SHM graphs and energy in oscillations

Displacement and velocity graphs

• A displacement-time graph of SHM is a cosine (or sine) curve. Its gradient is the velocity: zero at the peaks and troughs, greatest where the graph crosses the time axis.
• The velocity-time graph is a quarter of a cycle out of step with the displacement graph (v = -Aω sin ωt). Its gradient is the acceleration.

Acceleration graph

The acceleration-time graph (a = -Aω² cos ωt) is the displacement graph upside down and scaled by ω²: acceleration is in antiphase with displacement. Maximum displacement means maximum acceleration in the opposite direction.

Energy in SHM

For an undamped oscillator, total energy is conserved: kinetic energy and potential energy (elastic for a spring, gravitational for a pendulum) swap twice per cycle.

• At equilibrium: E_k is maximum, E_p is minimum.
• At maximum displacement: E_k = 0, E_p is maximum.

For a mass on a spring, total energy = (1/2)kA², so E_k = (1/2)k(A² - x²). In a damped oscillator, work is done against resistive forces, so the total energy and the amplitude decrease.

Worked example

A 0.25 kg mass on a spring (k = 40 N/m) oscillates with amplitude 0.050 m. Find its total energy and maximum speed.

1. Total energy = (1/2)kA² = 0.5 x 40 x 0.050² = 0.050 J
2. At equilibrium all of this is kinetic: (1/2)mv² = 0.050
3. v = sqrt(2 x 0.050 / 0.25)

Answer: 0.050 J; v_max = 0.63 m/s

Check yourself

1. A mass on a spring with k = 40 N/m oscillates with amplitude 0.050 m. Find the total energy of the oscillation in J.
   Show answer

   0.05 J. E = (1/2)kA² = 0.5 x 40 x 0.0025 = 0.050 J.

2. The mass in that oscillation (total energy 0.050 J) is 0.25 kg. Find its maximum speed in m/s, to 3 s.f.
   Show answer

   0.632 m/s. (1/2)mv² = 0.050, so v = sqrt(0.40) = 0.632 m/s.

3. For the same oscillation (k = 40 N/m, A = 0.050 m), find the kinetic energy when the displacement is 0.030 m, in J.
   Show answer

   0.032 J. E_k = (1/2)k(A² - x²) = 20 x (0.0025 - 0.0009) = 0.032 J.

4. On a displacement-time graph for SHM, where is the velocity zero?
   • Where the graph crosses the time axis
   • At the peaks and troughs, where the gradient is zero
   • Everywhere the displacement is positive
   • Only at t = 0
   Show answer

   At the peaks and troughs, where the gradient is zero. Velocity is the gradient of the displacement-time graph.

5. How does the acceleration-time graph of SHM compare with the displacement-time graph?
   • It is identical
   • It is a quarter of a cycle ahead
   • It is a straight line
   • It is in antiphase (upside down and scaled)
   Show answer

   It is in antiphase (upside down and scaled). a = -ω^2x, so a has the opposite sign to x at every moment.

6. Where is the kinetic energy of a simple pendulum greatest?
   • At the highest points of the swing
   • At the lowest point of the swing
   • Halfway up the swing
   • It is constant
   Show answer

   At the lowest point of the swing. Gravitational potential energy is smallest at the lowest point, so kinetic energy is greatest.

7. What happens to the total energy of a damped oscillator?
   • It stays constant
   • It increases
   • It changes from kinetic to potential only
   • It decreases, as work is done against resistive forces
   Show answer

   It decreases, as work is done against resistive forces. Energy is dissipated, so the amplitude falls.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)
• Pearson Edexcel IAL Physics Unit 5 (WPH15) question paper, June 2023: list of data, formulae and relationships

Last updated 7 November 2026. Spotted a mistake? Tell us.