Use dimensional analysis to see which of the following expressions is/are NOT allowed if p is a pressure, t is a time, m is a mass, r is a distance, v is a velocity and T is a temperature?

Question

TestHub · Accepted Answer

The argument of a logarithmic function must be dimensionless.

The dimensions are:

$p = [M L^{- 1} T^{- 2}]$

$t = [T]$

$m = [M]$

$r = [L]$

$v = [L T^{- 1}]$

$T = [Θ]$ (temperature)

A. $\frac{pt}{m r} = \frac{[ M L ^{- 1} T ^{- 2} ] [ T ]}{[ M ] [ L ]} = [L^{- 2} T^{- 1}]$ . NOT dimensionless.

B. $\frac{p r t ^{2}}{m} = \frac{[ M L ^{- 1} T ^{- 2} ] [ L ] [ T ^{2} ]}{[ M ]} = [1]$ . Dimensionless.

C. $\frac{p r ^{2}}{m t ^{2}} = \frac{[ M L ^{- 1} T ^{- 2} ] [ L ^{2} ]}{[ M ] [ T ^{2} ]} = [L T^{- 4}]$ . NOT dimensionless.

D. $\frac{p r}{m tT} = \frac{[ M L ^{- 1} T ^{- 2} ] [ L ]}{[ M ] [ T ] [ Θ ]} = [T^{- 3} Θ^{- 1}]$ . NOT dimensionless.

Therefore, A, C, and D are not allowed.

Physics - Unit & Dimension Question with Solution | TestHub

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