Problem 31 - Arihant AIEEE Physics

Question

Two small identical balls \(A\) and \(B\) lying on a horizontal smooth plane are connected by a massless spring. Ball \(A\) is fixed but ball \(B\) is free to move. When both balls are charged identically, then:(a) at the time of maximum separation between balls, magnitude of acceleration will be maximum (b) at the equilibrium position of \(B\), velocity of ball \(B\) will be maximum (c) the ball \(B\) executes simple harmonic motion (d) all of the above

Step-by-Step Solution

Answer

All the statements are correct; therefore, (d) "all of the above" is true.

Step 1: Understanding the System

Consider two balls, A and B, connected by a spring on a smooth plane. Ball A is fixed, and Ball B is free to move. When both balls are charged, there are two main forces acting on Ball B: the spring force and the electrostatic force due to the charges.

Step 2: Analyzing Forces at Maximum Separation

At maximum separation, the spring force that tries to pull Ball B back is at its maximum. According to Hooke's Law, the force is given by \( F = -kx \), where \( k \) is the spring constant and \( x \) is the displacement. The acceleration is maximum because acceleration is proportional to force (\( a = \frac{F}{m} \)). Thus, statement (a) is true.

Step 3: Analyzing Velocity at Equilibrium

At the equilibrium position, the spring is neither compressed nor stretched, meaning the net force on Ball B is zero. Ball B moves under the influence of any initial kinetic energy, which is maximum at this point, so the velocity of Ball B is maximum. This makes statement (b) true.

Step 4: Evaluating Simple Harmonic Motion

When Ball B moves, it undergoes periodic motion influenced by the restoring force of the spring. The system can be described by the equation for simple harmonic motion: \( a = -\omega^2 x \), where \( \omega \) is the angular frequency. Therefore, Ball B executes simple harmonic motion, confirming statement (c) is true.

Step 5: Conclusion

All the statements are verified to be correct within the context of the problem. Therefore, option (d) "all of the above" is the correct answer.

Key Concepts

Hooke's Law

Hooke's Law is a principle that describes how springs exert force. It provides an understanding of how springs stretch or compress in response to an applied force. The law is encapsulated in the equation \( F = -kx \), where \( F \) represents the force applied by the spring, \( k \) is the spring constant, and \( x \) is the displacement from the spring's original position.
The negative sign indicates that the force exerted by the spring is always opposite to the displacement - it's a restoring force, meaning it always acts to bring the system back to its equilibrium position.
Some essential points about Hooke's Law include:

• It is linear, meaning the force is directly proportional to the displacement.
• The spring constant \( k \) is a measure of the stiffness of the spring; higher \( k \) values mean a stiffer spring.
• Hooke's Law forms the basis for understanding simple harmonic motion in springs.

When applied to the problem, this law helps explain why the force—and therefore the acceleration—is greatest when the separation between balls is at its maximum. The spring is stretched to its largest extent, making Hooke's Law particularly relevant here.

Electrostatic Force

Electrostatic force is the force between two charged objects. This kind of force can be either attractive or repulsive, depending on the nature of the charges involved. When both objects have like charges, such as both being positively or negatively charged, the force is repulsive.
The equation governing electrostatic force is Coulomb's Law: \( F = k_e \frac{|q_1 q_2|}{r^2} \), where \( k_e \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the centers of the two charges.
The key aspects of electrostatic force include:

• It acts along the line joining the centers of the two charges.
• It is inversely proportional to the square of the distance between the charges, meaning it decreases quickly as distance increases.
• For two like-charged objects, like the balls in our problem, the electrostatic force pushes them apart.

Understanding this force helps in visualizing how the charged balls influence each other's positions and consequently, the behavior of the spring connecting them.

Equilibrium Position

The equilibrium position is a vital concept in physics, particularly in the context of oscillating systems like a spring. It is the position where net forces acting on the system are balanced, meaning there's no initial tendency for the object to move in any direction.
In the scenario of the balls connected by a spring, the equilibrium position is where the spring exerts no force because it is neither compressed nor stretched—hence, no net spring force.
Let's highlight important points about the equilibrium position:

• It is the natural, unforced length of the spring when no net external forces are acting.
• At this position, and if no external forces act besides initial kinetic energy, motion is purely due to momentum.
• The velocity of the object is at its maximum at the equilibrium position, as there is no spring force slowing it down.

This position is crucial because it signifies where kinetic energy is fully converted from potential energy, accounting for the maximum speed of Ball B in our exercise.