Question

A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 $\text {kg} \;\text m^2\; \text s^{-2}.$ 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, 1 calorie = 4.2 (1 $\text {kg}$) (1 $\text m^2$ ) (1 $\text s^{-2}$)
New unit of mass = $\alpha$ $\text {kg}$
Hence, in terms of the new unit, 1 $\text {kg}$ = $\frac{1}{\alpha}$ = $\alpha^{-1}$
In terms of the new unit of length, 1 $\text m$ = $\frac{1}{\beta}$= $\beta^{-1}$ or 1 $\text m^2$ =$\beta^{-2}$
And, in terms of the new unit of time,
1 s = $\frac{1}{\gamma}$= $\gamma^{-1}$
1 $\text s^2$ = $\gamma^{-2}$
1 $\text s^{-2}$ = $\gamma^2$
$\therefore$ 1 calorie = 4.2 (1 $\alpha^{-1}$) (1 $\beta^{-2}$) (1 $\gamma^2$ ) = 4.2 $\alpha^{-1}$ $\beta^{-2}$ $\gamma^2$