The upper end of a wire of radius `4 mm` and length `100 cm` is clamped and its other end is twisted through an angle of `30^@`. Then angle of shear is

Options

A. `12^@`
B. `0.12^@`
C. `1.2^@`
D. `0.012^@`

Explanation

To solve the problem, we will use the relationship between the shear angle (φ), the radius of the wire (r), the angle of twist (θ), and the length of the wire (L).

### Step-by-Step Solution:

1. **Identify the given values**:
- Radius of the wire, \( r = 4 \, \text{mm} = 4 \times 10^{-3} \, \text{m} \)
- Length of the wire, \( L = 100 \, \text{cm} = 1 \, \text{m} \)
- Angle of twist, \( \theta = 30^\circ \)

2. **Convert the angle of twist from degrees to radians**:
- Since the formula for shear angle often requires the angle in radians, we convert \( \theta \):
\[ \theta \text{ (in radians)} = \frac{30 \times \pi}{180} = \frac{\pi}{6} \, \text{radians} \]

3. **Use the formula for shear angle**:
- The formula for shear angle (φ) is given by:
\[ \phi = \frac{r \cdot \theta}{L} \]
- Substitute the values into the formula:
\[ \phi = \frac{(4 \times 10^{-3}) \cdot \left(\frac{\pi}{6}\right)}{1} \]

4. **Calculate φ**:
- First, calculate \( 4 \times 10^{-3} \cdot \frac{\pi}{6} \):
\[ \phi = \frac{4 \pi}{6} \times 10^{-3} = \frac{2 \pi}{3} \times 10^{-3} \approx 0.002094 \, \text{radians} \]

5. **Convert φ back to degrees**:
- To convert radians back to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \phi \text{ (in degrees)} = 0.002094 \times \frac{180}{\pi} \approx 0.120 \, \text{degrees} \]

6. **Final result**:
- The angle of shear φ is approximately \( 0.12^\circ \).

### Summary:
The angle of shear for the given wire when twisted through an angle of \( 30^\circ \) is approximately \( 0.12^\circ \).