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Question

(A) The volume flow rate $V$ (Poiseuille's Law) is given by the formula $V = \frac{\pi P r^4}{8 \eta l}$.To verify this dimensionally:Dimensions of $V

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$V = \frac{\pi P r^4}{8 \eta l}$

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(A) The volume flow rate $V$ (Poiseuille's Law) is given by the formula $V = \frac{\pi P r^4}{8 \eta l}$.To verify this dimensionally:Dimensions of $V = [L^3 T^{-1}]$.Dimensions of $P = [M L^{-1} T^{-2}]$.Dimensions of $r = [L]$.Dimensions of $\eta = [M L^{-1} T^{-1}]$.Dimensions of $l = [L]$.Substituting these into the $RHS$: $\frac{[M L^{-1} T^{-2}] [L^4]}{[M L^{-1} T^{-1}] [L]} = \frac{[M L^3 T^{-2}]}{[M T^{-1}]} = [L^3 T^{-1}]$.Since the dimensions of $LHS$ and $RHS$ match,the correct relation is $V = \frac{\pi P r^4}{8 \eta l}$.

$A$ dimensionally consistent relation for the volume $V$ of a liquid of coefficient of viscosity $\eta$ flowing per second through a tube of radius $r$ and length $l$,having a pressure difference $P$ across its ends,is:

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