Question

Derive the mathematical formula for the S.I. unit of power in terms of mass, length, and time.

Answer 1

Power is the rate of doing work:

\[ P = \frac{W}{t} \]

Since work equals force multiplied by displacement, and force equals mass multiplied by acceleration,

\[ P = \frac{F \times s}{t} = \frac{(m \times a) \times s}{t} \]

Acceleration has dimensions \([\mathrm{LT^{-2}}]\), displacement has dimensions \([\mathrm{L}]\), and time has dimensions \([\mathrm{T}]\).

Therefore,

\[ [P] = \frac{[\mathrm{M}][\mathrm{LT^{-2}}][\mathrm{L}]}{[\mathrm{T}]} = [\mathrm{ML^{2}T^{-3}}] \]

Thus, the dimensional formula of power is \([\mathrm{ML^{2}T^{-3}}]\), and its SI unit is

\[ 1\ \mathrm{W} = 1\ \mathrm{kg\,m^{2}\,s^{-3}} \]