### Correct Option: B. Acceleration is directly proportional to displacement and in the opposite direction. #### Detailed Explanation: In simple harmonic motion (SHM), the motion of an object is characterized by its oscillation about an equilibrium position. A classic example of SHM is a mass attached to a spring or a pendulum swinging back and forth. The key features of SHM include the relationship between displacement, velocity, and acceleration. 1. **Understanding Displacement**: - Displacement (denoted as \( x \)) is the distance and direction from the equilibrium position. In SHM, when the object is displaced from its equilibrium position, it experiences a restoring force that acts to bring it back to that position. 2. **Acceleration in SHM**: - The acceleration (\( a \)) of an object in SHM is defined as the rate of change of velocity. According to Hooke's Law, the restoring force (\( F \)) acting on the object is proportional to the displacement from the equilibrium position: \[ F = -kx \] where \( k \) is the spring constant (a measure of the stiffness of the spring) and \( x \) is the displacement. The negative sign indicates that the force acts in the opposite direction to the displacement. 3. **Newton's Second Law**: - According to Newton's second law, the acceleration of the object can be expressed as: \[ F = ma \] where \( m \) is the mass of the object. By substituting Hooke's Law into Newton's second law, we get: \[ ma = -kx \] Rearranging this gives: \[ a = -\frac{k}{m} x \] - This equation shows that the acceleration \( a \) is directly proportional to the displacement \( x \) but in the opposite direction (indicated by the negative sign). Thus, when the object is displaced to the right (positive \( x \)), the acceleration acts to the left (negative \( a \)), and vice versa. 4. **Conclusion**: - Therefore, the correct statement regarding the relationship between acceleration and displacement in simple harmonic motion is that acceleration is directly proportional to displacement and in the opposite direction. This is why option B is the correct answer. #### Why Other Options Are Incorrect: - **Option A**: Acceleration is directly proportional to displacement and in the same direction. - This is incorrect because, in SHM, the acceleration acts to restore the object to the equilibrium position, meaning it always acts in the opposite direction to the displacement. - **Option C**: Acceleration is inversely proportional to displacement. - This is incorrect because, in SHM, acceleration is directly proportional to displacement, not inversely. An inverse relationship would imply that as displacement increases, acceleration decreases, which contradicts the behavior observed in SHM. - **Option D**: Acceleration is independent of displacement. - This is incorrect because, in SHM, acceleration is explicitly dependent on displacement. The acceleration changes as the displacement changes, which is a fundamental characteristic of SHM. ### Revision Summary: - In simple harmonic motion, acceleration is directly proportional to displacement. - The acceleration acts in the opposite direction to the displacement from the equilibrium position. - The relationship is mathematically expressed as \( a = -\frac{k}{m} x \). - Understanding the direction of forces and accelerations is crucial in analyzing SHM.