Question

A rod of length $l$ and area of cross-section $A$, is heated from $0^\circ C$ to $100^\circ C$. The rod is so placed that it is not allowed to increase in length, then the force is developed in proportional to.
(A) 1
(B) ${1^{ - 3}}$
(C) $A$
(D) ${A^{ - 1}}$

Answer 1

Matter has a tendency to respond to temperature by changing the characteristics that define the shape and size of it. (Namely, shape, area, volume, and density).
Also, if a body is restrained and not allowed to expand in such conditions this builds an internal stress. This is found out by calculating the strain of the body if it were free to expand, and then finding the stress that would be required to reduce that strain to zero.
Formulas used: We will be using the formula $\dfrac{{\Delta L}}{L} = {\alpha _L}\Delta T$ which is used to calculate strain where $\dfrac{{\Delta L}}{L}$ is the strain experienced by the body, ${\alpha _L}$ is the linear coefficient of thermal expansion, and $\Delta T$ is the change in temperature.
We would also use $Y = \dfrac{{stress}}{{strain}}$ where $Y$ is the Young’s modulus.

Complete Step by Step Answer:
We know that every matter /body has the tendency to expand on heating or changing the body’s temperature by any amount. We call this ability of a body as the thermal expansion of the body. Also, if a body is restrained and applied force from any other direction it leads to building of internal stress in the body.
So here, the body is not allowed to increase its length and hence the body develops an internal stress. Considering the length of the body to be $l$ and the area of its cross section to be $A$ and the change in temperature to be $\Delta T$we can find the strain caused in the body due to internal stress to be, $\dfrac{{\Delta L}}{L} = {\alpha _L}\Delta T$.
We also know that thermal stress is given by, ${\sigma _{th}} = Y \times strain$, by substituting the value of stress, ${\sigma _{th}} = Y{\alpha _L}\Delta T$. We also know that stress can be given by, $\sigma = \dfrac{F}{A}$ .
Equating the two equations that give us the stress of a body, $Y{\alpha _L}\Delta T = \dfrac{F}{A}$. Solving the equation for $F$, we get, $A \times Y{\alpha _L}\Delta T = F$ .
Analysing the terms of the equation we can see that $Y$ and ${\alpha _L}$ are constants. Also, we know that here $\Delta T = 100^\circ c$. Since every other term in the equation is a constant, we can say that $F \propto A$ .
Thus, the force is proportional with the area of the cross section of the rod.

Hence the correct answer is option C.

Note: The system here is a restricted yet static environment hence the force applied on the body will be the same at all points of the rod. Thus, in such environments to find the stress developed we should be calculating the strain that might have been formed when the body was not restricted under the same conditions and work to equate the strain to zero.