### Correct Option: A. The restoring force is directly proportional to the maximum displacement. #### Detailed Explanation: In simple harmonic motion (SHM), the motion of an object is characterized by its oscillation around an equilibrium position. The key feature of SHM is that the restoring force acting on the object is always directed towards the equilibrium position and is proportional to the displacement from that position. 1. **Understanding Restoring Force**: - The restoring force (\(F\)) in SHM can be described by Hooke's Law, which states that the force exerted by a spring (or any elastic object) is proportional to the displacement (\(x\)) from its equilibrium position. Mathematically, this is expressed as: \[ F = -kx \] where: - \(F\) is the restoring force, - \(k\) is the spring constant (a measure of the stiffness of the spring), - \(x\) is the displacement from the equilibrium position (positive when the object is stretched and negative when compressed). 2. **Maximum Displacement**: - The maximum displacement from the equilibrium position in SHM is referred to as the amplitude (\(A\)). At this point, the restoring force is at its maximum because the displacement is at its maximum. 3. **Proportional Relationship**: - According to Hooke's Law, when the displacement increases, the restoring force also increases proportionally. This means that if you double the maximum displacement (amplitude), the restoring force will also double. This direct proportionality is a fundamental characteristic of SHM. 4. **Direction of the Force**: - The negative sign in Hooke's Law indicates that the restoring force acts in the opposite direction to the displacement. This is crucial because it ensures that the object will always be pulled back towards the equilibrium position, leading to oscillatory motion. #### Why Other Options Are Incorrect: - **Option B: The restoring force is inversely proportional to the maximum displacement.** - This option is incorrect because it suggests that as the displacement increases, the restoring force decreases, which contradicts the principle of SHM. In reality, the restoring force increases with increasing displacement. - **Option C: The restoring force is independent of the maximum displacement.** - This option is also incorrect. If the restoring force were independent of displacement, the object would not exhibit oscillatory motion. The very nature of SHM relies on the restoring force being dependent on the displacement. - **Option D: The restoring force is equal to the maximum displacement squared.** - This option is incorrect because it implies a quadratic relationship between the restoring force and displacement. In SHM, the relationship is linear (proportional), not quadratic. The force does not increase with the square of the displacement. ### Summary: - In simple harmonic motion, the restoring force is directly proportional to the displacement from the equilibrium position. - This relationship is described by Hooke's Law: \(F = -kx\). - The maximum restoring force occurs at maximum displacement (amplitude). - Understanding this relationship is crucial for analyzing oscillatory systems in physics. By grasping these concepts, students can better understand the dynamics of simple harmonic motion and apply this knowledge to various problems in physics.