Q 2.3: A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where . Suppose we employ a system of units in which the unit of mass equals , the unit of length equals , the unit of time is . Show that a calorie has a magnitude 4.2 in terms of the new units.

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)

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 $1 s = 1/\gamma = \gamma ^{-1}$

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