🪑 Mass-Spring System
⏱️ Simple Pendulum
⚡ Kinematics & Energy

The Physics of Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a periodic oscillation where the restoring force is directly proportional to the displacement from the equilibrium position. It is the foundation for understanding waves, alternating current, and molecular vibrations.

a = -ω²x

This defining equation states that acceleration is always directed toward the fixed equilibrium point and increases as the object moves further away.

Kinematic Relationships

Using calculus, we can derive equations for displacement, velocity, and acceleration at any given time (t):

  • Displacement: x = A cos(ωt) (assuming motion starts at max amplitude)
  • Velocity: v = -ωA sin(ωt)
  • Acceleration: a = -ω²A cos(ωt)

Energy in SHM

In an ideal system, energy continuously oscillates between Potential Energy (Eₚ) and Kinetic Energy (Eₖ). The total energy remains constant.

Eₜₒₜₐₗ = ½mω²A²
Exam Tip: Phase difference is often tested using graphs. Remember that the displacement graph and velocity graph are shifted by π/2 radians (90°), while displacement and acceleration are π radians (180°) out of phase.

Deep Dive: Worked Examples

✅ Example 1: Spring Period Calculation

A 500g mass is suspended from a vertical spring with a spring constant (k) of 45 N/m. Calculate the time period of the resulting oscillations.

Step 1: Identify Values m = 0.5 kg, k = 45 N/m
Step 2: Apply the Formula T = 2π√(m/k) = 2π√(0.5 / 45)
Step 3: Solve T = 2π√(0.0111) = 2π(0.105) ≈ 0.66 s

✅ Example 2: Maximum Acceleration

A piston in an engine oscillates with SHM at a frequency of 50 Hz and an amplitude of 4.0 cm. Calculate the maximum acceleration.

Step 1: Convert Units and Find ω f = 50 Hz, A = 0.04 m
ω = 2πf = 2π(50) = 314.16 rad/s
Step 2: Apply a_max Formula a_max = ω²A = (314.16)² × 0.04
Step 3: Solve a_max = 98696 × 0.04 ≈ 3948 m/s²

✅ Example 3: Energy Conservation

A system has a spring constant k = 100 N/m and amplitude A = 0.2 m. Calculate the potential energy when the displacement x = 0.1 m.

Step 1: Potential Energy Formula Eₚ = ½kx²
Step 2: Plug in values Eₚ = 0.5 × 100 × (0.1)²
Step 3: Solve Eₚ = 50 × 0.01 = 0.5 J
(Note: The total energy is ½kA² = 0.5 × 100 × 0.04 = 2.0 J)

✅ Example 4: Velocity at a Specific Point

An object oscillates with ω = 10 rad/s and amplitude A = 5 cm. Calculate its velocity when it is 3 cm from equilibrium.

Step 1: Choose Velocity Equation v = ±ω√(A² - x²)
Step 2: Plug in values (in meters) v = 10 × √((0.05)² - (0.03)²) = 10 × √(0.0025 - 0.0009)
Step 3: Solve v = 10 × √0.0016 = 10 × 0.04 = 0.4 m/s

✅ Example 5: Phase Difference

Pendulum A is at max positive displacement at t=0. Pendulum B is at equilibrium moving positive at t=0. Calculate the phase difference in radians.

Step 1: Map to Trigonometric Functions A starts at cos(0) = 1. B starts at sin(0) = 0.
Step 2: Check Phase Shift cos(θ) = sin(θ + π/2). This means cos lags sin by π/2, or sin leads cos by π/2.
Step 3: Conclusion Since B is at 0 moving positive, it is at phase 0 for a sine wave. A is at phase 0 for a cosine wave (which is π/2 for a sine). Difference = π/2 radians.