Simple Harmonic MotionEdit
Simple harmonic motion (SHM) describes a class of periodic motions in which the restoring force is proportional to the displacement from an equilibrium position and acts in the opposite direction. The canonical physical realization is a mass m attached to a light, ideal spring with constant k, obeying Hooke's law Hooke's law. In the absence of nonconservative forces, this setup exhibits perfectly periodic motion with a constant amplitude.
In SHM, the motion is characterized by a sinusoidal time dependence and a natural frequency set by the system's inertia and stiffness. The angular frequency is ω0 = sqrt(k/m), and the period is T = 2π/ω0. A typical solution for the displacement x(t) from equilibrium can be written as x(t) = A cos(ω0 t + φ), where A is the amplitude and φ is a phase constant. Because the energy is exchanged between kinetic and potential forms, SHM is often used as a simple, ideal model of vibrations in a wide range of physical systems. See also the general study of Oscillation and Frequency.
Mathematical formulation
Undamped SHM
The equation of motion for an undamped SHM system is m d^2x/dt^2 + k x = 0. This second-order linear differential equation yields sinusoidal solutions with the natural frequency ω0 = sqrt(k/m). The displacement, velocity, and acceleration are all related in a way that continually exchanges energy between kinetic energy Kinetic energy and potential energy Potential energy stored in the spring, so the total energy E = 1/2 k x^2 + 1/2 m (dx/dt)^2 remains constant.
Damped SHM
In real systems, nonconservative forces produce damping. The equation becomes m d^2x/dt^2 + c dx/dt + k x = 0, where c is a damping coefficient. The behavior depends on the damping ratio ζ = c / (2 sqrt(m k)):
- Underdamped (ζ < 1): the motion oscillates with a gradually decaying envelope. The damped frequency is ωd = ω0 sqrt(1 − ζ^2).
- Critically damped (ζ = 1): the system returns to equilibrium as quickly as possible without oscillating.
- Overdamped (ζ > 1): the return to equilibrium is slow and non-oscillatory.
Forced SHM
When a driving force acts, the equation becomes m d^2x/dt^2 + c dx/dt + k x = F0 cos(ωt), a driven harmonic oscillator. If the driving frequency ω is near the natural frequency ω0 and damping is small, the system can exhibit large steady-state oscillations. This phenomenon is described by resonance in physics Resonance (physics) and is a fundamental consideration in engineering design, from speakers to machinery. See also the concept of a driven system in Driven harmonic oscillator.
SHM in pendulums and small-angle motion
A simple pendulum of length L exhibits SHM in the small-angle limit. If the angle θ is small, θ'' + (g/L) θ = 0, giving an angular frequency ω0 = sqrt(g/L). This is a classical example of SHM that connects rotational motion to linear SHM descriptions. See Pendulum and Small-angle approximation.
Applications and interpretation
- Mass–spring systems: The textbook prototype for SHM is a mass on a spring, used in physics classrooms and engineering laboratories to illustrate energy exchange and resonance. See Spring (mechanics) and Mass (physics).
- Damped and driven systems: In real-world structures and machines, SHM concepts underpin vibration control, damping strategies, and resonance avoidance. See Damping and Resonance (physics).
- Electrical analogs: LC circuits and other electrical resonators behave as SHM systems in the appropriate regime, providing a useful parallel between mechanics and circuits. See LC circuit (where applicable).
- Pendulums and molecular vibrations: Small-angle pendulums and certain molecular vibrational modes can be modeled as SHM, enabling insights across physics and chemistry. See Pendulum and Molecular vibration.