Question

What is meant by the mechanical equivalent of heat?

Answer 1

Hint: Heat is a form of energy which is responsible for the change in temperature of any object. Like any other forms of energies, heat can neither be created nor destroyed. It can only be transferred from one object to another or can be converted from one form to another.

Complete step by step solution:
Heat is a form of energy by the virtue of which changes in temperature in an object takes place. Heat energy is actually possible because of the movement of molecules in the material. In solids, the molecules vibrate at their own places which creates their heat energy. In liquids, the motion followed by the molecules is known as convection, in which the molecules with higher heat energy tend to rise up towards the surface whereas the molecules with lower heat energy tend to sink down. In gases, the molecules follow Brownian motion to create heat energy.
According to Joule’s Law, whenever heat is converted into mechanical work or mechanical work is converted into heat, then the ratio of work done to heat produced always remains constant. Which means that,
$W \propto Q$, or
$ \Rightarrow W = JQ$
$ \Rightarrow \dfrac{W}{Q} = J$
Where,
$W = $Mechanical Work Done,
 $Q = $Heat produced, and
 $J = $Constant.
In Joule’s Law, $J$ is known as the mechanical equivalent of heat.
From $W = JQ$, we can infer that if $Q = 1$ then $J = W$. Hence the amount of work done necessary to produce a unit amount of heat is defined as the mechanical equivalent of heat.
$J$is neither a constant, nor a physical quantity rather it is a conversion factor which is used to convert Joule or erg into calorie or kilocalories, or vice-versa.
Value of $J = 4.2\dfrac{{Joule}}{{cal}} = 4.2 \times {10^7}\dfrac{{erg}}{{cal}} = 4200\dfrac{{Joule}}{{kcal}}$.

Note: When water in a stream falls from a height $h$, then its potential energy is converted into heat and the temperature of water rises slightly.
$W = JQ$
$ \Rightarrow mgh = J\left( {mC\Delta \theta } \right)$
Where, $m = $Mass of water,
              $g = $Acceleration due to gravity,
              $C = $Specific heat of water, and
               $\Delta \theta = $Change in temperature.