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 $\beta\; m$, 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.
Given that,
$1$ calorie $=4.2(1\, kg )\left(1 \,m ^{2}\right)\left(1\, s ^{-2}\right)$
New unit of mass $=\alpha kg$
Hence, in terms of the new unit, $1 \,kg =\frac{1}{\alpha}=\alpha^{-1}$ In terms of the new unit of length, $1\, m =\frac{1}{\beta}=\beta^{-1}$ or $1\, m ^{2}=\beta^{-2}$
And, in terms of the new unit of time, $1\, s =\frac{1}{\gamma}=\gamma^{-1}$
$1\, s ^{2}=\gamma^{-2}$
$1 \,s ^{-2}=\gamma^{2}$
$\therefore 1$ calorie $=4.2\left(1 \alpha^{-1}\right)\left(1 \beta^{-2}\right)\left(1 \gamma^{2}\right)=4.2 \alpha^{-1} \beta^{-2} \gamma^{2}$
$1$ calorie $=4.2(1\, kg )\left(1 \,m ^{2}\right)\left(1\, s ^{-2}\right)$
New unit of mass $=\alpha kg$
Hence, in terms of the new unit, $1 \,kg =\frac{1}{\alpha}=\alpha^{-1}$ In terms of the new unit of length, $1\, m =\frac{1}{\beta}=\beta^{-1}$ or $1\, m ^{2}=\beta^{-2}$
And, in terms of the new unit of time, $1\, s =\frac{1}{\gamma}=\gamma^{-1}$
$1\, s ^{2}=\gamma^{-2}$
$1 \,s ^{-2}=\gamma^{2}$
$\therefore 1$ calorie $=4.2\left(1 \alpha^{-1}\right)\left(1 \beta^{-2}\right)\left(1 \gamma^{2}\right)=4.2 \alpha^{-1} \beta^{-2} \gamma^{2}$
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