Question

A rod is made of uniform material and has non-uniform cross section. It is fixed at both ends as shown and heated at the mid-section. Which of the following statements are not correct?

A. Force of compression in the rod will be maximum at mid-section
B. Compressive stress in the rod will be maximum at left end
C. Since the rod is fixed at both the ends, its length will remain unchanged. Hence, no strain will be induced in it.
D. None of the above

Answer 1

Since the rod is held in between two walls, the force of compression will be constant. For calculating stress and strain qualitative, we can use the formula,
$$
Stress = \dfrac{{Force}}{{Area}} \\
Strain = \dfrac{{\Delta l}}{L} \\
\eta = \dfrac{{stress}}{{strain}} \\
Stress = \dfrac{{Force}}{{Area}} \\
Strain = \dfrac{{\Delta l}}{L} \\
\eta = \dfrac{{stress}}{{strain}} \\

Where, $\Delta l$ is the change in length (Increased length-original length) $\eta $ is the modulus of elasticity (Hooke’s Law) **Complete step by step answer:** This question is based on qualitative analysis of Hooke’s Law and concept of elasticity. Given are the following formulae from the mentioned concept: Now, going option by option. In option A, it is mentioned that force is maximum at the middle. It is wrong. Since we are applying heat at the centre does not mean the force will be expansion. The heat will be conducted throughout the material and also the rod is kept stationary between two walls implies that the force is constant throughout the rod. The compressive stress formula is mentioned above. Now the force is constant as explained above, the stress depends upon the area alone. Since inversely proportional to area, we find the compressive stress will be maximum at the left end. Therefore, option B is correct. The rod is fixed between two walls but it does not mean there will be no expansion when heat is applied. If there is no expansion the modulus of elasticity will be infinite which is impossible (Since ideal materials do not exist). On continuous heating, if the material is less elastic, cracks will be formed and the rod will be broken. If the material is very elastic it will bend in the form of an arc. Since heat is applied from below, the rod will move upwards so it will form an inverted parabola. **Note:** This question is a multiple answer type question (that too a qualitative one) so you have to study each option and find whether one is correct or not. In case of numerical, students think that this type of questions will not be asked but they are asked. The answer is in different units and dimensions there.