Simple Harmonic Motion

• Define Simple Harmonic Motion (SHM) and identify its characteristics.
• Understand the restoring force, amplitude, period, and frequency.
• Analyse springs and pendulums as SHM systems.
• Describe energy exchange in oscillations.

What is SHM?

Simple Harmonic Motion: Periodic motion where the restoring force is directly proportional to displacement and directed toward the equilibrium position.

F = −kx (Hooke's Law)

• F = restoring force (N)
• k = spring constant (N/m) — stiffness of the spring
• x = displacement from equilibrium (m)
• Negative sign: force opposes displacement

Key Quantities

Spring-Mass System

T = 2π√(m/k)

• m = mass (kg), k = spring constant (N/m)
• Heavier mass → longer period (slower)
• Stiffer spring → shorter period (faster)

Example: Mass 0.5 kg on spring with k = 200 N/m:

• T = 2π√(0.5/200) = 2π × 0.05 = 0.314 s

Simple Pendulum

T = 2π√(L/g)

• L = length of pendulum (m), g = 9.8 m/s²
• Period depends on length only — NOT on mass or amplitude (for small angles)

Example: Pendulum of length 1 m:

• T = 2π√(1/9.8) ≈ 2π × 0.319 ≈ 2.01 s

Energy in SHM

Energy continuously converts between kinetic and potential:

Total mechanical energy = ½kA² = constant (no friction)

KE = ½k(A² − x²), PE = ½kx²

Displacement vs Time Graph

• Sinusoidal wave: x = A cos(ωt) or x = A sin(ωt)
• Velocity leads displacement by 90°
• Acceleration is opposite to displacement

Damped Oscillations

In real systems, friction reduces amplitude over time:

• Underdamped: oscillates but amplitude decreases
• Critically damped: returns to equilibrium fastest without oscillating
• Overdamped: slowly returns without oscillating

• What is SHM? What is the direction of the restoring force?
• A spring has k = 400 N/m. A 1 kg mass is attached. Find the period.
• A pendulum has period 2 s. What is its length? (g = 9.8 m/s²)
• Where is kinetic energy maximum in SHM? Where is it zero?
• What happens to energy in a damped oscillation?

Open the Practice tab for graded questions on Simple Harmonic Motion.