Question

A simple pendulum of period \(\mathrm{T}\) has a metal bob which is negatively charged. If it is allowed to oscillate above a positively charged metal plate, its period will ...... (A) Remains equal to \(\mathrm{T}\) (B) Less than \(\mathrm{T}\) (C) Infinite (D) Greater than \(\mathrm{T}\)

Short answer

The period of the simple pendulum with a negatively charged bob oscillating above a positively charged metal plate will be less than T (Option B). This is because the presence of an electrostatic force between the bob and the plate affects the motion of the pendulum, leading to reduced acceleration during the downswing and increased acceleration during the upswing, causing the period to decrease from the original period T.

Step-by-step solution

Understand the forces acting on the bob

The main forces acting on a simple pendulum are gravity and tension in the string. When the bob is negatively charged and oscillating above a positively charged metal plate, another force is introduced - the electrostatic force between the bob and the metal plate.

Analyze the effect of the electrostatic force on the pendulum's motion

An electrostatic force between the bob and the metal plate affects both the upswing and downswing of the pendulum. When the pendulum moves towards the plate, the electrostatic force opposes gravity, thus having a net force less than gravitational force. On the other hand, when the pendulum moves away from the plate, the electrostatic force aids gravity, causing a net force greater than gravitational force. Overall, since the motion of the pendulum is affected by the electrostatic force, its period will change.

Determine how the period is affected

Given this additional upward force on the pendulum's bob due to electrostatic attraction, the bob experiences a diminished acceleration during the downswing (since the electrostatic force is opposing gravity). Moreover, during the upswing, it experiences an increased acceleration (since the electrostatic force is aiding gravity). Therefore, the time for each half of the oscillation is changed, as the bob spends more time near the charged plate. Since the acceleration during the downswing is reduced and the acceleration during the upswing is increased (on average), the period of the pendulum is expected to decrease from the original period T.

Select the correct answer

Based on our analysis, the correct answer is: (B) Less than \(\mathrm{T}\) As the presence of an electrostatic force between the negatively charged bob and positively charged plate results in a decrease in the time required to complete one full oscillation, causing the period to be less than T.