Physics

Problem

Consider the following four ratios that arise in different branches of physics: A.

\displaystyle \frac{F\,\ell}{m v^{2}}

where

F

is a force,

\ell

a length,

m

a mass and

v

a speed. B.

\displaystyle \frac{v^{2}}{g \ell}

where

g

is the acceleration due to gravity. C.

\displaystyle e^{\frac{-q V}{k T}}

where

q

is charge,

V

electric potential,

k

Boltzmann constant and

T

absolute temperature. D.

\displaystyle \frac{p}{\rho g h}

where

p

is pressure,

\rho

density and

h

a height. a. (i) For each ratio A–D, determine whether it is dimensionless. Show your working by expressing every quantity in terms of the fundamental dimensions

M,\,L,\,T

. b. (ii) For each ratio that is dimensionless, state one physical situation where it plays an important role and explain that role briefly. c. (iii) For any ratio found to possess dimensions, suggest a modification (for example, by multiplying or dividing by another quantity) that would make it dimensionless, and justify your suggestion.

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