Problem 21 - Arihant AIEEE Physics

Question

Two identical pendulums \(A\) and \(B\) are suspended from the same point. The bobs are given positive charges, with \(A\) having more charge than \(B\). They diverge and reach at equilibrium, with \(A\) and \(B\) making angles \(\theta_{1}\) and \(\theta_{2}\) with the vertical respectively : (a) \(\theta_{1}>\theta_{2}\) (b) \(\theta_{1}<\theta_{2}\) (c) \(\theta_{1}=\theta_{2}\) (d) the tension in \(A\) is greater than that in \(B\)

Step-by-Step Solution

Answer

The correct options are (a) \( \theta_1 > \theta_2 \) and (d) the tension in \( A \) is greater than that in \( B \).

Step 1: Understanding the Problem

We have two pendulums with charged bobs that diverge due to repulsion and reach a state of equilibrium. We need to determine which condition about the angles and tensions between these pendulums is correct.

Step 2: Analyzing the Forces on Each Bob

At equilibrium, the forces acting on each bob are gravity (acting vertically downward), tension (acting along the pendulum string), and electrostatic repulsion between the charges. Since bob \( A \) has more charge, it experiences a greater repulsive force than bob \( B \).

Step 3: Comparing the Angles

With bob \( A \) having more charge, it will be pushed further away due to stronger repulsion, causing it to make a larger angle \( \theta_1 \) with the vertical than bob \( B \), which makes angle \( \theta_2 \). Thus, \( \theta_1 > \theta_2 \).

Step 4: Evaluating Tensions in the Strings

The tension in each string must balance the components of gravitational force and electrostatic force. Since bob \( A \) is at a larger angle, its tension has to counteract both a larger component of gravitational force parallel to the pendulum and the increased electrostatic force. Therefore, the tension in pendulum \( A \) is greater than that in pendulum \( B \).

Step 5: Conclusion

Based on the analysis, both \( \theta_1 > \theta_2 \) and tension in \( A \) being greater than in \( B \) are true. Therefore, correct options are (a) \( \theta_1 > \theta_2 \) and (d) the tension in \( A \) is greater than that in \( B \).

Key Concepts

Electrostatic Forces

When dealing with electrostatics in pendulums, a key concept is the electrostatic force, which acts between charged objects. In our scenario, the pendulum bobs are given positive charges. Due to like charges repelling each other, these bobs experience a force pushing them apart.
This force is described by Coulomb's law, which states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it is given by \[ F = k \frac{|q_1 \, q_2|}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges
• \( r \) is the distance between the centers of the two charges
• \( k \) is Coulomb's constant

Understanding these principles helps explain why pendulum A, with a greater charge, experiences a stronger repulsive force compared to pendulum B. This difference is crucial as it influences how each bob swings away from the vertical and plays a role in determining the tension in the strings.

Equilibrium Conditions

Equilibrium for pendulums involves a balance of forces resulting in no net motion. In physics, equilibrium is reached when all the forces acting on an object sum to zero. For the pendulum bobs, they are in equilibrium when the pulling force of gravity, the pushing electrostatic force, and the tension in the string are balanced.
The gravitational force acts downward toward the Earth and is opposed by the tension in the string, which pulls upward and along the string's length. Simultaneously, the electrostatic force, due to charge repulsion, pushes the bobs apart, making them swing outwards to form angles \( \theta_1 \) and \( \theta_2 \) with the vertical.
Since bob A has more charge, it experiences a stronger electrostatic force and swings out further, resulting in \( \theta_1 > \theta_2 \). This visualization of forces and the understanding of equilibrium conditions provide insights into why pendulum bobs do not keep moving indefinitely but instead stabilize at certain angles.

Tension in Strings

In the context of pendulums, tension is crucial because it provides the necessary force to balance both gravitational and electrostatic influences on the bob.
Tension in the string arises due to its role in maintaining equilibrium. For pendulum bob A, which swings at a larger angle due to its greater charge, the string must handle a larger component of gravitational force pulling downward and a stronger electrostatic force pushing outward.
The tension force can be decomposed into:

• A vertical component, which balances the gravitational force \((mg)\)
• A horizontal component, which balances the electrostatic repulsion force

Given these components, the tension \((T)\) in the string of bob A must be greater than that in bob B to counteract the extensive forces acting upon it.Overall, understanding tension and its balance within these forces provides a clearer picture of the dynamics at play in electrostatic pendulums, explaining the relationships between angles, forces, and tensions.