First slide

Dimensions

Question

If dimensions of critical velocity $ν_{c}$ of a liquid flowing through a tube are expressed as $[η^{x} ρ^{y} r^{z}]$ where $η, ρ$ and r are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then the values of x, y and z are given by

Moderate

A

-1, -1, -1
B

1, 1, 1
C

1, -1, -1
D

-1, -1, 1

Solution

$[v_{c}] = [η^{x} ρ^{y} r^{2}] (given) . . . .(i)$
$W r i t i n g t h e d i m e n s i o n s o f v a r i o u s q u a n t i t i e s i n e q n . (i), w e g e t$
$[M^{0} {LT}^{- 1}] = {[{ML}^{- 1} T^{- 1}]}^{x} {[{ML}^{- 3} T^{0}]}^{y} {[M^{0} {LT}^{0}]}^{z}$
$= [M^{x + y} L^{- x - 3 y + z} T^{- x}]$
$A p p l y i n g t h e p r i n c i p l e o f h o m o g e n e i t y o f d i m e n s i o n s, w e g e t$
$x + y = 0; - x - 3 y + z = 1; - x = - 1$
$On solving, we get$
$x = 1, y = - 1, z = - 1$

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