Introduction
Simple Harmonic Motion (SHM) is oscillatory motion where restoring force is proportional to displacement: F = -kx. It is the simplest periodic motion and fundamental to understanding waves, molecular vibrations, and quantum mechanics.
Equation of Motion
From F = ma = -kx, we get d2x/dt2 + ω2x = 0, where ω = √(k/m) is angular frequency. This second-order linear differential equation describes SHM completely.
General Solution
x(t) = A cos(ωt + φ) = A sin(ωt + φ'). A is amplitude (maximum displacement), φ is phase constant determined by initial conditions, ω is angular frequency in rad/s.
Velocity and Acceleration
v(t) = dx/dt = -Aω sin(ωt + φ). Maximum velocity v_max = Aω at equilibrium. a(t) = dv/dt = -Aω2 cos(ωt + φ) = -ω2x. Acceleration always toward equilibrium, proportional to displacement.
Period and Frequency
Period T = 2π/ω = 2π√(m/k). Frequency f = 1/T = ω/(2π). T depends only on system properties (m, k), not amplitude. This is isochronism - characteristic of SHM.
\nIntroduction
Simple Harmonic Motion (SHM) is oscillatory motion where restoring force is proportional to displacement: F = -kx. It is the simplest periodic motion and fundamental to understanding waves, molecular vibrations, and quantum mechanics.
Equation of Motion
From F = ma = -kx, we get d2x/dt2 + ω2x = 0, where ω = √(k/m) is angular frequency. This second-order linear differential equation describes SHM completely.
General Solution
x(t) = A cos(ωt + φ) = A sin(ωt + φ'). A is amplitude (maximum displacement), φ is phase constant determined by initial conditions, ω is angular frequency in rad/s.
Velocity and Acceleration
v(t) = dx/dt = -Aω sin(ωt + φ). Maximum velocity v_max = Aω at equilibrium. a(t) = dv/dt = -Aω2 cos(ωt + φ) = -ω2x. Acceleration always toward equilibrium, proportional to displacement.
Period and Frequency
Period T = 2π/ω = 2π√(m/k). Frequency f = 1/T = ω/(2π). T depends only on system properties (m, k), not amplitude. This is isochronism - characteristic of SHM.
\nApplications
Applications: mass-spring systems, small-angle pendulums, LC circuits, molecular bonds, acoustic waves, atomic vibrations in solids. SHM is the foundation for understanding all oscillatory phenomena.
Solved Example: Mass-Spring System
A 0.5 kg mass attached to a spring with k = 200 N/m is displaced 0.1 m from equilibrium and released from rest. Find: (a) Period of oscillation, (b) Maximum velocity, (c) Position after 0.5 s. Solution: (a) ω = √(k/m) = √(200/0.5) = √400 = 20 rad/s. T = 2π/ω = 2π/20 = 0.314 s ≈ 0.31 s. (b) Released from rest at x = A = 0.1 m, so φ = 0. v_max = Aω = 0.1 × 20 = 2 m/s. (c) x(t) = A cos(ωt) = 0.1 cos(20 × 0.5) = 0.1 cos(10 rad). 10 rad = 10 × 180/π ≈ 573°. 573° - 360° = 213°. cos(213°) = -0.84. x(0.5) = 0.1 × (-0.84) = -0.084 m. Mass is 8.4 cm on opposite side of equilibrium after 0.5 seconds.
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