The volume of a liquid flowing out per second of a pipe of length $l$ and radius $r$ is written by a student as $V = \frac{\pi p r^4}{8 \eta l}$,where $p$ is the pressure difference between the two ends of the pipe and $\eta$ is the coefficient of viscosity of the liquid having the dimensional formula $[M^1 L^{-1} T^{-1}]$. Check whether the equation is dimensionally correct.

(A) The volume of a liquid flowing out per second of a pipe is given by $V = \frac{\pi p r^4}{8 \eta l}$.
Dimensional formula of $LHS$:
$[V] = \frac{[Volume]}{[Time]} = \frac{[L^3]}{[T]} = [L^3 T^{-1}]$.
Dimensional formula of $RHS$:
$[p] = [M L^{-1} T^{-2}]$
$[r] = [L]$
$[\eta] = [M L^{-1} T^{-1}]$
$[l] = [L]$
Substituting these into the $RHS$ expression:
$[RHS] = \frac{[M L^{-1} T^{-2}] \cdot [L^4]}{[M L^{-1} T^{-1}] \cdot [L]} = \frac{[M L^3 T^{-2}]}{[M T^{-1}]} = [L^3 T^{-1}]$.
Since $[LHS] = [RHS]$,the equation is dimensionally correct.