At the maximum positive displacement (x = x_max), which is true?

At the maximum positive displacement (x = x_max), which is true?

• A. Restoring force = max; acceleration = max; Potential energy = max; kinetic energy = 0; speed = 0
• B. Restoring force = 0
• C. Potential energy = max
• D. Kinetic energy = max

Answer: (A)

At the turning point, the mass is at its farthest displacement and momentarily at rest, so velocity and kinetic energy are zero. The restoring force is proportional to displacement and points toward equilibrium, so its magnitude is largest there. For a spring, potential energy is (1/2) k x^2, which means the potential energy is also at its maximum when x is at x_max. Since acceleration is F/m and the force is at its maximum in magnitude (a = -(k/m) x), the acceleration magnitude is likewise maximal at that point. So the full, true description at x = x_max is that the restoring force is at its maximum, the acceleration is at its maximum, the potential energy is maximum, and the kinetic energy (and speed) are zero. The other statements are incomplete or false: the force isn’t zero, kinetic energy isn’t max (it’s zero), and while potential energy is indeed max, that alone doesn’t capture the whole situation.

At the turning point, the mass is at its farthest displacement and momentarily at rest, so velocity and kinetic energy are zero. The restoring force is proportional to displacement and points toward equilibrium, so its magnitude is largest there. For a spring, potential energy is (1/2) k x^2, which means the potential energy is also at its maximum when x is at x_max. Since acceleration is F/m and the force is at its maximum in magnitude (a = -(k/m) x), the acceleration magnitude is likewise maximal at that point. So the full, true description at x = x_max is that the restoring force is at its maximum, the acceleration is at its maximum, the potential energy is maximum, and the kinetic energy (and speed) are zero. The other statements are incomplete or false: the force isn’t zero, kinetic energy isn’t max (it’s zero), and while potential energy is indeed max, that alone doesn’t capture the whole situation.