Explain why the total mechanical energy of a ball undergoing simple harmonic motion on a frictionless horizontal surface, attached to a spring fixed at one end, remains constant even though both its kinetic energy and elastic potential energy change continuously.

IB DP Physics Higher Level (2023 syllabus) — A.3 Work, energy and power · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

A ball of mass m is attached to a horizontal spring fixed to a wall. The ball oscillates back and forth on a frictionless surface with amplitude A, undergoing simple harmonic motion.

Model answer (4 marks)

The total mechanical energy is the sum of the kinetic energy (½mv²) and the elastic potential energy (½kA²). On a frictionless surface no non‑conservative forces act, so energy cannot be lost. When the ball moves away from the equilibrium position the spring stretches, doing negative work on the ball; this reduces the ball’s kinetic energy while the work is stored as elastic potential energy. When the ball moves back toward equilibrium the spring contracts, doing positive work on the ball; the stored elastic potential energy is converted back into kinetic energy by the same amount. Because the decrease in kinetic energy equals the increase in elastic potential energy at every instant, the sum of the two energies – the total mechanical energy – remains constant throughout the motion.

Examiner tips

  • Use the definition of total mechanical energy and state no dissipative forces. Explain the sign of the work done by the spring in each half‑cycle. Show that the energy changes cancel, giving a constant total.
  • common_mistakes
  • :
  • Forgetting to mention that no friction or other non‑conservative forces act. Confusing the direction of work done by the spring with the sign of the energy change. Failing to state that the energy changes are equal and opposite at each instant.

Mark scheme (4 marks)

  1. The total mechanical energy is the sum of kinetic energy and elastic potential energy, and no non-conservative (dissipative) forces act on the system.
  2. As the ball moves away from the equilibrium position, the spring does negative work on the ball, decreasing its kinetic energy while increasing the elastic potential energy stored in the spring.
  3. As the ball moves back towards the equilibrium position, the spring does positive work on the ball, converting elastic potential energy back into kinetic energy by the same amount.
  4. Since the decrease in kinetic energy exactly equals the increase in elastic potential energy at every instant (and vice versa), the total mechanical energy remains constant throughout the motion.

Key terms in this question

total mechanical energy · elastic potential energy · kinetic energy · simple harmonic motion

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