Question

Find the dimensions of mass, if energy $E$ , velocity $v$ , force $F$ are assumed to be fundamental quantities:

• A. $E{v}^2$
• B. $E{v}^{-2}$
• C. $F{v}^{-1}$
• D. $F{v}^{-2}$

Answer 1

Answer: (B)

Given that m depends on $E,v$ and $F$ .
So, $m\propto E^x{v}^y{F}^z$
By substituting the following dimensions:
$E=\left[M{L}^2{T}^{-2}\right],$ $v=\left[L{T}^{-1}\right],$ $F=\left[ML{T}^{-2}\right]$ ,

So, $M={\left[M{L}^2{T}^{-2}\right]}^x{\left[L{T}^{-1}\right]}^y{\left[ML{T}^{-2}\right]}^z$
And by equating the dimensions on both sides, $1=x+z,0=2x+y+z$ and $0=-2x-y-2z$
By solving these equations, we can get
$x=1,$ $y=-2$ and $z=0$ .
So, $m=\left[E{v}^{-2}\right]$