What is the dimensional formula of Power?

The dimensional formula of Power in terms of Mass [M], Length [L], and Time [T] is:

[ M1  L2  T-3 ]

By looking at this formula, we can identify that Power is made up of 3 fundamental quantities: Mass, Length, and Time.

Here, the powers of these fundamental quantities (1, 2, -3) are the dimensions.

Now, let’s see how we can derive this formula.


How to derive the dimensional formula of Power

For this derivation, we first need to know some of the formulas:

QuantityFormula
Power (P)Work (W) / Time (T)
Work (W)Force (F) * Displacement (D)
Force (F)Mass (m) * Acceleration (a)
Acceleration (a) Velocity (v) / Time (T)
Velocity (v)Displacement (D) / Time (T)
Requirements to compute the dimensional formula of Power

Now, let’s use these formulas to find the dimension of power.

Derivation

The dimension formula of Power includes 1 dimension of Mass, 2 dimension of Length and -3 dimension of Time.
Derivation of Dimension Formula of Power

Power = Work / Time

      = (Force * Displacement) / Time  {using formula of Work}

      = (Mass * Acceleration * Displacement) / Time  {using formula of Force}

      = (Mass * (Velocity / Time) * Displacement / Time  {using formula of Acceleration}

      = (Mass * (Displacement / (Time * Time) ) * Displacement ) / Time  {using the formula of Velocity}

      = (Mass * Displacement * Displacement ) / (Time * Time * Time)

      = (Mass * Displacement2 ) / Time3

Power = Mass * Displacement2 * Time-3

Now, let’s convert this formula in terms of fundamental quantity [M] (Mass), [L] (Length), and [T] (Time)

  • Mass is represented in terms of [M]
  • Displacement is represented in terms of [L]
  • Time is represented in terms of [T]

Hence, 

Dimensional Formula of Power = [M1 L2 T-3]

From the formula, the dimensions are

  • 1 for Mass
  • 2 for Length
  • -3 for Time

If you want to learn more about dimensional analysis, visit Dimensional Formula and Equations of Physical Quantity.