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 \(\alpha^{-1}\) \(\beta^{-2}\) \(\gamma^2\)