Ellen says that whenever the acceleration is directly proportional to the displacement of an object from its equilibrium position, the motion of the object is simple harmonic motion. Mary says this is true only if the acceleration is opposite in direction to the displacement. Which one, if either, is correct? A) Ellen, because ω² is directly proportional to the constant multiplying the displacement and to the mass. B) Ellen, because ω² is directly proportional to the mass. C) Mary, because ω² is directly proportional to the constant multiplying the displacement and to the mass. D) Mary, because ω² is directly proportional to the mass. E) Mary, because the second derivative of an oscillatory function like sin(ωt) or cos(ωt) is always proportional to the negative of the original function.

Question

Quizplus · Accepted Answer

Mary is correct because for simple harmonic motion, the acceleration (which is the second derivative of displacement with respect to time) must be directly proportional to the displacement and in the opposite direction. This is mathematically represented in the behavior of functions like sin(ωt) or cos(ωt), where the second derivative is indeed proportional to the negative of the original function, indicating the acceleration is opposite in direction to the displacement.

Ellen Says That Whenever the Acceleration Is Directly Proportional to the Displacement