A pendulum of length L carries a negative charge - q on the bob. A positive charge +q is held at the point of support . Then, the time period of the bob is

Options

A. Greater than `2pisqrt(L/g)`
B. Less than `2pisqrt(L/g)`
C. equal to `2pisqrt(L/g)`
D. Equal to `2pisqrt((2L)/g)`

Answer

To find the time period of a pendulum with a charged bob, we can follow these steps:

### Step 1: Understand the Forces Acting on the Bob
The bob of the pendulum carries a negative charge (-q) and is influenced by the electric force due to the positive charge (+q) at the point of support. The electric force (F_e) can be calculated using Coulomb's law:

\[ F_e = k \frac{|q_1 q_2|}{r^2} \]

where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges (which is equal to the length of the pendulum, L).

### Step 2: Calculate the Electric Force
For our case, the electric force acting on the bob is:

\[ F_e = k \frac{|-q| \cdot |+q|}{L^2} = k \frac{q^2}{L^2} \]

### Step 3: Determine the Effective Gravitational Force
The gravitational force acting on the bob is given by:

\[ F_g = mg \]

where \( m \) is the mass of the bob and \( g \) is the acceleration due to gravity.

### Step 4: Find the Net Force
The total effective force acting on the bob when it is displaced from its equilibrium position will be the combination of gravitational force and electric force. The effective force can be expressed as:

\[ F_{net} = mg - F_e = mg - k \frac{q^2}{L^2} \]

### Step 5: Set Up the Equation of Motion
For small angular displacements, the pendulum exhibits simple harmonic motion (SHM). The restoring force can be related to the displacement \( x \) from the equilibrium position:

\[ F_{net} = -k_{eff} x \]

where \( k_{eff} \) is the effective spring constant. The effective spring constant can be derived from the forces acting on the pendulum.

### Step 6: Calculate the Time Period
The time period \( T \) of a simple pendulum is given by the formula:

\[ T = 2\pi \sqrt{\frac{L}{g_{eff}}} \]

where \( g_{eff} \) is the effective acceleration due to gravity, which includes the effect of the electric force.

### Step 7: Substitute the Effective Gravity
The effective gravity can be expressed as:

\[ g_{eff} = g - \frac{k q^2}{m L^2} \]

Thus, the time period becomes:

\[ T = 2\pi \sqrt{\frac{L}{g - \frac{k q^2}{m L^2}}} \]

### Final Result
The time period of the pendulum bob is:

\[ T = 2\pi \sqrt{\frac{L}{g - \frac{k q^2}{m L^2}}} \]