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. `0.12^(@)`
B. `1.2^(@)`
C. `12^(2)`
D. `0.012^(@)`

Explanation

To find the angle of shear in the wire when it is twisted, we can use the formula for the angle of shear:

\[ \phi = \frac{R \cdot \theta}{L} \]

where:
- \( R \) is the radius of the wire,
- \( \theta \) is the angle of twist in radians,
- \( L \) is the length of the wire.

### Step 1: Convert the given values to appropriate units
- The radius \( R \) of the wire is given as 4 mm. We convert this to meters:
\[ R = 4 \, \text{mm} = 4 \times 10^{-3} \, \text{m} \]

- The length \( L \) of the wire is given as 100 cm. We convert this to meters:
\[ L = 100 \, \text{cm} = 1 \, \text{m} \]

- The angle of twist \( \theta \) is given as 30 degrees. We need to convert this to radians:
\[ \theta = 30^\circ = \frac{30 \times \pi}{180} \, \text{radians} = \frac{\pi}{6} \, \text{radians} \]

### Step 2: Substitute the values into the formula
Now we can substitute the values into the formula for the angle of shear:
\[ \phi = \frac{R \cdot \theta}{L} = \frac{(4 \times 10^{-3}) \cdot \left(\frac{\pi}{6}\right)}{1} \]

### Step 3: Calculate the angle of shear
Calculating the above expression:
\[ \phi = 4 \times 10^{-3} \cdot \frac{\pi}{6} \]
Using the approximate value of \( \pi \approx 3.14 \):
\[ \phi \approx 4 \times 10^{-3} \cdot \frac{3.14}{6} \approx 4 \times 10^{-3} \cdot 0.5233 \approx 2.0932 \times 10^{-3} \, \text{radians} \]

### Step 4: Convert the angle of shear back to degrees
To convert radians back to degrees:
\[ \phi \text{ (in degrees)} = \phi \text{ (in radians)} \cdot \frac{180}{\pi} \approx 2.0932 \times 10^{-3} \cdot \frac{180}{3.14} \approx 0.12^\circ \]

### Final Answer
The angle of shear \( \phi \) is approximately \( 0.12^\circ \).

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