Angular Acceleration Formula: α = Δω/Δt and α = τ/I

The angular acceleration formula is $\alpha = \frac{\omega_2 - \omega_1}{t}$ : the change in angular velocity divided by the time over which it happens. Its SI unit is the radian per second squared (rad/s²). When you know the turning effect instead, use the rotational form of Newton's second law, $\alpha = \frac{\tau}{I}$ .

Angular acceleration is to spinning what ordinary acceleration is to straight line motion. A potter's wheel winding up, a fan running down after you switch it off, and a car wheel as the driver brakes all have an α. The dimensional formula of angular acceleration page covers why it reduces to [T⁻²].

Quick facts

Definition
$\alpha = \frac{\Delta\omega}{\Delta t}$
From torque
$\alpha = \frac{\tau}{I}$
Link to linear
$a_t = r\alpha$
SI unit
rad/s²
Dimensional formula
[M⁰ L⁰ T⁻²]
Type
Vector along the axis
Negative α
Spin slowing down in the chosen sense

Applying the angular acceleration formula in four moves

Decide which sense of turning counts as positive; anticlockwise is the usual pick.
Change every rate into rad/s. For rpm, multiply by $\frac{2\pi}{60}$ .
Substitute in $\alpha = \frac{\omega_2 - \omega_1}{t}$ , keeping the signs.
A negative result means the rotation is slowing down (angular retardation), or speeding up in the negative sense.

The angular velocity inputs come from the angular velocity formula. If the question gives a torque and a moment of inertia instead, skip the definition and go straight to $\alpha = \frac{\tau}{I}$ .

Examples worked in full

Example 1: a grinding wheel speeding up

A wheel goes from 10 rad/s to 40 rad/s in 5 s. Find α.

$\alpha = \frac{40 - 10}{5} = 6\ \text{rad/s}^2$

Example 2: a fan switched off

A table fan running at 1500 rpm coasts to rest in 25 s after you switch it off. Find its angular acceleration.

Turn rpm into rad/s first: $\omega_1 = 1500 \times \frac{2\pi}{60} = 50\pi \approx 157.1\ \text{rad/s}$ , and the final value is $\omega_2 = 0$ .

$\alpha = \frac{0 - 157.1}{25} \approx -6.28\ \text{rad/s}^2$

The fan slows by about 6.28 rad/s every second.

Example 3: from torque

A net torque of 12 N m acts on a wheel with I = 3 kg m². Find α.

$\alpha = \frac{\tau}{I} = \frac{12}{3} = 4\ \text{rad/s}^2$

Rotational equations of motion

For constant α the same algebra used for straight lines works with angles. Replace u, v, a and s by ω₁, ω₂, α and θ.

Linear and rotational versions side by side
Straight line	Rotation
$v = u + at$	$\omega_2 = \omega_1 + \alpha t$
$s = ut + \frac{1}{2}at^2$	$\theta = \omega_1 t + \frac{1}{2}\alpha t^2$
$v^2 = u^2 + 2as$	$\omega_2^2 = \omega_1^2 + 2\alpha\theta$

For the fan in Example 2, $\theta = \frac{\omega_1 + \omega_2}{2} t = \frac{157.1}{2} \times 25 \approx 1963\ \text{rad}$ . Divide by 2π and you find the blades make about 312 turns before they stop. You could get the same angle from the third row: $\theta = \frac{0 - 157.1^2}{2 \times (-6.28)} \approx 1965\ \text{rad}$ , the small gap coming only from rounding α.

Angular and linear acceleration together

A point at distance r from the axis has a tangential acceleration $a_t = r\alpha$ along its path. This is separate from the inward centripetal acceleration $\omega^2 r$ , which exists even when α is zero. The torque link, $\tau = I\alpha$ , is set out on the torque page, and the rotation hub shows the full set of parallels.

Units and dimensions of α

rad/s divided by s gives rad/s². The radian is a ratio of an arc to a radius and carries no dimension, so α has dimensions of inverse time squared, [M⁰ L⁰ T⁻²]. Compare ordinary acceleration, [M⁰ L¹ T⁻²]: the missing L is the radius, which is why $a_t = r\alpha$ needs the factor r.

Never mix rpm and seconds in one formula. Convert rpm to rad/s at the start, every time.

Where marks go missing

Forgetting the minus sign for a body that is slowing down.
Using α = a/r with the centripetal acceleration instead of the tangential one.
Treating revolutions as radians. One revolution is 2π rad.
Applying the constant α equations when the torque changes with time.
Writing the unit as rad/s. Acceleration of any kind has one more 'per second' than the rate it changes, so the unit is rad/s².

A good habit: after each answer, ask whether the size is believable. A household fan or a bicycle wheel rarely has α above a few tens of rad/s². A result in the thousands usually means rpm went in where rad/s was needed.

Frequently asked questions

What is the formula of angular acceleration?

Angular acceleration is α = (ω₂ − ω₁) / t, the change in angular velocity per unit time. With a known torque, α = τ / I.

Which unit is used for angular acceleration?

Radian per second squared, rad/s². Its dimensions reduce to [T⁻²] because the radian has none.

How is angular acceleration related to linear acceleration?

A point at radius r has tangential acceleration aₜ = rα. The inward centripetal part, ω²r, is separate.

Is angular acceleration a vector?

It does have a direction, along the rotation axis. When the spin speeds up it points the same way as ω, and when the spin slows it points the opposite way.

Can angular acceleration be negative?

Yes. A negative α means the angular velocity is decreasing in the chosen positive sense, like a wheel coming to rest.

What is the formula for angular acceleration in terms of torque?

From Newton's second law for rotation, α = τ / I, where τ is the net torque in N m and I is the moment of inertia in kg m².