Q. Find the dimensions of mass, if energy $E$ , velocity $v$ , force $F$ are assumed to be fundamental quantities:
NTA AbhyasNTA Abhyas 2020
Solution:
Given that m depends on $E,v$ and $F$ .
So, $m \propto E^{x}v^{y}F^{z}$
By substituting the following dimensions:
$E=\left[ML^{2} T^{- 2}\right],$ $v=\left[LT^{- 1}\right],$ $F=\left[MLT^{- 2}\right]$ ,
So, $M=\left[ML^{2} T^{- 2}\right]^{x}\left[LT^{- 1}\right]^{y}\left[MLT^{- 2}\right]^{z}$
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, $m=\left[Ev^{- 2}\right]$
So, $m \propto E^{x}v^{y}F^{z}$
By substituting the following dimensions:
$E=\left[ML^{2} T^{- 2}\right],$ $v=\left[LT^{- 1}\right],$ $F=\left[MLT^{- 2}\right]$ ,
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, $m=\left[Ev^{- 2}\right]$