A simple pendulum oscillates slightly above a large horizontal metal plate. The bob is given a charge. The time period

Options

A. has no reffect, whatever be the nature of charge
B. always decreases, whatever be the nature of charge
C. always increases, whatever be the nature of charge
D. a increases or decreases depending upon the nature of charge.

Explanation

To solve the problem of the time period of a simple pendulum oscillating above a large horizontal metal plate with a charged bob, we can follow these steps:

### Step 1: Understand the System
- A simple pendulum consists of a bob suspended from a fixed point, which can swing back and forth.
- In this case, the bob is charged, and it is positioned above a large horizontal metal plate.

### Step 2: Analyze the Forces Acting on the Bob
- The forces acting on the bob include:
1. The gravitational force (weight) acting downwards, \( F_g = mg \), where \( m \) is the mass of the bob and \( g \) is the acceleration due to gravity.
2. An electrostatic force due to the interaction between the charged bob and the induced charges on the metal plate. Since the bob is positively charged, the plate will induce a negative charge on its upper surface, resulting in an attractive force.

### Step 3: Determine the Effective Gravitational Force
- The effective gravitational force acting on the bob can be expressed as:
\[ F_{\text{effective}} = mg + F_q \]
where \( F_q \) is the electrostatic force acting upwards.

### Step 4: Define the Effective Acceleration
- The effective acceleration \( g_{\text{effective}} \) can be defined as:
\[ g_{\text{effective}} = g + \frac{F_q}{m} \]
This indicates that the effective gravitational acceleration is increased due to the additional upward force from the electrostatic attraction.

### Step 5: Write the Formula for the Time Period
- The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum. However, since we have an effective gravitational acceleration, we replace \( g \) with \( g_{\text{effective}} \):
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} \]

### Step 6: Analyze the Effect of Increased \( g_{\text{effective}} \)
- Since \( g_{\text{effective}} > g \) due to the additional force \( F_q \), we can conclude that:
\[ T = 2\pi \sqrt{\frac{L}{g + \frac{F_q}{m}}} \]
This implies that as \( g_{\text{effective}} \) increases, the time period \( T \) will decrease.

### Step 7: Conclusion
- Regardless of whether the bob is positively or negatively charged, the presence of the metal plate will always increase the effective gravitational force acting on the bob, leading to a decrease in the time period of oscillation.

### Final Answer
The time period of the pendulum will decrease when the bob is charged and oscillating above the metal plate.

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