AP® Physics C: Mechanics · Unit 7 · Topic 7.4
Energy Graphs and Amplitude
Total mechanical energy in an oscillator is fixed and equals one half k A squared, so the amplitude is just the turning point where the energy parabola meets that fixed total.
Plot potential energy U = one half k x2 against position x and you get a parabola sitting in a bowl. Total energy E stays constant through the motion, so on this graph it is a flat horizontal line. Where that line crosses the parabola, kinetic energy hits zero: the oscillator has stopped turning around. Those crossing points are the turning points, and the distance from the center to either one is the amplitude A. Because U grows with x squared, doubling A means the crossing point moves out to where U is four times bigger, so E = one half k A2 says energy doubles twice when amplitude doubles once.
At any instant, energy splits between kinetic and potential: E = one half m v2 + one half k x2. Picture the vertical gap between the flat E line and the U parabola — that gap is KE, and it is biggest at the center where x = 0 and v is maximum, then shrinks to zero at the turning points. Both KE and U trace out curves that look like squared sine and cosine waves, so each one completes two full oscillations for every one full oscillation of position. That doubled frequency is a direct fingerprint of the x2 and v2 dependence.
This energy picture is what connects amplitude to speed and to everything resonance does later. Since E fixes A through one half k A2, and max speed v_max = A times √(k/m) comes from converting all that energy to kinetic at the center, amplitude is really just a stand-in for how much energy the system is carrying.
Key terms
- Amplitude
- The maximum displacement from equilibrium, reached at the turning points where velocity is zero.
- Total mechanical energy
- The constant sum of kinetic and potential energy in an oscillator, E = one half k A2.
- Turning point
- The position where all energy is potential and the oscillator momentarily stops before reversing direction.
- Angular frequency
- The quantity omega = √(k/m) that sets how fast the oscillator cycles, independent of amplitude.
Watch it explained
- How it worksHow do the potential energy parabola and a horizontal total energy line set the turning points?
- Worked exampleFind the amplitude of a 2 kg oscillator on a 50 N/m spring that carries 4 J of energy.
- How it worksHow do the kinetic and potential energy curves both oscillate at twice the motion's frequency?
- Going furtherWhat happens to the maximum speed when you double the amplitude at fixed mass and stiffness?
Check yourself
End of topic.