Dimensional Formula of Work [ML²T⁻²]: Derivation and Joule

The dimensional formula of work is [M¹ L² T⁻²], short form [ML²T⁻²]. Read it as one power of mass, two of length and an inverse square of time.

You reach it from W = Fs cos θ. Force supplies [MLT⁻²], the displacement supplies one more [L], and cos θ is a plain number. The bracket is identical to the one for every form of energy, which is why work and energy share the joule. Below you will find the derivation, the unit, the look alike quantities, a gas work check and the slips that lose marks. Problems with numbers are on the work formula page.

Quick facts

Dimensional formula
[M¹ L² T⁻²], short form [ML²T⁻²]
Defining formula
$W = F s \cos\theta$
SI unit
joule (J) = N m = kg m² s⁻²
CGS unit
erg; 1 J = 10⁷ erg
Scalar or vector
Scalar
Same dimensions as
Energy, heat, torque, PV

Derivation of the work done dimensional formula

Work done by a constant force: $W = F s \cos\theta$ .
Force: $[F] = [M L T^{-2}]$ . Displacement: $[s] = [L]$ .
$\cos\theta$ is a pure number, a length divided by a length.
Multiply: $[W] = [M L T^{-2}] \times [L]$ , giving $[M L^{2} T^{-2}]$ .

The same bracket appears if you start from the work energy theorem, W = ΔK, because kinetic energy is ½mv² and $[M][L T^{-1}]^{2}$ is again [ML²T⁻²]. Two independent routes giving one answer is a good sign the work done dimensional formula is right. The force page covers the first factor in detail.

Units that follow from the dimensional formula of work

Each letter becomes a base unit: kg for M, m² for L², s⁻² for T⁻². The SI unit of work is kg m² s⁻², named the joule (J).

$1\ \text{J} = 1\ \text{N} \times 1\ \text{m} = 1\ \text{kg m}^{2}\,\text{s}^{-2}$

The CGS unit is the erg, one dyne times one centimetre, or g cm² s⁻².

Convert 1 joule into ergs with the dimensional formula

Powers from [M¹ L² T⁻²]: $1\ \text{J} = (1000\ \text{g})(100\ \text{cm})^{2}(1\ \text{s})^{-2}$ .

The mass factor gives 10³ and the squared length factor gives 10⁴; together that is 10⁷.

So 1 J = 10⁷ erg. The square on L doubles the power of ten from the length factor.

Quantities with the same dimensions as work

Quantities sharing the work bracket [M¹ L² T⁻²]
Quantity	Formula	Note
Kinetic energy	$\frac{1}{2} m v^{2}$	Energy of motion
Potential energy	$m g h$	Stored energy
Heat	Q in joules	Energy in transfer
Torque	$\tau = r F \sin\theta$	Same bracket, but written N m, not J
Pressure × volume	$P V$	Work done by a gas

The torque row is the classic trap. Torque and work have the same dimensions, yet they are different physical quantities: work is a scalar transfer of energy, torque is a turning effect with a direction. Dimensions alone cannot separate them. Read the torque and energy pages for the full comparison.

Dimensional check on gas work

Check: work done by a gas at constant pressure, W = PΔV

Pressure: $[P] = [M L^{-1} T^{-2}]$ . Volume change: $[\Delta V] = [L^{3}]$ .

Product: $[M L^{-1} T^{-2}][L^{3}] = [M L^{2} T^{-2}]$ .

That is the work bracket, so the relation is consistent. A gas expanding by 0.002 m³ at 1.0 × 10⁵ Pa does 1.0 × 10⁵ × 0.002 = 200 J of work.

Dimensions check the form of an equation, not its numbers. W = 2PΔV would pass the same test.

Where students slip

Writing [MLT⁻²]. That is force alone. Work multiplies it by a distance.
Writing [ML²T⁻³]. That is power, work per second. Keep the time power at −2 for work.
Giving cos θ a dimension. Any trig function of an angle is a pure number.
Calling the unit of torque a joule. Same dimensions, but convention keeps N m for torque.

Work in the force, energy, power chain

Work connects force to energy and to power. Following that chain helps you rebuild the brackets during a test instead of memorizing each one.

From force to power
Step	Relation	Dimensions
Force	$F = ma$	[MLT⁻²]
Work	$W = F s$	[ML²T⁻²]
Power	$P = \frac{W}{t}$	[ML²T⁻³]

Multiply by length to go from force to work, divide by time to go from work to power. The rest of the mechanics list is on the dimensional formulas page.

Frequently asked questions

Is work a scalar or vector quantity?

Scalar. Force and displacement combine through a dot product, so work done has size and sign but no direction.

What is the dimensional formula of work done?

In brackets, work done is [M¹ L² T⁻²]. It comes from force [MLT⁻²] multiplied by displacement [L].

Which SI unit measures work done?

The joule. Written with base units it reads kg m² s⁻², or one newton metre.

Why do work and energy have the same dimensional formula?

Work done on a body changes its energy by the same amount, so the two must be measured in the same unit. Both reduce to [ML²T⁻²].

How do work and power differ in dimensions?

Power is work divided by time. Work is [ML²T⁻²] and power is [ML²T⁻³].