Question

Suppose speed of light $(c),$ force $(F)$ and kinetic energy $(K)$ are taken as the fundamental units, then the dimensional formula for mass will be

• A. $\left[K{c}^{-2}\right]$
• B. $\left[K{F}^{-2}\right]$
• C. $\left[c{K}^{-2}\right]$
• D. $\left[F{c}^{-2}\right]$

Answer 1

Answer: (A)

Let $m=k{c}^x{F}^y{K}^2$ where $k$ is a dimensional constant.
$\therefore \left[M^1{L}^0{T}^0\right]$
$={\left[L{T}^{-1}\right]}^x{\left[ML{T}^{-2}\right]}^y{\left[M{L}^2{T}^{-2}\right]}^z$
$=\left[M^{y+z}{L}^{x+y+2z}{T}^{-x-2y-2z}\right]$
Applying principle of homogeneity of dimensions,
$y+z=\mathrm{1...}(i)$
$x+y+2z=\mathrm{0...}(ii)$
$-x-2y-2z=\mathrm{0...}(iii)$
Adding (ii) and (iii), we get $-y=0$ or $y=0$
From (i) $z=1-y=1$ From
(ii) $x=-y-2z=0-2$
$\therefore m=c^{-2}{F}^0{K}^1=K{c}^{-2}$
Hence the dimensional formula for mass is $\left[K{c}^{-2}\right]$.