Angular Velocity Dimensional Formula: [M⁰ L⁰ T⁻¹] Explained

The angular velocity dimensional formula is [M⁰ L⁰ T⁻¹]; drop the zero powers and it shrinks to [T⁻¹]. Angular velocity has no mass or length dimension, only one over time.

The reason is that ω is an angle divided by time, and an angle in radians is an arc length divided by a radius, a length over a length. So the radian is dimensionless and drops out. That is why rad/s and s⁻¹ describe the same dimensions, even though writing rad/s tells the reader you mean rotation. Formulas and solved problems for ω are on the angular velocity formula page.

Quick facts

Dimensional formula
[M⁰ L⁰ T⁻¹], or [T⁻¹]
Defining formula
$\omega = \frac{\Delta\theta}{\Delta t}$
SI unit
rad/s, dimensionally s⁻¹
Other forms
$\omega = 2\pi f = \frac{2\pi}{T}$ , $\omega = \frac{v}{r}$
Type
Vector, along the axis by the right hand rule
Same dimensions as
Frequency, angular frequency, velocity gradient

Derivation of the angular velocity dimensional formula

Definition: $\omega = \frac{\Delta\theta}{\Delta t}$ .
An angle in radians is $\theta = \frac{s}{r}$ , arc length over radius, so $[\theta] = \frac{[L]}{[L]} = [M^{0}L^{0}T^{0}]$ .
Time has dimension [T].
Divide: $[\omega] = \frac{1}{[T]} = [T^{-1}]$ , which is [M⁰ L⁰ T⁻¹] once the zero powers are shown.

A quicker route uses $v = \omega r$ . Rearranged, $\omega = \frac{v}{r}$ , and $\frac{[LT^{-1}]}{[L]} = [T^{-1}]$ . Either derivation is acceptable in a Class 11 answer. Pick the one that matches the definition your board gives.

Why rad/s is fine even though the radian vanishes

Units and dimensions answer slightly different questions. The dimension [T⁻¹] says that ω scales like one over a time. The unit rad/s says the same, plus the useful reminder that the quantity measures turning. SI calls the radian a dimensionless derived unit for exactly this reason.

Example: a table fan's blades spin at 300 rpm. Find ω in rad/s.

One revolution is 2π rad and one minute is 60 s.

$\omega = 300 \times \frac{2\pi\ \text{rad}}{60\ \text{s}} = 10\pi = 31.4\ \text{rad/s}$ .

Example: the minute hand of a wall clock goes round once every hour. What is its angular velocity?

One turn is 2π rad and one hour is 3600 s, so $\omega = \frac{2\pi}{3600} = 1.75 \times 10^{-3}\ \text{rad/s}$ . The hand's length does not matter; ω is the same for every point on it.

Never put rpm straight into v = ωr. Convert to rad/s first.

Testing rotation formulas with [T⁻¹]

Example: check centripetal acceleration $a = \omega^{2} r$ .

$[T^{-1}]^{2}[L] = [LT^{-2}]$ , the bracket of acceleration. It passes.

Example: check rotational kinetic energy $K = \frac{1}{2} I\omega^{2}$ , where I is moment of inertia.

Moment of inertia: $[ML^{2}]$ . Then $[ML^{2}][T^{-2}] = [ML^{2}T^{-2}]$ , which is energy. The 1/2 is invisible to dimensions.

Angular momentum $L = I\omega$ gives $[ML^{2}T^{-1}]$ , matching the angular momentum dimensions.

Other quantities measured in [T⁻¹]

One bracket, several meanings
Quantity	Formula	Unit written as
Angular velocity	$\Delta\theta / \Delta t$	rad/s
Angular frequency of an oscillator	$2\pi f$	radian per second
Frequency	$1/T$	Hz
Velocity gradient	$dv/dx$	s⁻¹
Decay constant	$\lambda$ in $N = N_0 e^{-\lambda t}$	s⁻¹

Because these share a bracket, dimensions alone cannot tell them apart. The factor 2π between ω and f is a pure number, so an equation with a missing 2π still passes a dimensional check. Read the question for which one it wants. Compare too with angular acceleration, which carries an extra factor of time in its denominator.

Traps in angular velocity dimensions

Giving the radian a dimension. It is length over length, so it has none.
Writing [LT⁻¹]. That is linear velocity. Angular velocity has no L.
Mixing ω with frequency in numbers. Same bracket, but ω = 2πf, so the values differ by 2π.
Keeping rpm in a formula. Revolutions per minute is a valid rate but must be converted before use.

Rotation quantities appear together on the dimensional formulas hub, from ω and α to torque and moment of inertia.

Frequently asked questions

What is the dimensional formula of angular velocity?

It is [M⁰ L⁰ T⁻¹]. An angle in radians has no dimensions, so dividing it by time leaves only T⁻¹.

Is the radian a dimension?

No. A radian is arc length divided by radius, a length over a length, so it is dimensionless. It is kept in units like rad/s only to show that rotation is meant.

Do angular velocity and frequency have the same dimensional formula?

Yes, both are [T⁻¹]. They differ by the pure number 2π, because ω = 2πf, and by the unit name: rad/s for ω and hertz for f.

Which unit does angular velocity take in SI?

Radian per second, rad/s. Since the radian is dimensionless, this is dimensionally the same as s⁻¹.

Can angular velocity be negative if its bracket has no sign?

Yes. A bracket shows only how a quantity scales with mass, length and time. The sign of ω depends on the sense of rotation you call positive, which dimensions never record.

How do I change rpm into rad/s?

Multiply by 2π and divide by 60. For example, 1200 rpm becomes 1200 × 2π / 60 = 126 rad/s to three significant figures.