Question

A calorie is a unit of heat (energy in transit) and it equals about $4.2J$ where $1J=1kg{m}^2{s}^{-2}$. 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 $4.2{\alpha }^{-1}{\beta }^{-2}{\gamma }^2$ in terms of the new units.

Answer 1

We know that : $n_1{u}_1=n_2{u}_2$
$n_2=n_1\frac{u_1}{u_2}=n_1\frac{\left[M_1^a{L}_1^b{T}_1^c\right]}{\left[M_2^a{L}_2^b{T}_2^c\right]}$

$=n_1{\left[\frac{M_1}{M_2}\right]}^a{\left[\frac{L_1}{L_2}\right]}^b{\left[\frac{T_1}{T_2}\right]}^c$

SI System New system

$n_1=4.2$ $n_2=?$
$M_1=1kg$ $M_2=\alpha kg$
$L_1=1m$ $L_2=\beta m$
$T_1=1s$ $T_2=\gamma s$

$1cal=4.2J=4.2kg{m}^2{s}^{-2}\therefore a=1,b=2,c=-2$

$\therefore n_2=4.2{\left[\frac{1kg}{\alpha kg}\right]}^1{\left[\frac{1m}{\beta m}\right]}^2{\left[\frac{1s}{\gamma s}\right]}^{-2}$

$\therefore n_2=4.2{\alpha }^{-1}{\beta }^{-2}{\gamma }^2$

$\therefore 1cal=4.2{\alpha }^{-1}{\beta }^{-2}{\gamma }^2$ in new system