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Average energy over a cycle
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Average kinetic equals average potential energy, each half the total.
Over a full period the average kinetic and potential energy are each E/2. This is the equipartition used in thermal physics.
Time-average of cos² = ½.
Not equal at every instant.
In more detail
Over a complete cycle, the average kinetic energy and the average potential energy in SHM are each half the total energy. This is because both energies vary sinusoidally.
Real-world uses
- Estimating average power dissipation in vibrating systems.
- Calculating average energy in AC circuits for power analysis.
- A physicist averages the energy over many cycles of a mass on a spring to find each part is half the total.
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Frequently asked questions
Why are they equal on average?
Because the sine and cosine squared functions average to one half over a cycle.
Does the average depend on amplitude?
Yes, it scales with the square of the amplitude.
Why is the average kinetic energy half the total?
Over a full cycle the kinetic and potential energies are equal on average, so each takes half the total energy.
Related topics in Oscillations & SHM
Simple pendulum in elevator · Energy in SHM · Kinetic and potential energy in SHM · Differential equation of SHM · Damped oscillations · Underdamped, critical, overdamped