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

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