Question

Taking force, length and time to be the fundamental quantities find the dimensions of

(a) density,

(b) pressure,

(c) momentun and

(d) energy.

Answer 1

Taking force, length and time as fundamental quantity

(a) $\text{Density}=\frac{m}{v}\frac{\text{(force / acceleration)}}{\text{Volume}}$

$=\frac{\left[F/L{T}^{-2}\right]}{\left[L^3\right]}$

$=\frac{F}{L^4{T}^{-2}}=\left[F{L}^{-4}{T}^2\right]$

(b) Pressure = $\frac{F}{A}=\frac{F}{L^2}=\left[F{L}^{-2}\right]$

(c) Momentum = mv = (Force / acceleration) $\times \text{velocity}=\left[\frac{F}{L{T}^{-2}}\right]\times \left[L{T}^{-1}\right]=\left[FT\right]$

(d) Energy = $\frac{1}{2}m{v}^2$

$=\frac{\text{Force}}{\text{acceleration}}\times (velocity{)}^2$

$=\left[\frac{F}{L{T}^{-2}}\right]\times [L{T}^{-1}{]}^2$

$=\left[\frac{F}{L{T}^{-2}}\right]\times \left[L^2{T}^{-2}\right]=\left[FL\right]$