Problem 78 - Arihant AIEEE Physics

Question

A pendulum bob of mass ' \(m^{\prime}\) and charge ' \(q^{\prime}\) is suspended by a thread of length \(l\). The pendulum is placed in a region of a uniform electric field \(E\) directed vertically upward. If the electrostatic force acting on the sphere is less than that of gravitational force, the period with which the pendulum oscillates is : (Assume small oscillation) (a) \(T=2 \pi \sqrt{\frac{1}{g+\frac{q E}{m}}}\) (b) \(T=2 \pi \sqrt{\frac{1}{g-\frac{q E}{m}}}\) (c) \(T:=\pi \sqrt{\frac{1}{g-\frac{q E}{m}}}\) (d) \(T=\pi \sqrt{\frac{1}{g+\frac{q E}{m}}}\)

Step-by-Step Solution

Answer

The correct period of oscillation is given by \( T=2\pi \sqrt{\frac{l}{g - \frac{qE}{m}}} \), so the answer is (b).

Step 1: Understanding the Forces Acting

The pendulum bob experiences two main forces: the gravitational force, given by \( F_g = mg \), and the electrostatic force due to the electric field, given by \( F_e = qE \). Since \( F_e < F_g \), the effective gravitational acceleration is decreased by the electrostatic force’s effect.

Step 2: Calculating Effective Gravitational Acceleration

When the electrostatic force acts against gravity, the effective gravitational acceleration \( g' \) can be given by \( g' = g - \frac{qE}{m} \). This accounts for the reduction in the gravitational force perceived by the bob due to the upward electric force.

Step 3: Applying the Formula for Period of a Simple Pendulum

The period of a simple pendulum for small oscillations is given by: \[ T = 2\pi \sqrt{\frac{l}{g'}} \]Substituting \( g' \) from the previous step, it becomes:\[ T = 2\pi \sqrt{\frac{l}{g - \frac{qE}{m}}} \]

Step 4: Conclusion and Choice of Correct Answer

The correct formula for the oscillation period corresponds to the equation we derived: \( T=2\pi \sqrt{\frac{l}{g - \frac{qE}{m}}} \). Thus, the correct answer is option (b).

Key Concepts

Pendulum Motion

Pendulum motion is a fascinating example of periodic movement. A simple pendulum consists of a weight or bob attached to the end of a string or rod, which swings back and forth under the influence of gravity. In the absence of any external force, the restoring force is provided entirely by gravity. This force tries to bring the pendulum back to its equilibrium position.
However, when other forces, like an electrostatic force, interact with the pendulum, the motion becomes slightly more complex. This is because those forces can alter the effective force acting on the pendulum bob. Despite these complexities, pendulums often still trace out a regular arc, in a specific period attached to their length and the forces involved.

Electrostatic Force

Electrostatic force is a key concept in electromagnetism, describing the force between static electrically charged particles. In this exercise, the pendulum bob holds a charge.
When placed in an electric field, the bob experiences an electrostatic force that can influence its motion. The force is directly proportional to the strength of the electric field and the magnitude of the charge, expressed as \( F_e = qE \), where \( q \) is the charge and \( E \) is the electric field.
In the context of a pendulum, the electrostatic force can act against or with gravity, modifying the pendulum's motion. If it opposes gravity and is less than the gravitational force, it reduces the effective gravitational pull, altering the pendulum's oscillation dynamics.

Gravitational Force

Gravitational force is a fundamental force acting on a mass due to Earth's gravity. For a pendulum, this force pulls the pendulum bob toward the equilibrium point.
It is given by the formula \( F_g = mg \), where \( m \) is the mass of the bob and \( g \) is the acceleration due to gravity, typically \( 9.8 \, \text{m/s}^2 \). This force is responsible for the pendulum’s swinging motion.
In our scenario, the gravitational force competes with the electrostatic force. Even slight modifications in gravitational force due to external fields change the pendulum's timing significantly. The interaction between these forces needs careful balance for accurate period computations.

Oscillation Period

The oscillation period of a pendulum refers to the time it takes to complete one full swing back and forth. This period is influenced not just by the pendulum's length, but also the forces acting on it. For a basic pendulum, the formula is \( T = 2\pi \sqrt{\frac{l}{g}} \).
In this problem, the oscillation period alters because the effective gravitational force is modified by the electric field's presence. As such, the adjusted period for the pendulum becomes \( T = 2\pi \sqrt{\frac{l}{g - \frac{qE}{m}}} \), reflecting the net gravitational force's reduction.
Understanding this concept is essential for predictions about any pendulum-based systems where external forces might be at play.

Uniform Electric Field

A uniform electric field is one where the force experienced by a charged particle remains consistent, regardless of its position within the field.
In this exercise, the field is directed vertically upwards, impacting the pendulum’s motion. The uniform field exerts a constant electrostatic force on the pendulum bob throughout its movement. This uniformity makes calculations and predictions regarding the pendulum's oscillation more straightforward, as the electrostatic influence is predictable.
By affecting the effective gravitational acceleration, the uniform electric field provides a unique environment for studying dynamics of charged pendulums, offering insights into how electric forces interact with mechanical systems.