Lesson 1 · 14 min
Simple harmonic motion
The restoring force is linear. Nature then draws a sine wave, and a lot of machines depend on it.
The linear restoring force
If F = −kx, the motion is simple harmonic: x(t) = A cos(ωt + φ) with ω = √(k/m). Amplitude A is set by how far you pull it; the frequency is not. That independence is why a clock can keep time as the spring winds down (approximately), and why a mass-spring vibration isolator has a predictable natural frequency.
ω = √(k/m) T = 2π/ω T_pendulum ≈ 2π √(L/g)
The pendulum formula is a small-angle approximation (sinθ ≈ θ in radians). Large swings run slow.
Check
You quadruple the mass on a spring. The period
Drive the numbers. Then go back to the algebra.
Bench
Pendulum
T = 2π √(L/g)
- T small
- 2.20 s
- L
- 1.20 m
- g
- 9.81 m/s²
- θ₀
- 25.0°
Small-angle T ignores amplitude. Pull past ~15° and the clock runs slow — the sinθ ≈ θ story is breaking.