Question

Find the dimensional formulae of the following quantities:

1. the universal constant of gravitational $G$

2. the surface tension $S$,

3. the thermal conductivity $k$ and

4. the coefficient of viscosity $\eta$

Answer 1

Universal constant of gravitation $G$

Gravitational Force $=G\frac{m_1{m}_2}{r^2}\Rightarrow G=\frac{F.r^2}{m_1{m}_2}$

$\left[G\right]=\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[M\right]\left[M\right]}$

$\left[G\right]=M^{-1}{L}^3{T}^{-2}$

2) Surface tension $S$

$S=\frac{Force}{length}\Rightarrow \frac{\left[ML{T}^{-2}\right]}{\left[L\right]}$

$\left[S\right]=\left[M{L}^0{T}^{-2}\right]$

3) Thermal conductivity $K$

$\frac{\Delta Q}{\Delta t}=KA\frac{\Delta T}{\Delta x}\Rightarrow K=-\frac{\Delta Q}{\Delta t}\times \frac{\Delta x}{\Delta T}\times \frac{1}{A}$

$K=\frac{energy}{time}\times \frac{length}{temperature}\times \frac{1}{Area}$

$=\frac{M{L}^2{T}^{-2}}{T}\times \frac{L}{K}\times \frac{1}{L^2}$

$\left[K\right]=\left[ML{T}^{-3}{K}^{-1}\right]$

4) coefficient of viscosity $n$

$n=\frac{F}{A\left(dv/dx\right)}\Rightarrow \frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]\left[L{T}^{-1}\right]/\left[L\right]}$

$\left[n\right]=M{L}^{-1}{T}^{-1}$