Question

A steel rod of a cross section $1{\text{ m}}{{\text{m}}^2}$ is prevented from expansion by heating through ${10^0}C$. The thermal force developed in it is: $\left( {Y = 2 \times {{10}^{11}}{\text{ N/}}{{\text{m}}^2};\alpha = {{10}^{ - 5}}{{\text{/}}^{\text{0}}}{\text{C}}} \right)$
A) 20 N
B) 2 N
C) 200 N
D) 0.2 N

Answer 1

Hint:In this question, we need to determine the thermal force developed in a steel rod of a cross section $1{\text{ m}}{{\text{m}}^2}$ so as to prevent the expansion by heating through ${10^0}C$. For this, we will use the relation between young modulus of elasticity, area of the cross-section of the rod, coefficient of linear expansion, changes in the thermal temperature and thermal force which is given as ${F_T} = YA\alpha \vartriangle T$.

Complete step by step answer:

The product of young modulus of elasticity, area of the cross-section of the rod, coefficient of linear expansion and change in the thermal temperature of the rod results in the thermal force exerted on the rod.

Mathematically, ${F_T} = YA\alpha \vartriangle T$ where ‘Y’ is the young modulus of elasticity of the rod (in ${\text{N/}}{{\text{m}}^2}$), ‘A’ is the area of the cross-section of the rod (in ${{\text{m}}^2}$), $\alpha $ is the coefficient of linear expansion of the rod (in ${{\text{/}}^{\text{0}}}{\text{C}}$) and $\vartriangle T$ is the change in the temperature of the rod (in ${}^{\text{0}}{\text{C}}$).

Area of the cross-section of the steel rod has been given in square millimetres which should be in squares meters. So, convert square millimetres to square meters by dividing the numerical value by ${10^6}$. So, $1{\text{ m}}{{\text{m}}^2} = \dfrac{1}{{{{10}^6}}}{\text{ }}{{\text{m}}^2} = {10^{ - 6{\text{ }}}}{{\text{m}}^2}$.

Here, $Y = 2 \times {10^{11}}{\text{ N/}}{{\text{m}}^2};\alpha = {10^{ - 5}}{{\text{/}}^{\text{0}}}{\text{C}};A = 1{\text{ m}}{{\text{m}}^2} = {10^{ - 6{\text{ }}}}{{\text{m}}^2};\vartriangle T = {10^0}C$. So substitute these values in the formula ${F_T} = YA\alpha \vartriangle T$ to determine the thermal force developed in the rod due to temperature difference.

$
  {F_T} = YA\alpha \vartriangle T \\
   = 2 \times {10^{11}} \times {10^{ - 6}} \times {10^{ - 5}} \times 10 \\
   = 2 \times {10^{\left( {11 - 6 - 5 + 1} \right)}} \\
   = 2 \times 10 \\
   = 20{\text{ N}} \\
 $

Hence, the thermal force developed in a steel rod of a cross section $1{\text{ m}}{{\text{m}}^2}$ so as to prevent the expansion by heating through ${10^0}C$ is 20 Newton.

Option A is correct.

Note:The total kinetic energy of motion of all the particles that make up the body is known as the thermal energy of a body. Students should be careful while substituting the values of the given terms in the formula; all the terms should be in their SI unit only.