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Chapter 7: Problem 936

Two wires \(A \& \mathrm{~B}\) of same length and of the same material have the respective radius \(\mathrm{r}_{1} \& \mathrm{r}_{2}\) their one end is fixed with a rigid support and at the other end equal twisting couple is applied. Then what will we be the ratio of the angle of twist at the end of \(\mathrm{A}\) and the angle of twist at the end of \(\mathrm{B}\). (A) \(\left(\mathrm{r}_{1}^{2} / \mathrm{r}_{2}^{2}\right)\) (B) \(\left(\mathrm{r}_{2}^{2} / \mathrm{r}_{1}^{2}\right)\) (C) \(\left(\mathrm{r}_{2}^{4} / \mathrm{r}_{1}^{4}\right)\) (D) \(\left(\mathrm{r}_{1}^{4} / \mathrm{r}_{2}^{4}\right)\)

Short Answer

Expert verified
The short answer is: \(\mathrm{X}=\frac{\mathrm{r}_{2}^4}{\mathrm{r}_{1}^4}\), which corresponds to option (C).

Step by step solution

01

Recall the torsion formula

The torsion formula states that the angle of twist θ is proportional to the twisting couple T and the length of the wire L and inversely proportional to the polar moment of inertia J and the modulus of rigidity G. Mathematically, we can write it as: θ = TL / (GJ)
02

Determine the polar moment of inertia for both wires

For a circular wire with a radius r, the polar moment of inertia J can be calculated using the formula: \(J = \frac{πr^4}{2}\) Calculate the polar moment of inertia for wire A (with radius r1) and wire B (with radius r2) using the formula above. \(J_A = \frac{πr_1^4}{2}\) and \(J_B = \frac{πr_2^4}{2}\)
03

Set up the ratio of the angles of twist

The goal is to find the ratio of the angle of twist in wire A to the angle of twist in wire B. Let's denote this ratio as X. Mathematically, we can represent this as: X = θ_A / θ_B
04

Use the torsion formula to find the ratio of the angles of twist

As both wires have the same length, material, and twisting couple: X = (T_A * L) / (G * J_A) / ((T_B * L) / (G * J_B)) Since TA = TB and both wires have the same material (same G) and length L, the ratio simplifies as follows: X = J_B / J_A Substitute the previously calculated polar moment of inertia values for wire A and wire B: X = \(\frac{\frac{πr_2^4}{2}}{\frac{πr_1^4}{2}}\) The π and 2 terms will cancel each other out, giving the final expression for the ratio of the angles of twist: X = \(\frac{r_2^4}{r_1^4}\) Comparing this expression to the given options, we can see that the correct answer is: (C) \(\left(\mathrm{r}_{2}^{4} / \mathrm{r}_{1}^{4}\right)\)

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