Library / Foundations: Physics / FPH-64
Foundations: Physics
Kinetic fraction of the energy in simple harmonic motion
Share of the total energy that is kinetic when the oscillator is at displacement x from the centre and has amplitude A.
- Ref
- FPH-64
- Rev
- v1
- Field
- Foundations: Physics
- Result
- K/E
At displacement x, the potential energy is ½kx² and the total energy is ½kA², so U/E = x²/A² and K/E = 1 − x²/A². The two shares always add to 1. They are equal when |x| = A/√2, about 70.7 % of the amplitude.
Share of the total energy that is kinetic when the oscillator is at displacement x from the centre and has amplitude A. At the centre all of the energy is kinetic; at the ends of the swing, none is. The potential share is the remainder.
Calculator
- Potential share, x²/A²0.36
- Displacement where K = U, A/√20.03536 m
- Speed as a fraction of the peak speed, √(K/E)0.8
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 60 % of the amplitude, 64 % of the energy is kinetic and 36 % is stored in the spring.
- The shares cross at 70.7 % of the amplitude: further out than that, most of the energy is potential.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Displacement | −0.05 to 0.05 m | 0 | 1 | 1 | +0.01 m | −43.8 % |
| A Amplitude | 0.01 to 1 m | 0.5133 | 0.9991 | 0.4858 | +10% | +9.76 % |
Sweep of Displacement
| Displacement (m) | Kinetic share of the energy |
|---|---|
| −0.05 | 0 |
| −0.0375 | 0.4375 |
| −0.025 | 0.75 |
| −0.0125 | 0.9375 |
| 0 | 1 |
| 0.0125 | 0.9375 |
| 0.025 | 0.75 |
| 0.0375 | 0.4375 |
| 0.05 | 0 |
Sweep one variable
Variables and how each one contributes
Kinetic share of the energy
What the equation gives you. Everything else feeds it.
Displacement
The displacement from the centre, in m, with |x| not above A. Lowers the kinetic share, because more of the energy is stored in the spring further from the centre. Raising it by 10 % (to 0.033) moves the result by −11.8 %, from 0.64 to 0.5644.
Amplitude
The amplitude, in m. Raises the kinetic share: at the same displacement, a larger swing leaves more of the energy as motion. Raising it by 10 % (to 0.055) moves the result by +9.76 %, from 0.64 to 0.7025.
Worked example with the values above
- Ratio: x/A = 0.03 / 0.05 = 0.6.
- Square it: x²/A² = 0.36.
- K/E = 1 − 0.36 = 0.64.
Assumptions and limits
- Undamped simple harmonic motion, so total energy is constant.
- The displacement is measured from the equilibrium (centre) position.
Common mistakes
- Entering a displacement larger than the amplitude (the page then gives no result).
- Measuring x from the end of the swing instead of from the centre.
- Confusing the energy fraction with the speed fraction (which is its square root).
Statistics and uncertainty
K/E depends on the square of the ratio x/A, so a 1 % error in x near the ends of the swing changes the kinetic share by a large fraction of a small number.
Where it is used
- Finding where a mass on a spring has equal kinetic and potential energy.
- Checking energy sharing in vibration and pendulum problems.
Maths you need
- Energy in SHM: E = ½kA²
- Squares and ratios: x²/A²