**Correct Option: B. The acceleration is directly proportional to the displacement and always directed towards the equilibrium position.** ### Detailed Explanation: In simple harmonic motion (SHM), an object oscillates back and forth around an equilibrium position. The key characteristics of SHM involve the relationships between displacement, velocity, and acceleration. Let's break down the correct answer and why it is true. 1. **Understanding Displacement, Velocity, and Acceleration:** - **Displacement (x)**: This is the distance from the equilibrium position. It can be positive or negative depending on the direction from the equilibrium. - **Velocity (v)**: This is the rate of change of displacement. It indicates how fast and in which direction the object is moving. - **Acceleration (a)**: This is the rate of change of velocity. In SHM, acceleration is related to displacement. 2. **The Relationship in SHM:** - In SHM, the acceleration of the object is given by the formula: \[ a = -\omega^2 x \] where \( \omega \) is the angular frequency, and \( x \) is the displacement from the equilibrium position. The negative sign indicates that the acceleration is always directed towards the equilibrium position, opposing the displacement. 3. **Why Option B is Correct:** - The statement in Option B accurately reflects the nature of SHM. The acceleration is indeed directly proportional to the displacement (the larger the displacement, the larger the acceleration) and is always directed towards the equilibrium position (hence the negative sign in the formula). This means that if the object is displaced to the right (positive displacement), the acceleration will act to the left (negative direction), pulling it back towards the center. ### Why the Other Options are Incorrect: - **Option A: The velocity is always in the same direction as the displacement.** - This statement is false. In SHM, when the object is at maximum displacement (either positive or negative), its velocity is zero as it changes direction. Therefore, at these points, the velocity is not in the same direction as the displacement. - **Option C: The velocity is maximum when the displacement is maximum.** - This is also incorrect. In SHM, the velocity is maximum when the displacement is zero (at the equilibrium position). At maximum displacement, the object momentarily stops before reversing direction, resulting in zero velocity. - **Option D: The acceleration and velocity are always in the same direction.** - This statement is misleading. In SHM, acceleration and velocity are not always in the same direction. When the object is moving towards the equilibrium position, the velocity is in one direction while the acceleration is in the opposite direction (towards the equilibrium). This is a key characteristic of SHM. ### Summary of Key Points: - In SHM, acceleration is directly proportional to displacement and always directed towards the equilibrium position. - Velocity is maximum at the equilibrium position, not at maximum displacement. - Displacement and velocity can be in opposite directions at certain points in the motion. - Understanding the relationships between these quantities is crucial for analyzing simple harmonic motion. This thorough understanding of SHM will help you tackle related problems and questions effectively in your exams.