The Memory Trick
💡 Restoring Force ∝ Displacement
Simple harmonic motion (SHM) describes any oscillating system where the restoring force — the force pulling the object back toward equilibrium — is directly proportional to how far the object has moved from that equilibrium, and always points opposite to the displacement.
Why It Works
This proportional relationship is exactly what produces the smooth, repeating back-and-forth motion characteristic of pendulums and springs — the farther you pull it, the harder it's pulled back, creating a predictable, periodic oscillation.
Step by Step
Two Classic SHM Systems
1
The simple pendulum
Period: T = 2π√(L/g). Remarkably, this depends ONLY on the pendulum's length and gravitational acceleration — not on mass, and not on amplitude (for small swing angles).
A 1 m pendulum on Earth (g=9.8 m/s²) has a period T = 2π√(1/9.8) ≈ 2.0 seconds, regardless of whether the pendulum bob is light or heavy.
2
The mass-spring system
Period: T = 2π√(m/k), where k is the spring constant. Unlike the pendulum, this DOES depend on mass — more mass means a longer period.
Doubling the mass on a spring increases the period by a factor of √2, since mass is inside the square root.
3
Restoring force always opposes displacement
Whether it's gravity pulling a pendulum back toward vertical or a spring pulling back toward its natural length, the restoring force always acts in the direction opposite to how far the object has moved.
Stretch a spring further, and it pulls back harder — that's the proportional restoring force in action.
🏥 Worked Example
A pendulum has a length of 2.5 m. What is its period on Earth?
1
Apply the pendulum period formula: T = 2π√(L/g).
2
Plug in values: T = 2π√(2.5/9.8) = 2π√(0.255) ≈ 2π × 0.505 ≈ 3.17 seconds.
3
Note what's absent: the mass of the pendulum bob was never needed — period depends only on length and gravity.
📌 Exam Application
Exams test correctly applying the pendulum and spring period formulas, and specifically knowing that pendulum period is independent of mass and amplitude (for small angles), while spring period does depend on mass.
⚠️ Most Common Simple Harmonic Motion Mistakes
The most common trap is assuming a heavier pendulum bob swings with a different period than a lighter one — for a simple pendulum, mass cancels out of the equation entirely and has no effect on period.
✓ Quick Self-Test
1) What defines simple harmonic motion? A restoring force directly proportional to displacement from equilibrium, always acting opposite to that displacement. 2) Write the period formula for a simple pendulum. T = 2π√(L/g). 3) Does the mass of the pendulum bob affect its period? No — mass cancels out; period depends only on length and gravitational acceleration. 4) Write the period formula for a mass on a spring, and state whether mass affects it. T = 2π√(m/k); yes, mass does affect the period here. 5) If a pendulum's length is quadrupled, what happens to its period? The period doubles (since period depends on the square root of length).
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