Energy in Simple Harmonic Motion
Energy in Simple Harmonic Motion
- In an undamped oscillator, energy moves between kinetic and potential stores. The total stays constant.
Follow the energy through one cycle
Either extreme
Kinetic energy is zero.
Potential energy is maximum.
Between
Kinetic energy is changing.
Potential energy is changing.
Equilibrium
Kinetic energy is maximum.
Potential energy is minimum.
- As the object moves towards equilibrium, it speeds up. Potential energy changes into kinetic energy.
- After it passes equilibrium, it slows down. Kinetic energy changes back into potential energy.
- For a spring, the potential store is elastic. For a pendulum, it is gravitational. The pattern is the same.
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- Against displacement, potential energy is an upward curve and kinetic energy is a downward curve.
- Against time, each energy curve repeats twice during one oscillation because the object reaches each energy condition twice.
Calculate the total energy
Symbols
- = total oscillation energy (J)
- = oscillating mass (kg)
- = angular frequency (rad s⁻¹)
- = amplitude (m)
- Total energy is proportional to the square of amplitude. Doubling amplitude makes the energy four times larger.
- The same value is the maximum kinetic energy at equilibrium.
Your turnquick check
The mass and angular frequency stay fixed while the amplitude is tripled. By what factor does the total energy change?
Show worked answer
Total energy is proportional to amplitude squared, so the factor is the square of three.
Worked example
Work backwards from energy
A 0.130 kg mass has total energy 6.4 mJ and amplitude 0.015 m. Find the period.
- Substitute SI values into the energy relation.
- Solve for angular frequency.
- Convert angular frequency to period.
Answer
This is the method from (9702/42/F/M/23 Q3(b)).
Common mistake
Convert grams to kilograms and centimetres to metres before using the energy relation.