Question

A calorie is a unit of heat or energy 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

Given that

$1Calorie=4.2J=4.2kg{m}^2{s}^{-2}$ (i)

As new unit of mass = $\alpha$ kg, 1 kg = $1/\alpha$ new unit of mass

Similarly, $1m={\beta }^{-1}$ new unit of length,

$1s={\gamma }^{-1}$ new unit of time

Putting these values in (i), we get

$1calorie=4.2\times \left({\alpha }^{-1}newunitofmass\right)\times ({\beta }^{-1}newunitoflength{)}^2\times ({\gamma }^{-1}newunitoftime{)}^{-2}$

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