Delta s = r\,\Delta\theta$ (that is the definition of a radian). Divide by $\Delta t$ and take the limit:

Linear speed(7.4)

v = r\omega

Differentiate once more to see how fast the speed itself changes:

Tangential acceleration(7.5)

a_t = \frac{dv}{dt} = r\alpha

Worked example

Spinning a stone on a string

A stone moves on a circle of radius 30 cm at a steady speed of 12 m/s. Find its angular velocity.

Worked example

Speeding up on a circle

A toy car on a circular track of radius 50 cm speeds up uniformly from 4.0 m/s to 7.0 m/s in 1.5 s. Find its angular acceleration.

What to remember

1 $\omega = d\theta/dt$ , $\alpha = d\omega/dt$ ; angles in radians.
2Constant α gives rotational copies of the three equations of motion.
3 $v = r\omega$ and $a_t = r\alpha$ .
4aₜ is the rate of change of speed only, not the total acceleration.

Check yourself

A wheel turns at 120 revolutions per minute. Its angular velocity is…

Two points on a spinning disc, at 5 cm and 10 cm from the centre, have…

Keep exploring

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

WatchUniform Circular MotionCrash Course Physics #7 ReadSpeed and velocity in circular motionThe Physics Classroom ReadUniform circular motionOpenStax University Physics §4.4 ReadCentripetal forceHyperPhysics

Next Centripetal Acceleration

Angles, Angular Velocity & Angular Acceleration

Describe circling motion with an angle, and everything else follows.

11 minUpdated 21 October 2026angular positionangular velocityangular accelerationv = rωtangential acceleration

Lesson at a glanceShow the key points

Angular position θ locates a particle on a circle; ω = dθ/dt is angular velocity (rad/s); α = dω/dt is angular acceleration.
For constant α: θ = ω₀t + ½αt², ω = ω₀ + αt, ω² = ω₀² + 2αθ, mirroring the linear equations.
Linear speed v = rω (with θ in radians).
The rate of change of speed is aₜ = dv/dt = rα: the tangential acceleration, not the whole acceleration.

A fan blade, a merry-go-round, a satellite: for anything going round a circle, one number, an angle, tells you where it is.

Angular position and angular velocity

\omega = \lim_{\Delta t\to0}\frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}, \qquad \alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}

Angles are measured in radians, so ω is in rad/s and α in rad/s².

Constant angular acceleration

Constant angular acceleration(7.1–7.3)

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

Interactive

Angle, angular velocity, angular acceleration

t

0.00s

\theta

0.00 rad

\omega = \omega_0 + \alpha t

0.50rad/s

v = r\omega

1.00m/s

a_t = r\alpha

0.60m/s²

Initial angular velocity

\omega_0

0.50rad/s

Angular acceleration

\alpha

0.30rad/s²

Radius

r

2.0m

From angles back to metres

In time $\Delta t$ the particle moves along the arc a distance $\Delta s = r\,\Delta\theta$ (that is the definition of a radian). Divide by $\Delta t$ and take the limit:

Linear speed(7.4)

v = r\omega

Differentiate once more to see how fast the speed itself changes:

Tangential acceleration(7.5)

a_t = \frac{dv}{dt} = r\alpha

Worked example

Spinning a stone on a string

A stone moves on a circle of radius 30 cm at a steady speed of 12 m/s. Find its angular velocity.

Worked example

Speeding up on a circle

A toy car on a circular track of radius 50 cm speeds up uniformly from 4.0 m/s to 7.0 m/s in 1.5 s. Find its angular acceleration.

What to remember

1 $\omega = d\theta/dt$ , $\alpha = d\omega/dt$ ; angles in radians.
2Constant α gives rotational copies of the three equations of motion.
3 $v = r\omega$ and $a_t = r\alpha$ .
4aₜ is the rate of change of speed only, not the total acceleration.

Check yourself

A wheel turns at 120 revolutions per minute. Its angular velocity is…

Two points on a spinning disc, at 5 cm and 10 cm from the centre, have…

Keep exploring

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

Next Centripetal Acceleration

Angular position and angular velocity#

Constant angular acceleration#

From angles back to metres#

Check yourself

Keep exploring

Angular position and angular velocity#

Constant angular acceleration#

From angles back to metres#

Check yourself

Keep exploring

Angular position and angular velocity

Constant angular acceleration

From angles back to metres

Angular position and angular velocity

Constant angular acceleration

From angles back to metres