The back-and-forth motion of a spring or pendulum is simple harmonic motion (Simple Harmonic Motion). Fully master the phase relationships among position, velocity, and acceleration, as well as the period formulas.
Basic Knowledge
Displacement in simple harmonic motion: x = A sin(ωt + φ) (A: amplitude, ω: angular frequency, φ: initial phase)
Velocity: v = Aω cos(ωt + φ) (maximum velocity Aω: maximum at the center)
Acceleration: a = −Aω² sin(ωt + φ) = −ω²x (maximum at the endpoints, zero at the center)
Restoring force: F = −mω²x = −kx (proportional to displacement and directed toward the center)
Spring oscillation (spring constant k):
Period: T = 2π√(m/k), angular frequency ω = √(k/m)
Simple pendulum (length L, small amplitude):
Period: T = 2π√(L/g) (independent of mass and amplitude!)
Spring oscillation period: T = 2π√(m/k)
Simple pendulum period: T = 2π√(L/g)
Relationship between acceleration and displacement: a = −ω²x
Maximum velocity: v_max = Aω (achieved at the equilibrium position)
Conservation of energy: ½mv² + ½kx² = ½kA² (constant)
Further Study (Background and Significance)
The most important feature of simple harmonic motion is that acceleration is proportional to displacement and opposite in direction (a = −ω²x). Motion satisfying this condition is simple harmonic motion, and the period formulas are derived from this condition.
The period T = 2π√(m/k) of a spring oscillation is independent of amplitude. The period does not change whether the oscillation is large or small (isochronism). This is also the principle of a pendulum clock (isochronism holds for a simple pendulum as well).
Phase relationship: velocity leads displacement by π/2 (velocity is zero when displacement is maximum, and velocity is maximum when displacement is zero). Acceleration is in opposite phase to displacement (a difference of π).
From the energy perspective, kinetic energy and potential energy continuously interchange during simple harmonic motion, while their total remains constant (conservation of mechanical energy). At x = 0, all the energy is kinetic energy; at x = ±A, all the energy is potential energy.
- a = −ω²x is the definition of simple harmonic motion
- Isochronism independent of amplitude (for both springs and pendulums)
- Maximum velocity: at the center (x = 0)
- Zero velocity: at the endpoints (x = ±A)
- T = 2π√(m/k)、T = 2π√(L/g)
- Mechanical energy is conserved
- Velocity and displacement differ in phase by π/2
Points to Note (Common Confusions)
① The period of spring oscillation is determined only by mass and k (g is not needed). ② The period of a simple pendulum depends only on L and g (it is independent of mass and amplitude). ③ Maximum velocity Aω is achieved at the center. At the endpoints, velocity is zero. ④ The minus sign in F = −kx indicates “toward the center”; the magnitude is kx.
Practice
- Find the period of a spring oscillation with spring constant k = 200 N/m and mass m = 0.50 kg.
- Find the period of a simple pendulum of length L = 1.0 m (g = 10 m/s²).
- Find the maximum velocity of simple harmonic motion with amplitude A = 0.10 m and angular frequency ω = 10 rad/s.