**Correct Option: B. Displacement; opposite direction** ### Detailed Explanation: In simple harmonic motion (SHM), the key characteristics revolve around how an object moves back and forth around an equilibrium position. The fundamental relationship in SHM is that the acceleration of the object is directly proportional to its displacement from the equilibrium position. 1. **Understanding Displacement and Acceleration**: - Displacement (denoted as \( x \)) is the distance and direction from the equilibrium position. In SHM, when the object is displaced from this position, it experiences a restoring force that pulls it back toward the equilibrium. - According to Hooke's Law, the restoring force \( F \) is proportional to the displacement \( x \) and is given by the formula: \[ F = -kx \] where \( k \) is the spring constant (a measure of stiffness). The negative sign indicates that the force acts in the opposite direction to the displacement. 2. **Relating Force to Acceleration**: - According to Newton's second law, the force acting on an object is also related to its mass \( m \) and acceleration \( a \): \[ F = ma \] - By combining these two equations, we can express acceleration in terms of displacement: \[ ma = -kx \implies a = -\frac{k}{m}x \] - This shows that the acceleration \( a \) is directly proportional to the displacement \( x \) and acts in the opposite direction (indicated by the negative sign). 3. **Direction of Acceleration**: - When the object is displaced to the right (positive \( x \)), the acceleration is negative, meaning it acts to the left (toward the equilibrium). Conversely, if the object is displaced to the left (negative \( x \)), the acceleration is positive, acting to the right. This consistent behavior reinforces that the acceleration always acts in the opposite direction to the displacement. ### Why Other Options Are Incorrect: - **Option A: Velocity; opposite direction**: - This option is incorrect because acceleration is not directly proportional to velocity in SHM. Instead, acceleration is proportional to displacement. While velocity does change direction, it does not determine the acceleration's direction in SHM. - **Option C: Mass; same direction**: - This option is misleading. While mass is a factor in determining the acceleration (as seen in \( F = ma \)), it is not the factor that is directly proportional to acceleration in SHM. Moreover, acceleration does not act in the same direction as mass; rather, it is dependent on displacement. - **Option D: Frequency; same direction**: - Frequency relates to how often the motion occurs but does not influence the acceleration directly. The acceleration in SHM is not proportional to frequency; instead, it is determined by the displacement from the equilibrium position. ### Summary of Key Points: - In simple harmonic motion, acceleration is directly proportional to displacement from the equilibrium position. - The acceleration acts in the opposite direction to the displacement, ensuring the object returns to equilibrium. - The relationship can be mathematically expressed as \( a = -\frac{k}{m}x \). - Understanding the relationship between displacement and acceleration is crucial for analyzing SHM behavior. This thorough understanding of SHM will help you tackle related problems and concepts effectively in your studies!