### Correct Option: A. a = -kx ### Detailed Explanation: In simple harmonic motion (SHM), the behavior of an object, such as a mass attached to a spring, can be described by a few fundamental principles. One of the key characteristics of SHM is that the acceleration of the object is directly proportional to its displacement from the equilibrium position, but in the opposite direction. This relationship can be mathematically expressed as: \[ a = -\frac{k}{m} x \] Where: - \( a \) is the acceleration of the object, - \( k \) is the spring constant (a measure of the stiffness of the spring), - \( m \) is the mass of the object, - \( x \) is the displacement from the equilibrium position. In the context of the options provided, we can simplify the equation by focusing on the proportionality aspect. The negative sign indicates that the acceleration is directed towards the equilibrium position, meaning if the object is displaced to the right (positive \( x \)), the acceleration will be to the left (negative \( a \)), and vice versa. #### Why Option A is Correct: - **Proportionality**: The equation \( a = -kx \) shows that acceleration is directly proportional to displacement \( x \) but in the opposite direction (hence the negative sign). - **Physical Interpretation**: If you stretch or compress a spring, the force exerted by the spring (and thus the acceleration of the mass) will always act to restore the mass back to its equilibrium position. This is a fundamental characteristic of SHM. ### Why the Other Options are Incorrect: **Option B: a = kx** - This option suggests that acceleration is directly proportional to displacement without the negative sign. This is incorrect because it does not account for the direction of the acceleration. In SHM, the acceleration must always act in the opposite direction to the displacement. **Option C: a = -kx²** - This option implies that acceleration is proportional to the square of the displacement. In SHM, the relationship is linear (not quadratic). The acceleration should change linearly with displacement, not as the square of the displacement. **Option D: a = kx²** - Similar to Option C, this option suggests a quadratic relationship between acceleration and displacement. This is not consistent with the principles of SHM, where the acceleration is linearly proportional to displacement. ### Common Pitfalls: - **Ignoring the Negative Sign**: Students often forget the negative sign, which is crucial for indicating the direction of acceleration. - **Confusing Linear and Non-linear Relationships**: It's important to recognize that in SHM, the relationship between acceleration and displacement is linear, not quadratic. ### Revision Summary: - In simple harmonic motion, acceleration is directly proportional to displacement from equilibrium, expressed as \( a = -kx \). - The negative sign indicates that acceleration acts in the opposite direction to displacement. - The spring constant \( k \) measures the stiffness of the spring and affects the magnitude of acceleration. - Understanding the directionality and linearity of the relationship is key to mastering SHM concepts.