Find the dimensional formulae of following quantities:
(a) The surface tension S,
(b) The thermal conductivity k and
(c) The coefficient of viscosity `eta`.
Some equation involving these quantities are `S=(rho g r h)/(2) " " Q=k(A(theta_(2)-theta_(1))t)/(d) and " " F=-etaA(v_(2)-v_(1))/(x_(2)-x_(1)) `,
where the symbols have their usual meanings. (`rho`-density, g- acceleration due to gravity , r-radius,h-height, A-area, `theta_(1)&theta_(2)`- temperatures, t-time, d-thickness, `v_(1)&v_(2)` -velocities, `x_(1)& x_(2)` - positions.)

Explanation

(a)`S=(rho g rh)/(2)`
or `[S]=[rho][g]L^(2)=(M)/L^(2).(L)/(T^(2)). L^(2)=MT^(-2)`.
(b)`Q=k(A(theta_(2)-theta_(1))t)/(d)`
`k=(Qd)/(A(theta_(2)-theta_(1))t)`.
Here, Q is the heat energy having dimension `ML^(2)T^(-2),theta_(2)-theta_(1)` is temperature, A is area, d is thickness and t is time . Thus,
`[k]=(ML^(2)T^(-2))/(L^(2)KT)=MLT^(-3)K^(-1)`
(d) `F=-hA(v_(2)-v_(1))/(x_(2)-x_(1))`
or `MLT^(-2)=[eta]L^(2)(L//T)/(L)=[eta](L^(2))/(T)`
or, `[eta]=ML^(-1)T^(-1)`.