At x = -x_max on a spring, which statement is true?

At x = -x_max on a spring, which statement is true?

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

Answer: (C)

At the extreme left turning point, the mass is momentarily at rest and the spring is stretched to its maximum amount, so the velocity is zero and the kinetic energy is zero. The potential energy of a spring is (1/2) k x^2, which is largest when |x| is as big as possible, so at x = -x_max the potential energy is maximum. The restoring force is F = -kx, so its magnitude at this point is k|x_max|, the largest possible, and it points toward the equilibrium (toward x = 0). The acceleration is F/m, so its magnitude is kx_max/m, also maximum and directed toward the center. Putting these together, the statement that describes maximum restoring force, maximum acceleration, maximum potential energy, and zero kinetic energy and speed at the turning point is correct.

At the extreme left turning point, the mass is momentarily at rest and the spring is stretched to its maximum amount, so the velocity is zero and the kinetic energy is zero. The potential energy of a spring is (1/2) k x^2, which is largest when |x| is as big as possible, so at x = -x_max the potential energy is maximum. The restoring force is F = -kx, so its magnitude at this point is k|x_max|, the largest possible, and it points toward the equilibrium (toward x = 0). The acceleration is F/m, so its magnitude is kx_max/m, also maximum and directed toward the center. Putting these together, the statement that describes maximum restoring force, maximum acceleration, maximum potential energy, and zero kinetic energy and speed at the turning point is correct.