Question

A new system of units is proposed in which unit of mass is $\alpha kg$, unit of length is $\beta m$ and unit of time is $\gamma s$. What will be value of $5J$ in this new system ?

• A. $5a{\beta }^2{\gamma }^{-2}$
• B. $5a^{-1}{\beta }^{-2}{\gamma }^2$
• C. $5a^{-2}{\beta }^{-1}{\gamma }^{-2}$
• D. $5a^{-1}{\beta }^2{\gamma }^2$

Answer 1

Answer: (B)

Joule is a unit of energy.

$SI$	New system
$n_1=5$	$n_2=?$
$M_1=1kg$	$M_2=\alpha kg$
$L_1=1m$	$L_2=\beta m$
$T_1=1s$	$T_2\gamma s$

Dimensional formula of energy is $[M{L}^2{T}^{-2}]$. Comparing with $[M{L}^2{T}^{-2}]$, we get
$a=1$, $b=2$, $c=-2$
As $n_2=n_1{\left(\frac{M_1}{M_2}\right)}^a{\left(\frac{L_1}{L_2}\right)}^b{\left(\frac{T_1}{T_2}\right)}^c$
$=5{\left(\frac{1kg}{\alpha kg}\right)}^1{\left(\frac{1m}{\beta m}\right)}^2{\left(\frac{1s}{\gamma s}\right)}^{-2}$
$=\frac{5{\gamma }^2}{\alpha {\beta }^2}=5{\alpha }^{-1}{\beta }^{-2}{\gamma }^2$