The correct option is **B. The restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.** ### Detailed Explanation: 1. **Understanding Simple Harmonic Motion (SHM)**: - Simple Harmonic Motion is a type of periodic motion where an object oscillates back and forth around an equilibrium position. The key characteristic of SHM is that the motion is sinusoidal in time and demonstrates a restoring force that is proportional to the displacement from the equilibrium position. 2. **Restoring Force**: - The restoring force is the force that brings the oscillating object back towards its equilibrium position. In SHM, this force is crucial because it determines how the object behaves as it moves away from equilibrium. 3. **Mathematical Representation**: - The restoring force \( F \) can be expressed mathematically as: \[ F = -kx \] where: - \( F \) is the restoring force, - \( k \) is the spring constant (a measure of stiffness), - \( x \) is the displacement from the equilibrium position. - The negative sign indicates that the force acts in the opposite direction to the displacement. If the object is displaced to the right (positive \( x \)), the restoring force acts to the left (negative \( F \)), and vice versa. 4. **Why Option B is Correct**: - Option B accurately describes the relationship between the restoring force and displacement. It states that the restoring force is directly proportional to the displacement and acts in the opposite direction, which aligns perfectly with the mathematical model of SHM. ### Why the Other Options are Incorrect: - **Option A: The restoring force is always directed away from the equilibrium position.** - This statement is incorrect because the restoring force always acts towards the equilibrium position, not away from it. If the object is displaced to the right, the restoring force pulls it back to the left, towards equilibrium. - **Option C: The restoring force is constant regardless of the displacement from the equilibrium position.** - This is false because the restoring force varies with displacement. According to Hooke's Law, the force changes linearly with displacement. If the displacement increases, the restoring force increases proportionally. - **Option D: The restoring force is independent of mass and only depends on the amplitude of oscillation.** - This statement is misleading. While the restoring force does not depend on mass directly, it is influenced by the spring constant \( k \) and the displacement \( x \). The amplitude of oscillation does not affect the restoring force directly; rather, it affects the maximum displacement. The restoring force is fundamentally related to the displacement from equilibrium, not just the amplitude. ### Common Pitfalls: - Students often confuse the direction of the restoring force, thinking it pushes the object away from equilibrium rather than pulling it back. - Misunderstanding the relationship between force and displacement can lead to incorrect conclusions about the nature of SHM. ### Revision Summary: - In SHM, the restoring force is proportional to the displacement from equilibrium and acts in the opposite direction. - The mathematical expression for the restoring force is \( F = -kx \). - The restoring force is not constant; it varies with displacement. - The restoring force is not solely dependent on amplitude but on the displacement from equilibrium.