Two rods A and B of the same material and length have radii `r_(1)` and `r_(2)`respective. When they are rigidly fixed at one end and twisted by the same torque applied at the other end, the ratio `[("the angle of twist at the end of A")/("the angle of twist at the end of B")]` equal to

Options

A. `(r_(1)^(2))/(r_(2)^(2))`
B. `(r_(1)^(3))/(r_(2)^(3))`
C. `(r_(2)^(4))/(r_(1)^(4))`
D. `(r_(1)^(4))/(r_(2)^(4))`

Explanation

To solve the problem, we need to find the ratio of the angles of twist for two rods A and B, which are subjected to the same torque. The angle of twist is given by the formula:

\[ \theta = \frac{2 \tau L}{\pi r^4 \eta} \]

where:
- \(\theta\) is the angle of twist,
- \(\tau\) is the applied torque,
- \(L\) is the length of the rod,
- \(r\) is the radius of the rod,
- \(\eta\) is the shear modulus of the material.

### Step 1: Write the formula for the angle of twist for both rods

For rod A:
\[ \theta_A = \frac{2 \tau L}{\pi r_1^4 \eta} \]

For rod B:
\[ \theta_B = \frac{2 \tau L}{\pi r_2^4 \eta} \]

### Step 2: Find the ratio of the angles of twist

We need to find the ratio \(\frac{\theta_A}{\theta_B}\):

\[ \frac{\theta_A}{\theta_B} = \frac{\frac{2 \tau L}{\pi r_1^4 \eta}}{\frac{2 \tau L}{\pi r_2^4 \eta}} \]

### Step 3: Simplify the ratio

The terms \(2 \tau L\) and \(\pi \eta\) cancel out:

\[ \frac{\theta_A}{\theta_B} = \frac{r_2^4}{r_1^4} \]

### Step 4: Write the final answer

Thus, the ratio of the angles of twist at the ends of rods A and B is:

\[ \frac{\theta_A}{\theta_B} = \left(\frac{r_2}{r_1}\right)^4 \]

### Final Result

The final result is:
\[ \frac{\theta_A}{\theta_B} = \left(\frac{r_2}{r_1}\right)^4 \]

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