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Differential equation of SHM
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The standard second-order equation.
ẍ + ω²x = 0ẍ + ω²x = 0, with general solution x = A cos ωt + B sin ωt. Any system with this equation oscillates sinusoidally.
LC circuit: q̈ + q/(LC) = 0.
Initial conditions fix A and B.
In more detail
The equation of SHM, m x'' + k x = 0, is a second-order linear differential equation. Its solutions are sinusoids, and the constants come from initial conditions.
Real-world uses
- Engineers solve the equation of motion to predict vibration of machines.
- Quantum mechanics uses the harmonic oscillator, the same mathematics.
- Engineers write the equation of motion for a vibrating system as this second-order equation, then solve it for a given set of conditions.
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Frequently asked questions
What are the initial conditions?
The starting position and velocity, which set the amplitude and phase.
Why is SHM solvable?
Because the equation is linear, so its solutions can be written in sinusoidal form.
Why is this called second-order?
It contains the second derivative of the displacement with respect to time, which is the acceleration.
Related topics in Oscillations & SHM
Energy in SHM · Kinetic and potential energy in SHM · Average energy over a cycle · Damped oscillations · Underdamped, critical, overdamped · Quality factor