Question

A calorie is a unit of heat (energy in transit) and it equals about 4.2J where \[1J=1kg{{m}^{2}}{{s}^{-2}}\]. Suppose we employ a system of units in which the unit of mass equals $\alpha kg$ ,The unit of length equals $\beta m$ , the unit of time is $\gamma s$.show that a caloric has a magnitude $4.2{{\alpha }^{-1}}{{\beta }^{-2}}{{\gamma }^{2}}$ in terms of the new units.

Answer 1

Hint: Heat is a form of energy. As a result of temperature difference, heat energy is transferred from one body to another body .Heat flows from hotter body to cooler body. Temperature is the measure of the amount of heat energy present in a body.SI unit of heat is joule (J) and calorie is also used as the unit of heat.


Complete step-by-step solution:
Work is a form of heat. Work and energy are directly proportional to each other and energy can be converted into different forms. Energy which cannot be created nor destroyed but it can be transformed .SI units of energy are also joules so don’t confuse work and energy.
In SI unit mass is measured in kilogram (kg), length is measured in meter(m) and time is measured in second (s). So ${{M}_{1}}=1kg$ , ${{L}_{1}}=1m$ and ${{T}_{_{1}}}=1s$.
In the new system mass is measured in ($\alpha kg$), length is measured in ($\beta m$) and time is measured in ($\gamma s$). So ${{M}_{2}}=\alpha kg$ , ${{L}_{2}}=\beta m$ and ${{T}_{_{2}}}=\gamma s$.
By the standard unit conversion formula
$\dfrac{Given Unit({{n}_{2}})}{Newunit({{n}_{1}})}={{\left[ \dfrac{{{M}_{1}}}{{{M}_{2}}} \right]}^{x}}{{\left[ \dfrac{{{L}_{1}}}{{{L}_{2}}} \right]}^{y}}{{\left[ \dfrac{{{T}_{1}}}{{{T}_{2}}} \right]}^{z}}$
$1cal=4.2kg{{m}^{2}}{{s}^{-2}}$
So ${{n}_{1}}=4.2$
Dimensional formula of heat is $[{{M}^{1}}{{L}^{2}}{{T}^{-2}}]$ is compared with $[{{M}^{x}}{{L}^{y}}{{T}^{z}}]$ then we get the values of
X=1, y=2 and z=-2
After substituting
\[\begin{align}
  & {{n}_{2}}={{n}_{1}}{{\left[ \dfrac{{{M}_{1}}}{{{M}_{2}}} \right]}^{x}}{{\left[ \dfrac{{{L}_{1}}}{{{L}_{2}}} \right]}^{y}}{{\left[ \dfrac{{{T}_{1}}}{{{T}_{2}}} \right]}^{z}} \\
 & {{n}_{2}}=4.2{{\left[ \dfrac{1kg}{\alpha kg} \right]}^{1}}{{\left[ \dfrac{1m}{\beta m} \right]}^{2}}{{\left[ \dfrac{1s}{1s} \right]}^{-2}} \\
 & {{n}_{2}}=4.2{{\alpha }^{1}}{{\beta }^{2}}{{\gamma }^{-2}} \\
\end{align}\]
caloric has a magnitude $4.2{{\alpha }^{-1}}{{\beta }^{-2}}{{\gamma }^{2}}$ in terms of the new units is obtained.

Note:Student’s heat transfer is a branch of thermal engineering in that we are going to study generation, heat transfer and conversion. Heat transfer is the flow of heat due a temperature difference between the bodies. There are three mechanisms of heat transfer and they are conduction, convection and radiation.