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.

Answers (1)
S safeer


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 units1 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}