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

[ M^{1} L^{2} 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:

Quantity | Formula |

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) |

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

### Derivation

```
```**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 * Displacement**^{2} ) / Time^{3}
**Power = Mass * Displacement**^{2} * 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 = [M`^{1} L^{2} 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.