Question 2.3: Conversion of units for a calorie
A calorie is a unit of heat or energy and it equals about 4.2 J where 1 J = 1 kg m2 s-2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, and the unit of time is γ s. Show that a calorie has a magnitude 4.2 α-1 β-2 γ2 in terms of the new units.
Solution:
Given: \( 1 \, \text{cal} = 4.2 \, \text{J} \)
\[
1 \, \text{J} = 1 \, \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}
\]
In the new system of units:
1 unit of mass = \( \alpha \, \text{kg} \)
1 unit of length = \( \beta \, \text{m} \)
1 unit of time = \( \gamma \, \text{s} \)
We need to express 1 J in terms of the new units:
\[
1 \, \text{J} = 1 \, \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2} = \left( \frac{1 \, \text{kg}}{\alpha \, \text{kg}} \right) \cdot \left( \frac{1 \, \text{m}}{\beta \, \text{m}} \right)^2 \cdot \left( \frac{1 \, \text{s}}{\gamma \, \text{s}} \right)^{-2}
\]
\[
= \alpha^{-1} \cdot \beta^{-2} \cdot \gamma^{2}
\]
Therefore,
\[
4.2 \, \text{J} = 4.2 \cdot \alpha^{-1} \cdot \beta^{-2} \cdot \gamma^{2}
\]
Thus, a calorie has a magnitude of \( 4.2 \alpha^{-1} \beta^{-2} \gamma^{2} \) in terms of the new units.