Understanding Simple Harmonic Motion
What is Simple Harmonic Motion?
Simple Harmonic Motion (SHM) is a type of periodic oscillation where the restoring force is directly proportional to displacement from equilibrium and acts in the opposite direction: F = -kx for springs or F ≈ -(mg/L)x for small-angle pendulums. This produces sinusoidal motion described by x(t) = A·cos(ωt + φ).
Mass-Spring Systems
A mass attached to an ideal spring oscillates with angular frequency ω = √(k/m), where k is the spring constant and m is the mass. The period T = 2π√(m/k) depends only on these two parameters—not on how far you stretch the spring (amplitude). Stiffer springs and lighter masses oscillate faster.
Simple Pendulums
A simple pendulum consists of a point mass on a massless string. For small oscillations (θ < 15°), the period is T = 2π√(L/g), where L is the pendulum length and g is gravitational acceleration. The period is independent of mass and amplitude in this approximation. On Earth (g ≈ 9.81 m/s²), a 1-meter pendulum has a period of about 2 seconds.
Key Equations
- Position: x(t) = A·cos(ωt + φ)
- Velocity: v(t) = -Aω·sin(ωt + φ), with v_max = Aω
- Acceleration: a(t) = -Aω²·cos(ωt + φ) = -ω²x, with a_max = Aω²
- Mass-Spring: ω = √(k/m), T = 2π√(m/k)
- Pendulum: ω = √(g/L), T = 2π√(L/g)
- Relations: ω = 2πf = 2π/T, f = 1/T
Energy in SHM
Total mechanical energy in ideal SHM is conserved: E = ½kA² for springs. At maximum displacement, all energy is potential. At equilibrium, all energy is kinetic (½mv²_max = ½m(Aω)² = ½kA²). The energy oscillates between these forms without loss in the ideal case.
Phase and Initial Conditions
The phase angle φ determines the initial state at t = 0. With φ = 0, the oscillator starts at x = A (maximum displacement, zero velocity). With φ = π/2, it starts at x = 0 moving in the negative direction. Any initial condition (x₀, v₀) can be achieved by choosing appropriate A and φ.
Limitations and Real-World Effects
This calculator models ideal SHM. Real oscillators experience damping (air resistance, friction) that gradually reduces amplitude. Real springs have mass and may not follow Hooke's law at large extensions. Real pendulums deviate from SHM at large angles. Despite these limitations, the ideal model provides excellent approximations for many practical situations.
Frequently Asked Questions
Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. The motion is sinusoidal and can be described by x(t) = A·cos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is phase. Common examples include mass-spring systems and simple pendulums (for small angles).
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