Angular Momentum Dimensional Formula [ML²T⁻¹] with Derivation

The angular momentum dimensional formula is [M¹ L² T⁻¹], written short as [ML²T⁻¹]. It is the same bracket you get for Planck's constant, which is no accident: both measure "action", an energy multiplied by a time.

For a particle, angular momentum is the moment of linear momentum about a point, L = r × p. A spinning rigid body gives the same result through L = Iω. Both routes are worked below, followed by the SI unit, a check on a quantum rule and the usual slips. The numbers and vector rules live on the angular momentum formula page.

Quick facts

Dimensional formula
[M¹ L² T⁻¹], short form [ML²T⁻¹]
Defining formulas
$L = mvr$ , $L = I\omega$
SI unit
kg m²/s = J s
Vector or scalar
Vector, along the rotation axis
Same dimensions as
Planck's constant, angular impulse
Chapter
Rotational motion, Class 11

Deriving the angular momentum dimensional formula

Two definitions appear in Class 11 rotational motion. The dimension formula of angular momentum has to come out the same either way, so doing both is a good self check.

Route 1: moment of linear momentum

$L = r \times p$ , with $p = mv$ .
$[r] = [L]$ and $[p] = [M][L T^{-1}] = [M L T^{-1}]$ .
Multiply the two: $[L] \cdot [M L T^{-1}]$ gives $[M L^{2} T^{-1}]$ .

Route 2: moment of inertia times angular velocity

$L = I\omega$ .
$[I] = [M L^{2}]$ , from $I = \sum m r^{2}$ .
Angle is a ratio of arc to radius, so $[\omega] = [T^{-1}]$ .
Product: $[M L^{2}]$ times $[T^{-1}]$ is again $[M L^{2} T^{-1}]$ .

The letter L is doing two jobs here: the symbol for angular momentum and the dimension of length. Keep them apart by always writing dimensions inside square brackets. More detail on the factors is on the moment of inertia and angular velocity pages.

SI unit and its three spellings

Turn the bracket into base units letter by letter. M becomes kg, L² becomes m², T⁻¹ becomes s⁻¹. The SI unit of angular momentum is therefore kg m² s⁻¹, usually printed kg m²/s.

Because a joule is kg m² s⁻², multiplying it by one second gives kg m² s⁻¹. So the unit can also be written J s, which is how it is quoted next to Planck's constant. A third form, N m s, follows from torque (N m) times time. All three are the same unit.

There is no special named unit for angular momentum. If an answer key shows J s, kg m²/s or N m s, each is correct.

Same bracket, different physics

Quantities sharing [M¹ L² T⁻¹]
Quantity	Relation	Why the bracket matches
Planck's constant h	$h = \frac{E}{f}$	Energy [ML²T⁻²] divided by frequency [T⁻¹]
Reduced constant ħ	$\hbar = \frac{h}{2\pi}$	2π carries no dimensions
Angular impulse	$\tau \, \Delta t$	Torque [ML²T⁻²] times time [T]
Action	energy × time	The general name for this bracket

The match with h matters in modern physics. In the Bohr model, the allowed orbits are those where angular momentum is a whole number times h/2π. That rule only makes sense because both sides carry the same dimensions. The dimensional formula of Planck's constant page derives h on its own.

Checking a rule with dimensions

Check the Bohr condition mvr = nh/(2π)

Left side: $[M][L T^{-1}][L] = [M L^{2} T^{-1}]$ .

Right side: n and 2π are pure numbers, so it equals $[h] = [M L^{2} T^{-1}]$ .

The two sides match, so the quantization rule is dimensionally sound.

Convert 1 kg m²/s into CGS units

Use the powers from [M¹ L² T⁻¹]: $(10^{3}\ \text{g})(10^{2}\ \text{cm})^{2}(1\ \text{s})^{-1}$ .

$10^{3} \times 10^{4} = 10^{7}$ , so 1 kg m²/s = 10⁷ g cm²/s.

This is also 10⁷ erg s, since one joule is 10⁷ erg.

Slips students make

Writing [MLT⁻¹]. That is linear momentum. Angular momentum carries one extra length, from the lever arm.
Writing [ML²T⁻²]. That is torque or energy. Angular momentum has only one power of time in the denominator.
Giving ω the dimensions [LT⁻¹]. The radian is a ratio, so angular velocity is just [T⁻¹].
Mixing the symbol L with the dimension [L]. Use brackets for dimensions every time.

Linear and rotational brackets side by side

The rotational quantities line up with linear ones by adding powers of length. Seeing the pattern makes the brackets easy to rebuild in an exam.

Linear and rotational partners
Linear	Dimensions	Rotational partner	Dimensions
Mass m	[M]	Moment of inertia I	[ML²]
Momentum p	[MLT⁻¹]	Angular momentum L	[ML²T⁻¹]
Force F	[MLT⁻²]	Torque τ	[ML²T⁻²]

Each rotational partner is the linear one multiplied by a length, except moment of inertia, which needs a length squared. Since τ = dL/dt, dividing [ML²T⁻¹] by [T] gives the torque bracket at once. Compare the momentum page, while the dimensional formulas hub gives every other quantity its own bracket.

Frequently asked questions

What is the SI unit of angular momentum?

kg m²/s, which can also be written J s or N m s. All three come from the same base units.

What is the dimensional formula for angular momentum?

It is [M¹ L² T⁻¹]. You get it from L = mvr: mass [M], speed [LT⁻¹] and radius [L] multiply to [ML²T⁻¹].

Why do angular momentum and Planck's constant have the same dimensions?

Both equal an energy times a time. Planck's constant is energy divided by frequency, and frequency is one over time, so h ends up as [ML²T⁻¹], exactly like angular momentum.

Is angular momentum a vector?

Yes. Its direction is along the axis of rotation by the right hand rule. The dimensional formula only records size, not direction.

What is the difference between the dimensions of momentum and angular momentum?

Linear momentum is [MLT⁻¹]. Angular momentum multiplies it by a distance, so it is [ML²T⁻¹].