Question

Two wires A and B of same length and of the same material have the respective radii ${r_1}$ and ${r_2}$ . Their one end is fixed with a rigid support, and at the other end equal twisting couple is applied. Then the ratio of the angle of twist at the end of A and the angle of twist at the end of B will be
A. $\dfrac{{{r_1}^2}}{{{r_2}^2}}$
B. $\dfrac{{{r_2}^2}}{{{r_1}^2}}$
C. $\dfrac{{{r_2}^4}}{{{r_1}^4}}$
D. $\dfrac{{{r_1}^4}}{{{r_2}^4}}$

Answer 1

Hint: In this case, when a problem is based on Shear and Bulk Modulus in Mechanical properties of solids, we know that twisting couple is defined as $\dfrac{{\pi \eta {r^4}\theta }}{{2l}}$ plays a significant role in establishing a relationship between couple and angle of twist hence, use this formula to calculate the ratio in order to provide an accurate solution.

Formula Used:
 Couple = $C = \dfrac{{\pi \eta {r^4}\theta }}{{2l}}$

Complete answer:
We know that Couple per unit angle of twist is $\dfrac{{\pi \eta {r^4}}}{{2l}}$.
If angle of twist is denoted by $\theta $.
Then, Couple = $C = \dfrac{{\pi \eta {r^4}\theta }}{{2l}}$
According to the given question, the lengths of wires A and B are equal. Also, the material of A and B are the same and an equal twisting couple is applied on both the wires.
Since $\eta ,C\;and{\text{ }}l$ are same.
Therefore, ${r^4}\theta $ = constant.
i.e., the angle of twist varies inversely with the radius of the wire.
As the radii of two wires A and B are ${r_1}$ and ${r_2}$ respectively (given)
$ \Rightarrow \dfrac{{{\theta _A}}}{{{\theta _B}}} = \dfrac{{{r_2}^4}}{{{r_1}^4}}$
Thus, the ratio of the angle of twist at the end of A and the angle of twist at the end of B will be $\dfrac{{{r_2}^4}}{{{r_1}^4}}$.

Hence, the correct option is (C) $\dfrac{{{r_2}^4}}{{{r_1}^4}}$ .


Note: Since this is a problem related to stress-strain analysis in mechanical properties of solids hence, given conditions must be analyzed very carefully and quantities that are required to calculate the ratio of the angle of twist must be identified on a prior basis as it gives a better understanding of the problem.