Oscillations, resonance and energy exchange
An oscillation is motion or variation around an equilibrium state. Begin by identifying the displacement variable, the restoring mechanism and the approximation that makes the model linear. A periodic trace is not automatically simple harmonic motion: the defining dynamical relation for ideal SHM is acceleration proportional to displacement and opposite in sign, a = -omega squared times x. The equilibrium position is the origin of x, not necessarily the unstretched spring position.
For an ideal horizontal mass-spring system with negligible friction and a massless spring, Newton's law gives m times x double-prime = -kx. Thus omega = sqrt(k/m), the period is 2pi sqrt(m/k), and frequency is the reciprocal of period. The solution may be written x = A cos(omega t + phi). Differentiation gives velocity and acceleration. The speed is greatest at equilibrium and zero at the turning points; acceleration has the opposite pattern of magnitudes. Amplitude independence of period is a property of this ideal linear model, not every oscillator.
Energy provides an independent check. Spring potential energy is kx squared divided by two, kinetic energy is mv squared divided by two, and their sum is kA squared divided by two in the undamped model. Consequently maximum speed is omega A. For example, a hypothetical spring has k = 50 N/m and carries 0.20 kg with amplitude 0.030 m. Omega is sqrt(250), approximately 15.81 radians per second. The period is approximately 0.397 s, the total energy is 0.0225 J, and maximum speed is approximately 0.474 m/s. Changing the mass to 0.40 kg multiplies the period by sqrt(2), rather than doubling it.
A small-angle simple pendulum is another approximation: T is approximately 2pi sqrt(length/g). Large amplitudes require additional treatment. Damping transfers mechanical energy into other forms; a driven oscillator receives energy from an external periodic source. Its response depends on drive frequency and damping. Resonance describes an enhanced response near a natural frequency, with the exact peak depending on the measured quantity and damping.
An ideal LC circuit similarly exchanges capacitor electric energy and inductor magnetic energy. Its angular frequency is 1/sqrt(LC); circuit resistance introduces damping. For sinusoidal currents in the ideal component model, inductive reactance has magnitude omega L and capacitive reactance has magnitude 1/(omega C). In a series RLC circuit, reactive contributions cancel at omega = 1/sqrt(LC); the remaining impedance is the resistance in that idealized model. Natural undriven oscillation and steady driven response are related but distinct problems. Distinguish this natural frequency from the frequency of an externally imposed alternating source. State the idealizations before using either analogy.