A rod of length `l` and radius `r` is joined to a rod of length `l//2` and radius `r//2` of same material. The free end of small rod is fixed to a rigid base and the free end of larger rod is given a twist of `theta^(@)`, the twist angle at the joint will be

Options

A. `theta//4`
B. `theta//2`
C. `5 theta//6`
D. `8 theta//9`

Answer

To solve the problem, we need to find the twist angle at the joint of two rods made of the same material. We have a larger rod of length \( l \) and radius \( r \) joined to a smaller rod of length \( \frac{l}{2} \) and radius \( \frac{r}{2} \). The free end of the larger rod is twisted by an angle \( \theta \), and we want to determine the twist angle at the joint, which we will denote as \( \theta_0 \).

### Step-by-Step Solution:

1. **Identify the Torque Relationship**:
The torque \( \tau \) developed in a rod due to twisting is given by:
\[ \tau = C \cdot \theta \]
where \( C \) is the torsional rigidity of the rod and \( \theta \) is the twist angle.

2. **Torsional Rigidity**:
The torsional rigidity \( C \) for a circular rod is given by:
\[ C = \frac{\pi n r^4}{2l} \]
where \( n \) is the modulus of rigidity, \( r \) is the radius, and \( l \) is the length of the rod.

3. **Torque for the Larger Rod**:
For the larger rod (length \( l \), radius \( r \)):
\[ \tau_1 = \frac{\pi n r^4}{2l} \cdot \theta \]

4. **Torque for the Smaller Rod**:
For the smaller rod (length \( \frac{l}{2} \), radius \( \frac{r}{2} \)):
\[ \tau_2 = \frac{\pi n \left(\frac{r}{2}\right)^4}{2 \cdot \frac{l}{2}} \cdot \theta_0 = \frac{\pi n \frac{r^4}{16}}{l} \cdot \theta_0 \]

5. **Equating the Torques**:
Since both rods are joined and experience the same torque, we set \( \tau_1 = \tau_2 \):
\[ \frac{\pi n r^4}{2l} \cdot \theta = \frac{\pi n \frac{r^4}{16}}{l} \cdot \theta_0 \]

6. **Canceling Common Terms**:
We can cancel \( \pi n r^4 \) and \( l \) from both sides:
\[ \frac{1}{2} \cdot \theta = \frac{1}{16} \cdot \theta_0 \]

7. **Solving for \( \theta_0 \)**:
Rearranging gives:
\[ \theta_0 = 8 \cdot \theta \]

8. **Finding the Relation**:
From the previous steps, we can express \( \theta \) in terms of \( \theta_0 \):
\[ \theta = \frac{9}{8} \cdot \theta_0 \]
Thus, we can substitute back to find \( \theta_0 \):
\[ \theta_0 = \frac{8}{9} \cdot \theta \]

### Final Answer:
The twist angle at the joint \( \theta_0 \) is:
\[ \theta_0 = \frac{8}{9} \theta \]