A famous relation in physics relates 'moving mass' $m$ to the 'rest mass' $m_{0}$ of a particle in terms of its speed $v$ and the speed of light, $c .$ (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant $c$. He writes:

$m=\frac{m_{0}}{\left(1-v^{2}\right)^{1 / 2}}$

Guess where to put the missing $c$

Vedclass pdf generator app on play store
Vedclass iOS app on app store

Given the relation, $m=\frac{m_{0}}{\left(1-v^{2}\right)^{\frac{1}{2}}}$

Dimension of $m= M ^{1} L ^{0} T ^{0}$

Dimension of $m_{0}= M ^{1} L ^{0} T ^{0}$

Dimension of $v= M ^{0} L ^{1} T ^{-1}$

Dimension of $v^{2}= M ^{0} L ^{2} T ^{-2}$

Dimension of $c= M ^{0} L ^{1} T ^{-1}$

The given formula will be dimensionally correct only when the dimension of L.H.S is the same as that of R.H.S.

This is only possible when the factor, $\left(1-v^{2}\right)^{1 / 2}$ is dimensionless i.e., $\left(1-v^{2}\right)$ is dimensionless. This is only possible if $v^{2}$ is divided by $c^{2} .$

Hence, the correct relation is

$m=\frac{m_{0}}{\left(1-\frac{v^{2}}{c^{2}}\right)^{\frac{1}{2}}}$

Similar Questions

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0^\beta h^7 c^5$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by

  • [IIT 2024]

convert $1\; newton$ ($SI$ unit of force) into $dyne$ ($CGS$ unit of force) 

The density of a material in $SI\, units$ is $128\, kg\, m^{-3}$. In certain units in which the unit of length is $25\, cm$ and the unit of mass $50\, g$, the numerical value of density of the material is

  • [JEE MAIN 2019]

The dimensions of physical quantity $X$ in the equation Force $ = \frac{X}{{{\rm{Density}}}}$ is given by

Match List $I$ with List $II$ and select the correct answer using the codes given below the lists :

List $I$ List $II$
$P.$ Boltzmann constant $1.$ $\left[ ML ^2 T ^{-1}\right]$
$Q.$ Coefficient of viscosity $2.$ $\left[ ML ^{-1} T ^{-1}\right]$
$R.$ Planck constant $3.$ $\left[ MLT ^{-3} K ^{-1}\right]$
$S.$ Thermal conductivity $4.$ $\left[ ML ^2 T ^{-2} K ^{-1}\right]$

Codes: $ \quad \quad P \quad Q \quad R \quad S $ 

  • [IIT 2013]