Question

A calorie is a unit of heat or energy and it equals about $4.2$J, where $1$ J = $1kg{m}^2{s}^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha$kg, the unit of length equals $\beta$m, the unit of time is $\gamma s$. Show that a calorie has a magnitude $4.2{\alpha }^{-1}{\beta }^{-1}{\gamma }^2$ in terms of the new units.

Answer 1

$1$ cal = $4.2kg{m}^2{s}^{-2}$

SI system	NEW system
$M_1=1kg$	$M_2=1kg$
$L_1=1m$	$L_2=1cm$
$T_1=1s$	$T_2=1s$
$n_1=4$	$n_2=?$

dimensional formula of energy is $\left[M{L}^2{T}^{-2}\right]$
Comparing with $\left[M^a{L}^b{T}^c\right]$, we find that $a=1,b=2,c=-2$.
Now, $n_2=n_1{\left[\frac{M_1}{M_2}\right]}^a{\left[\frac{L_1}{L_2}\right]}^b{\left[\frac{T_1}{T_2}\right]}^c$
$=4.2{\left[\frac{1kg}{\alpha kg}\right]}^1{\left[\frac{1m}{\beta m}\right]}^2{\left[\frac{1s}{\gamma s}\right]}^{-2}$$=4.2{\alpha }^{-1}{\beta }^{-2}{\gamma }^2$