In a conical pendulum arrangement, a string of length 1 m is fixed at one end with a bob of mass 100 g and the string makes `(2)/(pi) mvs^(-1)`around a vertical axis through a fixed point. The angle of inclination of the string with vertical is: (Take `g = 10 ms^(-1)`)

Options

A. `tan^(-1) ((5)/(8))`
B. `tan^(-1) ((3)/(5))`
C. `cos^(-1) ((8)/(5))`
D. `cos^(-1) ((5)/(8))`

Explanation

To solve the problem of finding the angle of inclination of the string with the vertical in a conical pendulum arrangement, we can follow these steps:

### Step 1: Understand the Forces Acting on the Bob
In a conical pendulum, the forces acting on the bob are:
- The tension \( T \) in the string, which has two components:
- \( T \cos \theta \) acting vertically upwards (balancing the weight)
- \( T \sin \theta \) acting horizontally (providing the centripetal force)
- The weight of the bob \( mg \) acting downwards.

### Step 2: Set Up the Equations
From the balance of forces, we can write the following equations:
1. **Vertical Force Balance**:
\[ T \cos \theta = mg \]
2. **Horizontal Force (Centripetal Force)**:
\[ T \sin \theta = m \frac{v^2}{r} \]
where \( r \) is the horizontal radius of the circular path traced by the bob.

### Step 3: Express \( r \) in Terms of \( \theta \)
The radius \( r \) can be expressed in terms of the length of the string \( L \) and the angle \( \theta \):
\[ r = L \sin \theta \]

### Step 4: Substitute \( r \) into the Centripetal Force Equation
Substituting \( r \) into the horizontal force equation gives:
\[ T \sin \theta = m \frac{v^2}{L \sin \theta} \]

### Step 5: Divide the Two Equations
Dividing the horizontal force equation by the vertical force equation:
\[ \frac{T \sin \theta}{T \cos \theta} = \frac{m \frac{v^2}{L \sin \theta}}{mg} \]
This simplifies to:
\[ \tan \theta = \frac{v^2}{gL} \]

### Step 6: Substitute the Given Values
We know:
- \( L = 1 \, \text{m} \)
- \( m = 0.1 \, \text{kg} \) (100 g)
- \( g = 10 \, \text{ms}^{-2} \)
- The speed \( v = \frac{2}{\pi} \, \text{m/s} \)

Substituting \( v \):
\[ \tan \theta = \frac{\left(\frac{2}{\pi}\right)^2}{10 \times 1} \]
Calculating \( v^2 \):
\[ v^2 = \left(\frac{2}{\pi}\right)^2 = \frac{4}{\pi^2} \]
Thus,
\[ \tan \theta = \frac{\frac{4}{\pi^2}}{10} = \frac{4}{10\pi^2} = \frac{2}{5\pi^2} \]

### Step 7: Find \( \theta \)
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}\left(\frac{2}{5\pi^2}\right) \]

### Step 8: Calculate the Angle
Using a calculator, we can find the approximate value of \( \theta \).

### Final Result
The angle of inclination \( \theta \) can be approximated numerically.