### Question 2.3

NCERT Class XI Physics

A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 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, 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 in nutshell

First of all write conversions of mass, length and time from MKS [SI] to the new units in terms of α, β and γ. Then put them in 4.2 kg m2 s-2 to simplify and obtain the answer.

### Solution in detail

Given $\displaystyle \alpha \text{ kg } \equiv \text{new unit}$

$\displaystyle \implies 1 \:\text{kg} \equiv \frac{\text{new unit}}{ \alpha}$

$ \begin{aligned} \implies \text{kg} &\equiv \alpha^{-1}\\ \\ \text{similarly, }\text{m} &\equiv \beta^{-1}\\ \\ \text{s} &\equiv \gamma ^{-1}\\ \\ \end{aligned} $

$ \begin{aligned} \therefore 1 \text{ cal } &= \frac{4.2 \text{ kg } \times \text{ m } ^2}{\text{s}^2}\\ \\ &= \frac{4.2 \: \alpha^{-1} \times \beta ^{-2} } { \gamma^{-2} }\\ \\ &= 4.2 \:\alpha^{-1} \beta ^{-2} \gamma^{2} \text{ new units}\:\underline{ QED} \end{aligned} $

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