Energy of a simple harmonic oscillator — OCR A-Level Physics
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Energy of a simple harmonic oscillator explained
In an undamped simple harmonic oscillator, total energy is conserved while kinetic and potential energy interchange continuously.
Read the full explanation
Kinetic energy is maximum at the equilibrium position, where speed is greatest, and zero at maximum displacement, where the oscillator momentarily stops. Potential energy is zero at equilibrium and maximum at maximum displacement. For a mass–spring system, potential energy is elastic, stored in the spring; for a pendulum, it is gravitational. The total energy E = ½kA² = ½mω²A² remains constant, where A is amplitude. At any displacement x, kinetic energy is ½mω²(A² − x²) and potential energy is ½mω²x², so their sum is constant.
(b) energy-displacement graphs for a simple harmonic oscillator
Energy–displacement graphs show how kinetic and potential energy vary with displacement x for a simple harmonic oscillator. The potential energy curve is a parabola, Ep = ½mω²x², with a minimum of zero at x = 0 and maximum ½mω²A² at x = ±A. The kinetic energy curve is an inverted parabola, Ek = ½mω²(A² − x²), maximum at x = 0 and zero at x = ±A. Adding the two curves at every displacement gives a horizontal straight line at the constant total energy ½mω²A². The graphs are symmetric about x = 0 because displacement can be positive or negative. Reading the graphs shows that energy interchange is continuous and that the total energy depends on amplitude squared.
Your focus
- Describe how kinetic and potential energy interchange during simple harmonic motion.
- Calculate kinetic and potential energy at a given displacement using ½mω²(A² − x²) and ½mω²x².
- Explain why total energy remains constant in an undamped oscillator.
Show all 6 objectives
- Sketch and interpret energy–displacement graphs for kinetic and potential energy.
- Explain why the potential energy curve is parabolic and the total energy line is horizontal.
- Use energy–displacement graphs to determine amplitude, total energy and angular frequency.
Energy of a simple harmonic oscillator exam tips
Marking Points
- Total energy of an undamped simple harmonic oscillator is constant and equals ½mω²A².
- Kinetic energy is maximum at equilibrium (x = 0) and zero at maximum displacement (x = ±A).
- Potential energy is zero at equilibrium and maximum at maximum displacement.
- Kinetic energy at displacement x is ½mω²(A² − x²) and potential energy is ½mω²x².
- The interchange is continuous, with kinetic energy converting to potential energy and back each cycle.
- For a mass–spring oscillator the potential energy is elastic; for a pendulum it is gravitational.
- Potential energy varies as Ep = ½mω²x², giving a parabola with minimum zero at x = 0.
- Kinetic energy varies as Ek = ½mω²(A² − x²), giving an inverted parabola with maximum at x = 0.
- At x = ±A, kinetic energy is zero and potential energy equals the total energy ½mω²A².
- The sum of kinetic and potential energy at every displacement is the constant total energy ½mω²A².
- The graphs are symmetric about x = 0 and show that total energy is proportional to A².
- Graphical features such as intercepts, maxima and the constant-sum line can be used to find ω, A or energy values.
Examiner Tips
- 💡Write the total energy as ½mω²A² and subtract the potential energy to find kinetic energy at any displacement.
- 💡Check the position first: equilibrium means maximum kinetic energy, extremes mean maximum potential energy.
- 💡Keep amplitude A distinct from displacement x in every expression.
- 💡Label the axes with energy and displacement, and mark the key values 0 and ±A clearly.
- 💡Draw kinetic and potential energy on the same axes, then add them to show the constant total.
- 💡Use the intercepts and maxima to read off A and the total energy rather than recalculating from scratch.
Common Mistakes
- Thinking total energy changes during the cycle: total energy is constant when damping is negligible.
- Believing kinetic energy is maximum at maximum displacement: speed is zero there, so kinetic energy is zero.
- Assuming potential energy is maximum at equilibrium: potential energy is zero at equilibrium and maximum at the extremes.
- Using E = ½mv² with v as the maximum speed at all positions instead of using the displacement-dependent expression.
- Drawing the potential energy graph as a straight line: it is a parabola because Ep depends on x².
- Showing the total energy curve varying with displacement: the total is constant, so it is a horizontal line.
- Placing the kinetic energy maximum at x = ±A: kinetic energy is maximum at x = 0.
- Forgetting that the graphs are symmetric for negative displacement as well as positive.