A long wire hangs vertically with its upper end clam `A` torque of `8 Nm` applied to the free end twists it through`45^@`. The potential energy of the twisted wire is

Options

A. `piJ`
B. `(pi)/2J`
C. `(pi)/4J`
D. `(pi)/8J`

Explanation

To find the potential energy of the twisted wire, we can follow these steps:

### Step 1: Identify the given values
- Torque (τ) = 8 Nm
- Angle of twist (φ) = 45 degrees

### Step 2: Convert the angle from degrees to radians
Since the formula for potential energy requires the angle in radians, we convert 45 degrees to radians:
\[ \phi = 45^\circ = \frac{\pi}{4} \text{ radians} \]

### Step 3: Use the formula for potential energy
The potential energy (PE) stored in the twisted wire can be calculated using the formula:
\[ PE = \frac{1}{2} \times \tau \times \phi \]
Substituting the known values into the formula:
\[ PE = \frac{1}{2} \times 8 \, \text{Nm} \times \frac{\pi}{4} \]

### Step 4: Simplify the expression
Calculating the expression step-by-step:
1. Calculate \( \frac{1}{2} \times 8 = 4 \)
2. Now multiply by \( \frac{\pi}{4} \):
\[ PE = 4 \times \frac{\pi}{4} = \pi \, \text{Joules} \]

### Step 5: Conclusion
The potential energy of the twisted wire is:
\[ PE = \pi \, \text{Joules} \]