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