Question

If discharge rate is given by $V=\frac{\pi P{r}^4}{8\eta l}$, then the dimensions of $\eta$, by taking velocity $(v)$, time $(T)$ and mass $(M)$ as fundamental units, are $\left[M^a{\nu }^b{T}^c\right]$ Find $(a+b-c)$.
(Here, $P=$ Pressure, $r=$ Radius and $l=$ Length)

Answer 1

Answer: 2

$\therefore \eta =\frac{\pi P{r}^4}{8Vl}$
$\therefore$ Dimensions of $\eta =\frac{\left[ML{T}^{-2}{L}^{-2}\right]\left[L^4\right]}{\left[L^3{T}^{-1}\right][L]}$
$=\left[M{L}^{-1}{T}^{-1}\right]$
Also, $v=\left[L{T}^{-1}\right];L=[vT]$
$\therefore$ Dimension of $\eta$ are $\left[M(vT{)}^{-1}{T}^{-1}\right]$
$=\left[M{v}^{-1}{T}^{-2}\right]$