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