Angular Acceleration Dimensional Formula: [M⁰ L⁰ T⁻²]

The angular acceleration dimensional formula, [M⁰ L⁰ T⁻²], shrinks to [T⁻²] once the zero powers are dropped. It contains no mass and no length, just time to the power minus two.

Angular acceleration α tells you how fast the angular velocity of a spinning body is changing. Angular velocity is already [T⁻¹], and dividing by one more time gives [T⁻²]. Compare that with linear acceleration, [LT⁻²]: the length is missing because angles in radians are dimensionless. Formulas and problem types for α are on the angular acceleration formula guide.

Quick facts

Dimensional formula
[M⁰ L⁰ T⁻²], or [T⁻²]
Defining formula
$\alpha = \frac{\Delta\omega}{\Delta t}$
SI unit
rad/s², dimensionally s⁻²
Other forms
$\alpha = \frac{\tau}{I}$ , $\alpha = \frac{a_t}{r}$
Type
Vector along the rotation axis
Linear partner
Tangential acceleration, [LT⁻²]

Two derivations of the angular acceleration dimensional formula

From the definition.

$\alpha = \frac{\Delta\omega}{\Delta t}$ .
Angular velocity is angle over time, and the angle has no dimension, so $[\omega] = [T^{-1}]$ .
Divide by time: $[\alpha] = \frac{[T^{-1}]}{[T]} = [T^{-2}]$ .

From the tangential link. A point at radius r on a rotating wheel has tangential acceleration $a_t = \alpha r$ . So $\alpha = \frac{a_t}{r} = \frac{[LT^{-2}]}{[L]} = [T^{-2}]$ . Both routes agree. The first one relies on the dimensions of ω, which is worth revising alongside this page.

rad/s² in a real problem

Since the radian is length over length, rad/s² and s⁻² carry identical dimensions. Keep the word rad in answers anyway, because it tells the examiner you mean an angular quantity.

Example: a ceiling fan running at 600 rpm is switched off and stops evenly in 20 s. Find α.

Convert first: $\omega_0 = 600 \times \frac{2\pi}{60} = 62.8\ \text{rad/s}$ . Final $\omega = 0$ .

$\alpha = \frac{0 - 62.8\ \text{rad/s}}{20\ \text{s}} = -3.14\ \text{rad/s}^{2}$ .

The minus sign shows angular retardation: the fan is slowing down.

Dimensions never show the sign. Decide it from whether ω is growing or shrinking.

Testing rotational equations with [T⁻²]

Example: check Newton's second law for rotation, $\tau = I\alpha$ .

Moment of inertia: [ML²]. Multiplying, $[ML^{2}] \times [T^{-2}]$ gives the bracket of torque, [ML²T⁻²]. It passes.

Example: check $\theta = \omega_0 t + \frac{1}{2}\alpha t^{2}$ .

Left side: θ is dimensionless. First term: $[T^{-1}][T]$ , dimensionless. Second term: $[T^{-2}][T^{2}]$ , dimensionless. All three terms are pure numbers, and the equation holds up.

Notice that both checks say nothing about the factor 1/2. The torque dimensions page has more practice with rotational brackets.

Quantities that share [T⁻²]

Brackets equal to time to the minus two
Quantity or ratio	Where you meet it
Angular acceleration α	Rotating wheels, fans, motors
Square of angular frequency ω²	Simple harmonic motion, a = −ω²x
Spring constant over mass, k/m	Mass on a spring
g/l	Simple pendulum

This is why ω² can replace k/m or g/l in oscillation formulas without breaking any dimensions. It also explains a common slip: ω² and α share a bracket but are completely different ideas. Linear acceleration differs by a factor of L.

Avoidable slips on rotational acceleration

Writing [LT⁻²]. That is linear acceleration. Angles carry no L.
Writing [T⁻¹]. That is angular velocity. You have divided by time only once.
Leaving rpm in the formula. Convert revolutions per minute to rad/s before dividing by time.
Confusing α with the α of thermal expansion. Same Greek letter, different quantity, [K⁻¹].

For every rotational bracket in one table, see the dimensional formulas hub, where α sits between ω and torque.

Frequently asked questions

Which dimensional formula does angular acceleration have?

It is [M⁰ L⁰ T⁻²]. Angular velocity is [T⁻¹], and angular acceleration divides it by time once more.

What is the unit of angular acceleration in SI?

Radian per second squared, rad/s². Dimensionally this is the same as s⁻², because the radian has no dimension.

Is angular acceleration a vector?

It is. Its direction lies on the spin axis, matching ω while the rotation speeds up and opposing ω while it slows.

How does the bracket of α connect to tangential acceleration?

Tangential acceleration equals αr. Multiplying [T⁻²] by the radius bracket [L] gives [LT⁻²], the ordinary linear acceleration of a point on the rim. The radius supplies the missing length.

Can a body have angular acceleration while its angular velocity is zero?

Yes, for an instant. A pendulum bob at the end of its swing has ω = 0 but a nonzero α that starts it moving back. Dimensions cannot show this, because they never describe values, only how a quantity scales.

Do α and ω² have the same dimensions?

Indeed, both reduce to [T⁻²]. They mean different things: α is the rate of change of ω, while ω² appears in centripetal and oscillation formulas.