$A$ calorie is a unit of heat or energy and it equals about $4.2 \; J$ where $1 \; J = 1 \; kg \; m^2 \; s^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha \; kg$, the unit of length equals $\beta \; m$, and the unit of time equals $\gamma \; s$. Show that a calorie has a magnitude $4.2 \; \alpha^{-1} \beta^{-2} \gamma^2$ in terms of the new units.
(A) Given that, $1 \; \text{calorie} = 4.2 \; (1 \; kg) (1 \; m^2) (1 \; s^{-2})$.
Let the new units be $M' = \alpha \; kg$, $L' = \beta \; m$, and $T' = \gamma \; s$.
Then, $1 \; kg = \frac{1}{\alpha} \; M' = \alpha^{-1} \; M'$.
$1 \; m = \frac{1}{\beta} \; L' = \beta^{-1} \; L'$, so $1 \; m^2 = \beta^{-2} \; (L')^2$.
$1 \; s = \frac{1}{\gamma} \; T' = \gamma^{-1} \; T'$, so $1 \; s^{-2} = (\gamma^{-1})^{-2} \; (T')^{-2} = \gamma^2 \; (T')^{-2}$.
Substituting these into the expression for a calorie:
$1 \; \text{calorie} = 4.2 \times (\alpha^{-1} \; M') \times (\beta^{-2} \; (L')^2) \times (\gamma^2 \; (T')^{-2})$.
Therefore, the magnitude of a calorie in the new system is $4.2 \; \alpha^{-1} \beta^{-2} \gamma^2$.
Let the new units be $M' = \alpha \; kg$, $L' = \beta \; m$, and $T' = \gamma \; s$.
Then, $1 \; kg = \frac{1}{\alpha} \; M' = \alpha^{-1} \; M'$.
$1 \; m = \frac{1}{\beta} \; L' = \beta^{-1} \; L'$, so $1 \; m^2 = \beta^{-2} \; (L')^2$.
$1 \; s = \frac{1}{\gamma} \; T' = \gamma^{-1} \; T'$, so $1 \; s^{-2} = (\gamma^{-1})^{-2} \; (T')^{-2} = \gamma^2 \; (T')^{-2}$.
Substituting these into the expression for a calorie:
$1 \; \text{calorie} = 4.2 \times (\alpha^{-1} \; M') \times (\beta^{-2} \; (L')^2) \times (\gamma^2 \; (T')^{-2})$.
Therefore, the magnitude of a calorie in the new system is $4.2 \; \alpha^{-1} \beta^{-2} \gamma^2$.
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