Question

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

• A. ${KC}^{-2}$
• B. ${KF}^{-2}$
• C. ${CK}^{-2}$
• D. ${FC}^{-2}$

Answer 1

The correct option is A ${KC}^{-2}$

$C=\left[L{T}^{-1}\right]$
$F=\left[M^1{L}^1{T}^{-2}\right]$
$K=\left[M^1{L}^2{T}^{-2}\right]$
$M=Constant\times C^{\alpha }{F}^{\beta }{K}^{\gamma }$

$\Rightarrow$ $\left[M^1{L}^0{T}^0\right]={\left[L{T}^{-1}\right]}^{\alpha }{\left[M^1{L}^1{T}^{-2}\right]}^{\beta }{\left[M^1{L}^2{T}^{-2}\right]}^{\gamma }$

or, $\left[M^1{L}^0{T}^0\right]=\left[M^{\beta +\gamma }{L}^{\alpha +\beta +2\gamma }{T}^{-\alpha -2\beta -2\gamma }\right]$

$\Rightarrow$ $\beta +\gamma =1,\alpha +\beta +2\gamma =0,\alpha +2\beta +2\gamma =0$

$\Rightarrow \beta =0,\gamma =1,\alpha =-2$

$\Rightarrow \left[M\right]=\left[C^{-2}{K}^1\right]$