Dimensional Formula of Torque [ML²T⁻²] and Moment of Force

Torque comes out as [ML²T⁻²] in short form, so the dimensional formula of torque in full is [M¹ L² T⁻²]. This is also the dimensional formula of moment of force, since torque and moment of force are two names for one quantity.

Torque is a force times its perpendicular lever arm, τ = rF sin θ. The force gives [MLT⁻²] and the lever arm adds one length. The result has exactly the bracket of work and energy, which raises a fair question: why is torque not measured in joules? That answer, the derivation, two checks and the usual slips are below. Spanner and door problems with numbers are on the torque formula page.

Quick facts

Dimensional formula
[M¹ L² T⁻²], short form [ML²T⁻²]
Defining formula
$\tau = r F \sin\theta$
SI unit
N m (not written as J)
Other name
Moment of force
Vector or scalar
Vector, along the axis
Same dimensions as
Work and energy

r and F give the dimensional formula of torque

Start from $\tau = r F \sin\theta$ .
Lever arm: $[r] = [L]$ . Force: $[F] = [M L T^{-2}]$ .
The trig ratio sin θ carries no dimensions at all.
Product: [L] × [MLT⁻²] gives [ML²T⁻²] for torque.

The vector form τ = r × F gives the same result, since a cross product multiplies sizes. A couple, two equal and opposite forces a distance d apart, has torque F × d and again [ML²T⁻²]. The force step is derived on the force page.

Newton metre, not joule

Read the bracket as kg m² s⁻². Grouped as force times distance, that is the newton metre, N m, the SI unit of torque.

A joule is also kg m² s⁻², but by convention torque is never written in joules. Work is force times the displacement along the force; torque is force times a distance at right angles to it. One is a scalar energy transfer, the other a turning vector. Writing torque in N m and energy in J stops anyone adding the two by mistake.

A spanner 0.25 m long is pushed with 40 N at right angles. Find the torque.

$\tau = r F \sin 90^\circ = 0.25 \times 40 \times 1 = 10\ \text{N m}$ .

In CGS, 1 N m = 10⁵ dyne × 10² cm = 10⁷ dyne cm, so this torque is 1.0 × 10⁸ dyne cm.

Torque, work and energy compared

Same bracket as torque, [M¹ L² T⁻²]
Quantity	Formula	Is it torque?
Moment of force	$\tau = r F \sin\theta$	Yes, another name
Moment of a couple	$F d$	Yes, a special case
Work	$W = F s \cos\theta$	No, scalar energy transfer
Gravitational potential energy	$m g h$	No

This table is the reason dimensional analysis cannot tell work from torque. If a question asks which pair has identical dimensions, work and torque is a standard correct answer. See the work and energy pages.

Rotational laws checked

Check: τ = Iα

Moment of inertia [ML²] times angular acceleration [T⁻²] gives [ML²T⁻²].

Torque comes out, so τ = Iα, the turning version of F = ma, is sound.

Check: τ = dL/dt

Angular momentum over time: $[M L^{2} T^{-1}] \div [T] = [M L^{2} T^{-2}]$ .

Again torque. It changes angular momentum over time in the way a force changes linear momentum.

A third check: rotational power P = τω gives [ML²T⁻²][T⁻¹] = [ML²T⁻³], the bracket of power. The angular momentum page has the middle bracket.

Slips with moment of force

Writing [MLT⁻²]. That is force alone; the lever arm adds a length.
Giving the unit as J. Dimensionally equal, but the accepted unit is N m.
Thinking moment of force has its own bracket. It is torque under another name.
Writing [ML²T⁻¹]. That is angular momentum, one power of time short.

The rotational set, from moment of inertia to power, is laid out together on the dimensional formulas page.

Frequently asked questions

Is the dimensional formula of moment of force the same as torque?

Yes. The two terms mean one quantity, so the bracket is [M¹ L² T⁻²] either way.

Torque and work have equal dimensions, so are they the same quantity?

No. They share [ML²T⁻²], but work is a scalar measured in joules and torque is a vector measured in newton metres. Equal dimensions do not make equal quantities.

Which SI unit goes with torque?

The newton metre, N m. In base units it is kg m² s⁻².

Is torque a vector?

Yes. Point the right hand fingers along r and sweep them towards F; τ lies along the axis where your thumb ends up.

How does torque relate to angular momentum in dimensions?

Torque equals angular momentum divided by time. [ML²T⁻¹] over [T] gives [ML²T⁻²].