Explain why the total mechanical energy of an object undergoing simple harmonic motion remains constant, and describe how the kinetic and potential energies vary with displacement from the equilibrium position.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
An object of mass m oscillates in simple harmonic motion with amplitude A and angular frequency ω.
Model answer (4 marks)
In ideal SHM no dissipative forces act, so energy cannot be lost to the surroundings and the total mechanical energy is conserved.
The kinetic energy is
\[K=\tfrac12 m v^2=\tfrac12 m(\omega A\cos\omega t)^2=\tfrac12 m\omega^2A^2\cos^2\omega t\]
and is maximum when the displacement is zero (equilibrium) and zero when the displacement is ±A.
The potential energy is
\[U=\tfrac12 kx^2=\tfrac12 m\omega^2(A\sin\omega t)^2=\tfrac12 m\omega^2A^2\sin^2\omega t\]
and is maximum at ±A and zero at the equilibrium position.
At any instant the two energies are complementary: as one increases the other decreases by the same amount, so
\[E=K+U=\tfrac12 m\omega^2A^2\] is constant.
The kinetic energy is
\[K=\tfrac12 m v^2=\tfrac12 m(\omega A\cos\omega t)^2=\tfrac12 m\omega^2A^2\cos^2\omega t\]
and is maximum when the displacement is zero (equilibrium) and zero when the displacement is ±A.
The potential energy is
\[U=\tfrac12 kx^2=\tfrac12 m\omega^2(A\sin\omega t)^2=\tfrac12 m\omega^2A^2\sin^2\omega t\]
and is maximum at ±A and zero at the equilibrium position.
At any instant the two energies are complementary: as one increases the other decreases by the same amount, so
\[E=K+U=\tfrac12 m\omega^2A^2\] is constant.
Examiner tips
- Mention that no friction or air resistance means no energy loss; use the formula for total energy ½mω²A²; show the complementary nature of K and U; keep the answer concise and use correct symbols.
Common mistakes
- Confusing displacement with velocity; writing K∝x² instead of v²; forgetting that total energy is constant and giving a non‑constant expression.
Mark scheme (4 marks)
- No dissipative (resistive/frictional) forces act in ideal SHM, so no energy is lost to the surroundings, meaning total mechanical energy is conserved.
- Kinetic energy is maximum at the equilibrium position (zero displacement) and zero at maximum displacement (amplitude).
- Potential energy is maximum at maximum displacement (amplitude) and zero at the equilibrium position.
- At any displacement, kinetic and potential energies are complementary — as one increases the other decreases by the same amount — so their sum (total mechanical energy = ½mω²A²) remains constant.
Key terms in this question
simple harmonic motion · equilibrium position · total mechanical energy
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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