A simple pendulum is released from rest with the string in horizontal position. The vertical component of the velocity of the bob becomes maximum, when the string makes an angle `theta` with the vertical. The angle `theta` is equal to

Options

A. `(a)(pi)/(4)`
B. `(b)cos^(-1)((1)/(sqrt(3)))`
C. `(c)sin^(-1)((1)/(sqrt(3)))`
D. `(d)(pi)/(3)`

Explanation

To solve the problem of finding the angle θ at which the vertical component of the velocity of a pendulum bob is maximum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A simple pendulum is released from a horizontal position. We need to find the angle θ with the vertical at which the vertical component of the velocity of the bob is maximum.

2. **Identify the Components of Velocity**:
- When the pendulum swings down, it has both a tangential component of velocity (V₀) and a vertical component of velocity (V). The vertical component of velocity can be expressed in terms of the tangential velocity.

3. **Using Geometry**:
- When the pendulum makes an angle θ with the vertical, the vertical component of the velocity can be expressed as:
\[ V = V₀ \sin(θ) \]
- Here, V₀ is the tangential velocity of the bob.

4. **Finding the Expression for V₀**:
- From the conservation of energy, when the pendulum is released from the horizontal position, the potential energy converts into kinetic energy.
- The height (h) from which the bob falls when it swings to angle θ is given by:
\[ h = r(1 - \cos(θ)) \]
- The potential energy lost is equal to the kinetic energy gained:
\[ mgh = \frac{1}{2} mv₀^2 \]
- Substituting h:
\[ mg(r(1 - \cos(θ))) = \frac{1}{2} mv₀^2 \]
- Simplifying gives:
\[ v₀^2 = 2gr(1 - \cos(θ)) \]

5. **Substituting V₀ into V**:
- Now, substitute V₀ into the expression for V:
\[ V = \sqrt{2gr(1 - \cos(θ))} \sin(θ) \]

6. **Maximizing the Vertical Component of Velocity**:
- To find the maximum value of V, we differentiate V with respect to θ and set the derivative to zero:
\[ \frac{dV}{dθ} = 0 \]
- This requires using the product and chain rules of differentiation.

7. **Applying the Product Rule**:
- The expression for V can be simplified and differentiated. After differentiation and simplification, we find:
\[ \frac{dV}{dθ} = 0 \implies \text{solving gives } \cos(θ) = \frac{1}{2} \]

8. **Finding θ**:
- Thus, we find:
\[ θ = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

### Final Answer:
The angle θ at which the vertical component of the velocity of the bob becomes maximum is:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \]