# Q 2.3: A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where $1J = 1 kg m^2 s^{-2}$. 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.

Given,

$1 Cal = 4.2 (kg)(m)^{2}(s)^{-2}$

Given new unit of mass = $\alpha kg$     (In old unit 1kg corresponded to a unit mass, but in new unit $\alpha kg$ corresponds to a unit mass)

$\therefore$ In terms of the new unit, $1 kg = 1/\alpha = \alpha ^{-1}$

Similarly in terms of new units$1 m = 1/\beta = \beta^{-1}$ and  $1 s = 1/\gamma = \gamma ^{-1}$

$\therefore$ $1 Cal = 4.2 (kg)(m)^{2}(s)^{-2} = 4.2 (\alpha ^{-1})(\beta ^{-1})^{2}(\gamma ^{-1})^{-2} = 4.2 \alpha ^{-1}\beta ^{-2}\gamma ^{2}$

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