A calorie is a unit of heat or energy and it equals about
Question.
A calorie is a unit of heat or energy and it equals about $4.2 \mathrm{~J}$ where $1 \mathrm{~J}=1 \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha \mathrm{kg}$, the unit of length equals $\beta \mathrm{m}$, the unit of time is $\mathrm{y} \mathrm{s}$. Show that a calorie has a magnitude $4.2 \alpha^{-1} \beta^{-2} \gamma^{2}$ in terms of the new units.

solution:

Given that,

1 calorie $=4.2(1 \mathrm{~kg})\left(1 \mathrm{~m}^{2}\right)\left(1 \mathrm{~s}^{-2}\right)$

New unit of mass $=\alpha \mathrm{kg}$

Hence, in terms of the new unit, $1 \mathrm{~kg}=\frac{1}{\alpha}=\alpha^{-1}$

In terms of the new unit of length,

$1 \mathrm{~m}=\frac{1}{\beta}=\beta^{-1}$ or $1 \mathrm{~m}^{2}=\beta^{-2}$

And, in terms of the new unit of time,

$1 \mathrm{~s}=\frac{1}{\gamma}=\gamma^{-1}$

$1 \mathrm{~s}^{2}=\gamma^{-2}$

$\therefore 1$ calorie $=4.2\left(1 \alpha^{-1}\right)\left(1 \beta^{-2}\right)\left(1 \mathrm{y}^{2}\right)=4.2 \alpha^{-1} \beta^{-2} \mathrm{~V}^{2}$
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