Dimensional Formula of Energy [ML²T⁻²]: KE, PE and E = mc²

The dimensional formula of energy is [M¹ L² T⁻²], the short form being [ML²T⁻²]. Every kind of energy, kinetic, potential, heat, electrical or the energy locked in mass, reduces to this one bracket.

That is a strong statement, and it is useful. Energy changes form all the time, but the total is conserved, which only works if all forms are measured in the same unit, the joule. Below the bracket is derived three ways, from motion, from height and from mass. Then come the unit, the electron volt, a check on E = hf and the slips. Numerical problems sit on the energy formula page.

Quick facts

Dimensional formula
[M¹ L² T⁻²], short form [ML²T⁻²]
Kinetic energy
$\frac{1}{2} m v^{2}$
Potential energy
$m g h$
SI unit
joule (J) = kg m² s⁻²
Scalar or vector
Scalar
Same dimensions as
Work, heat, torque

Dimensional formula of energy from three formulas

From kinetic energy

$K = \frac{1}{2} m v^{2}$ . The half has no dimensions, so $[K] = [M][L T^{-1}]^{2} = [M L^{2} T^{-2}]$ .

From gravitational potential energy

$U = m g h$ . Mass [M], g an acceleration [LT⁻²], height [L]. Together: $[M][L T^{-2}][L]$ , which collects to $[M L^{2} T^{-2}]$ .

From the mass energy relation

$E = m c^{2}$ . With c a speed, $[E] = [M][L^{2} T^{-2}]$ , and the result repeats.

Three unrelated formulas, one answer. That agreement is the reason a single dimensional formula of kinetic energy and of potential energy can be quoted for all energy. The kinetic energy and potential energy pages work numbers for each.

The joule and other energy units

Read the bracket as base units: kg m² s⁻². That combination is the joule. One joule is the energy transferred when a force of one newton acts through one metre.

Energy units met in Class 11 and 12
Unit	Value in joules	Where it is used
erg (CGS)	10⁻⁷ J	Older problems
calorie	4.186 J	Heat
kilowatt hour	3.6 × 10⁶ J	Electricity bills
electron volt	1.602 × 10⁻¹⁹ J	Atoms and nuclei

Each row is still energy, so each still carries [ML²T⁻²]. A unit can hide its base units, as the kilowatt hour does, but the dimensions do not change.

Unit conversion and an equation check

Show that 1 kWh = 3.6 × 10⁶ J

A kilowatt hour is a power of 1000 W kept up for 3600 s.

$E = P t = 1000\ \text{J/s} \times 3600\ \text{s} = 3.6 \times 10^{6}\ \text{J}$ .

Power is [ML²T⁻³] and time is [T], so the product is [ML²T⁻²], as energy must be.

Check: is the photon relation E = hf consistent?

Planck\'s constant: $[h] = [M L^{2} T^{-1}]$ . Frequency: $[f] = [T^{-1}]$ .

Product: $[M L^{2} T^{-1}][T^{-1}] = [M L^{2} T^{-2}]$ , an energy. The relation passes.

Same dimensions, different quantities

Quantities that share the energy bracket
Quantity	Typical formula	Physical meaning
Work	$W = F s \cos\theta$	Energy transferred by a force
Heat	$Q = m c \Delta T$	Energy transferred by temperature difference
Torque	$\tau = r F \sin\theta$	Turning effect, not energy
Thermal energy per particle	$k_B T$	Scale of molecular energy

Work and heat are energy in transit, so their match is expected. Torque is the odd one out: it shares [ML²T⁻²] by coincidence of form, which is why its unit is left as N m. See the dimensional formula of work page for that comparison.

Frequent errors

Writing [ML²T⁻³]. That is power. Energy has only T⁻².
Treating the ½ in ½mv² as a dimension. Numbers never change a dimensional formula.
Thinking potential energy has different dimensions because it can be negative. The sign comes from the zero chosen, not from the dimensions.
Using the electron volt as if it were a volt. eV is an energy; V is energy per charge.

Dividing energy to get new brackets

Energy is the hub of the mechanics chain. Divide it by time and you get power, [ML²T⁻³]. Divide by volume and you get energy density, [ML⁻¹T⁻²], the same as pressure. Divide by mass and you get specific energy, [L²T⁻²], which is a speed squared.

Those three moves explain a lot of brackets you would otherwise memorize separately. Try them on the energy density page, then use the dimensional formulas list for the rest.

Frequently asked questions

Do kinetic and potential energy have the same dimensions?

Yes. ½mv² and mgh both reduce to [ML²T⁻²]. They must, because one form turns into the other when a body falls.

What is the dimensional formula of kinetic energy?

Kinetic energy, ½mv², comes to [M¹ L² T⁻²]. The ½ is dropped because pure numbers have no dimensions.

What is the dimensional formula of potential energy?

Gravitational potential energy mgh multiplies [M], [LT⁻²] and [L], so it is also [M¹ L² T⁻²].

Which SI unit measures energy?

The joule, equal to kg m² s⁻² in base units. Other units such as the calorie or electron volt are fixed multiples of it.

Is energy a vector?

No. Energy is a scalar. It can be added as plain numbers whatever the directions of motion.