A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its potential energy is a maximum?

Question

BlackTom AI · Accepted Answer

For a mass on a vertical spring the relevant potentials are elastic (spring) potential and gravitational potential. Use the equilibrium extension x0 found from k x0 = mg and measure the small oscillation displacement y from that equilibrium. This simplifies the algebra and shows how the two potentials combine.

Step-by-step derivation:
1. Let x be total extension from the spring's natural length. Equilibrium extension x0 satisfies k x0 = mg.
2. If the mass is displaced by y measured from equilibrium (positive downward), then x = x0 + y.
3. Spring potential: Us = (1/2) k x^2 = (1/2) k (x0 + y)^2.
   Change relative to equilibrium: ΔUs = (1/2) k[(x0 + y)^2 − x0^2] = (1/2) k (2 x0 y + y^2) = k x0 y + (1/2) k y^2.
4. Gravitational potential change for displacement y downward: ΔUg = − m g y.
5. Total potential change relative to equilibrium: ΔU = ΔUs + ΔUg = (k x0 y + (1/2) k y^2) − m g y.
6. Substitute k x0 = m g: the linear terms cancel, leaving ΔU = (1/2) k y^2.

Thus the total potential energy measured from equilibrium is U(y) = constant + (1/2) k y^2, a quadratic with minimum at y = 0 (the equilibrium). The maximum potential energy during the motion occurs at the largest magnitude of y (the turning points), where the kinetic energy is zero and all mechanical energy is stored as potential: Umax = (1/2) k A^2 (A = amplitude).

Why the correct answer is “at either A or B”: Points A and B are the turning points of the bob (the endpoints of its motion). At those points the displacement y = ±A is largest and the total potential is maximal. The midpoint between A and B is y = 0, where potential is minimum and kinetic energy is maximum, so “midway” is incorrect. “One-fourth of the way” is an intermediate position with neither maximum potential nor zero kinetic energy. “None of these” is also incorrect.

Therefore the potential energy is maximum at either A or B, matching the given correct answer.

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its potential energy is a maximum?单项选择题

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