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A2 · Practice questions · Energy in simple harmonic motion

Kinetic and potential, trading places.

Six original Cambridge-style questions on the interchange of kinetic and potential energy in simple harmonic motion and the total energy E = ½mω²x₀².

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Keep these straight

The total is fixed by the amplitude squared.

01
Calculation
[3 marks]

A mass of 0.150 kg oscillates with simple harmonic motion of amplitude 25 mm and period 0.50 s. Calculate the total energy of the oscillation and the kinetic energy of the mass when its displacement is 10 mm.

  • ω = 2π / T = 2π / 0.50 = 12.6 rad s⁻¹. ✓
  • E = ½mω²x₀² = 0.5 × 0.150 × 12.6² × 0.025² = 7.4 × 10⁻³ J. ✓
  • KE = ½mω²(x₀² − x²) = 0.5 × 0.150 × 158 × (0.025² − 0.010²) = 6.2 × 10⁻³ J, with the remaining 1.2 × 10⁻³ J potential. ✓
02
Analysis
[2 marks]

The amplitude of an undamped oscillator is reduced to one third of its original value while its frequency is unchanged. State the factor by which the total energy of the oscillation changes, and explain why the maximum speed does not change by the same factor.

  • E = ½mω²x₀² ∝ x₀², so the total energy falls to (1/3)² = 1/9 of its original value. ✓
  • The maximum speed v₀ = ωx₀ is proportional to the amplitude itself, so it falls only to one third; it is the kinetic energy ½mv₀², proportional to v₀², that falls to one ninth. ✓
03
Calculation
[3 marks]

A body oscillates with simple harmonic motion of amplitude 60 mm. Determine the displacement at which its kinetic energy is three times its potential energy, and state what fraction of the total energy is potential energy at that displacement.

  • KE = 3 × PE gives ½mω²(x₀² − x²) = 3 × ½mω²x², so x₀² = 4x². ✓
  • x = x₀ / 2 = 30 mm, on either side of the centre. ✓
  • There PE = ½mω²x² = ¼ × ½mω²x₀², so one quarter of the total energy is potential and three quarters kinetic. ✓
04
Analysis
[3 marks]

A mass on a spring oscillates with frequency f. Explain why its kinetic energy varies with time at a frequency of 2f, and state how its potential energy and its total energy vary with time.

  • The velocity is v = v₀ cos ωt, so the kinetic energy ½mv² = ½mv₀² cos² ωt, which is never negative and rises to its maximum whenever the velocity is at either peak. ✓
  • The mass passes through the centre twice in each oscillation, once in each direction, so the kinetic energy peaks twice per period: it repeats every T/2, a frequency of 2f. ✓
  • The potential energy ½mω²x² also varies at 2f, in antiphase with the kinetic energy, while the total energy stays constant. ✓
05
Calculation
[3 marks]

A trolley of mass 0.80 kg oscillates between two springs with simple harmonic motion of amplitude 0.12 m. Its maximum speed is 0.36 m s⁻¹. Calculate the total energy of the oscillation, the angular frequency, and the potential energy stored in the springs when the displacement is 0.090 m.

  • At the centre all the energy is kinetic: E = ½mv₀² = 0.5 × 0.80 × 0.36² = 5.2 × 10⁻² J. ✓
  • v₀ = ωx₀, so ω = 0.36 / 0.12 = 3.0 rad s⁻¹. ✓
  • PE = ½mω²x² = 0.5 × 0.80 × 3.0² × 0.090² = 2.9 × 10⁻² J, leaving 2.3 × 10⁻² J as kinetic energy. ✓
06
Calculation
[4 marks]

A mass of 0.25 kg hangs from a spring and oscillates vertically with simple harmonic motion of frequency 1.6 Hz and amplitude 40 mm.

(a) Calculate the angular frequency of the motion. (b) Calculate the total energy of the oscillation. (c) Calculate the speed of the mass when its displacement is 24 mm, and show that the kinetic and potential energies at that point add up to the total.

  • (a) ω = 2πf = 2π × 1.6 = 10.1 rad s⁻¹. ✓
  • (b) E = ½mω²x₀² = 0.5 × 0.25 × 10.1² × 0.040² = 2.0 × 10⁻² J. ✓
  • (c) v = ω√(x₀² − x²) = 10.1 × √(0.040² − 0.024²) = 10.1 × 0.032 = 0.32 m s⁻¹. ✓
  • KE = ½mv² = 0.5 × 0.25 × 0.322² = 1.3 × 10⁻² J and PE = ½mω²x² = 0.5 × 0.25 × 101 × 0.024² = 0.73 × 10⁻² J; their sum, 2.0 × 10⁻² J, is the total found in (b). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.