Question

A calorie is a unit of heat or energy and it equals about $4.2J$ where $1J$=$1kg{m}^2{s}^{-2}$.Suppose we employ a system of units in which the units of mass equals $\alpha kg$, the units 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.

Answer 1

By standard formula of unit conversion,

$\frac{Givenunit}{Newunit}={\left(\frac{M_1}{M_2}\right)}^2{\left(\frac{L_1}{L_2}\right)}^2{\left(\frac{T_1}{T_2}\right)}^2$

Dimension formula of heat $=\left[M^1{L}^2{T}^{-2}\right]$

$\therefore$ $x=1,y=2,z=2$

Since $M_1=1kg$, $L_1=1m$, $T_1=1s$

and $M_2=\alpha$ kg, $L_2=\beta m$, $T_2=\gamma s$

As 1 calorie $=4.2$ Joule

and $1$ Joule $=1kg$ $m^2{s}^{-2}$

$\Rightarrow$ $\frac{calorie}{Newunit}=4.2{\left(\frac{1kg}{\alpha kg}\right)}^1{\left(\frac{1m}{\beta m}\right)}^2{\left(\frac{1s}{\gamma s}\right)}^{-2}$

$\therefore$ Calorie $=4.2{\alpha }^{-1}$ ${\beta }^{-2}$ ${\gamma }^2$ New unit