Question

A calorie is a unit of heat energy and its value is 4.18 J where $1J=1kg{m}^2{s}^{-2}$. 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 then

• 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

The correct option is A 4.18 $\frac{{\gamma }^2}{\alpha {\beta }^2}$

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