Question

Use dimensional analysis to see which of the following expressions is allowed if P is a pressure, t is a time, m is a mass, r is a distance, v is a velocity and I is a temperature. a. log (pta/mr)
b. log(prt^2/m)
c. log(pr^2/mt^2)
d. log(pr/mtT) ​

Answer 1

Answer: a PV2T2

Explanation:

Let F∝pxVyTz

by subsituting the following dimensions:

[P]=[ML−1T−2][V]=[LT−1],[T]=[T]

and comparing the dimension of both sides

x=1,y=2,z=2, so F=PV2T2

Answer 2

We know that anything in the brackets of the logarithm function should be dimensionless.

$[P]=[M^{1} L^{-1} T^{ -2}]$   $[t]=[T^{1}]$  $[m]=[M^{1}]$  $[r]=[L^{1}]$  $[v]=[L^{1}T^{-1}]$  $[T]=[K^{1}]$

b. $[\frac{prt^{2}}{m}]=[\frac{[M^{1} L^{-1} T^{ -2}]\times [L^{1}]\times [T^{2}]}{[M^{1}]}=[M^{0}L^{0}T^{0}]$

$\therefore$The correct option is b.

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