Question

A calorie is a unit of heat energy and its value is $4.18\, J$ where $1 \,J = 1 \,kg\, m^2\, s^{-2}$. Suppose we use a new system of units in which unit of mass equals $\alpha\, kg$, the unit of length equals $\beta\, m$ and the unit of the time is $\gamma\, sec$. Then the value of a calorie in the new system of units is

• A. $4.18 \frac{\gamma^{2}}{\alpha\beta^{2}}$
• B. $4.18 \frac{\alpha\beta^{2}}{\gamma^{2}}$
• C. $4.18 \frac{\gamma^{2}}{\alpha}$
• D. $4.18 \frac{\beta^{2}}{\alpha\gamma^{2}}$

Answer 1

Answer: (A)

$1\,J=\left(1\,kg\right)\left(1\,m\right)^{2}\left(1\,sec\right)^{-2}$
$1x=\left(\alpha\,kg\right)\left(\beta\,m\right)^{2}\left(\gamma\,sec\right)^{-2}$
$\therefore \frac{1\,J}{1x}=\left(\frac{1}{\alpha}\right)\left(\frac{1}{\beta}\right)^{2}\left(\gamma\right)^{2}=\frac{\gamma^{2}}{\alpha\beta^{2}}$
$\therefore 1\,J=\frac{\gamma^{2}}{\alpha\beta^{2}}x$ or $1\,cal=4.18 \frac{\gamma^{2}}{\alpha\beta^{2}}$