Physics › 🚀 Mechanics

Simple Harmonic Motion

Why springs, swings, and pendulums all secretly move to the same beat.

High schoolAP Physics 1 levelUni Year 1
💡
The big idea: Simple harmonic motion happens whenever a restoring force grows in exact proportion to displacement and always points back toward equilibrium. That one rule — F = -kx — is enough to ensure smooth, repeating oscillation with a period that doesn't depend on amplitude. Springs and pendulums look like completely different systems, but underneath, they're playing the same rhythm.
🎯 By the end, you'll be able to
  • Recognize the restoring-force condition F = -kx that defines simple harmonic motion.
  • Calculate the period of a mass-spring system using T = 2π√(m/k).
  • Calculate the period of a simple pendulum using T = 2π√(L/g), and explain why mass doesn't matter.
  • Track how energy shifts between kinetic and potential energy over one oscillation cycle.
📎 You should already know
  • Newton's Second Law (F = ma)
  • Hooke's Law and spring forces
  • Kinetic and potential energy

Why things wobble back

Push a swing, pluck a guitar string, tap a mass hanging from a spring — nudge any of these away from where they want to rest, and they don't just drift off. They swing back, overshoot, swing back again, and settle into a steady, repeating rhythm. That rhythm has a name: simple harmonic motion (SHM). Once you see the one idea underneath it, pendulums, springs, and even vibrating atoms all start to look like variations on the same theme.

🔑 The one rule that makes it all work

Simple harmonic motion happens whenever the force pulling something back toward equilibrium is directly proportional to how far it's been displaced — and always points the opposite way. Double the displacement, double the restoring force. That relationship is captured in one compact equation, and it's the seed everything else in this lesson grows from.

\[ F = -kx \]
The restoring force F is proportional to displacement x and points opposite to it. The minus sign is doing real work: it's what pulls the system back rather than pushing it further away. k is a stiffness constant (for a spring, the spring constant).

From force to rhythm

Feed \( F = -kx \) into Newton's second law and solve, and something remarkable falls out: the motion is automatically periodic, and the time for one full cycle — the period — depends only on the physical properties of the system, never on how far you initially displaced it. A gentle push and a hard yank take the same amount of time to complete a swing.

\[ T = 2\pi\sqrt{\dfrac{m}{k}} \]
Period of a mass m oscillating on a spring of stiffness k. Heavier mass slows it down; a stiffer spring speeds it up.
\[ T = 2\pi\sqrt{\dfrac{L}{g}} \]
Period of a simple pendulum of length L, for swings small enough that the restoring force stays proportional to displacement. Notice: no mass in this formula at all.
🎮 Interactive: Mass-Spring Oscillator LIVE
Drag the mass, adjust the spring stiffness or the mass value, and watch how the period and the kinetic/potential energy trade-off change in real time.

Energy trades places, but never disappears

As the mass swings through equilibrium, it's moving fastest — all of the system's energy is kinetic. At the extremes of the swing, it stops for an instant to reverse direction — all of that energy is now stored as potential energy. Add them up at any point in the cycle and the total stays exactly constant: \( E = \tfrac{1}{2}kA^2 \), where A is the amplitude. SHM is really just energy continuously handing itself back and forth between two forms, on a perfectly regular schedule.

📝 Worked example: A 2.0 kg mass sits on a frictionless horizontal surface, attached to a spring with spring constant k = 200 N/m. It's pulled aside to an amplitude of 5.0 cm and released. Find the period of oscillation and the total mechanical energy.
  1. Period depends only on mass and spring constant: \( T = 2\pi\sqrt{m/k} \)
  2. Plug in: \( T = 2\pi\sqrt{2.0/200} = 2\pi\sqrt{0.010} = 2\pi(0.100) \)
  3. \( T \approx 0.63\ \text{s} \)
  4. Total energy comes from the amplitude: \( E = \tfrac12 k A^2 = \tfrac12(200)(0.050)^2 \)
  5. \( E = \tfrac12(200)(0.0025) = 0.25\ \text{J} \)
✓ T ≈ 0.63 s; total mechanical energy ≈ 0.25 J, and it stays constant throughout the motion, just changing form.
📝 Worked example: A pendulum clock uses a pendulum 1.0 m long. Estimate its period near Earth's surface (g = 9.8 m/s²), and predict what happens to the period if the length is quadrupled to 4.0 m.
  1. \( T = 2\pi\sqrt{L/g} \)
  2. \( T = 2\pi\sqrt{1.0/9.8} = 2\pi\sqrt{0.102} = 2\pi(0.319) \)
  3. \( T \approx 2.0\ \text{s} \)
  4. Since \( T \propto \sqrt{L} \), quadrupling L multiplies T by \( \sqrt{4} = 2 \)
✓ T ≈ 2.0 s at 1.0 m; quadrupling the length to 4.0 m doubles the period to about 4.0 s. Mass and amplitude never enter the calculation.
⚠️ Two assumptions worth knowing

These clean formulas rely on some fine print. The pendulum equation only holds for small swing angles (roughly under 15°) — swing it wildly and the restoring force stops being proportional to displacement, so the period creeps longer than the formula predicts. Also notice that the pendulum's mass never appears in \( T = 2\pi\sqrt{L/g} \) — a bowling ball and a golf ball on identical strings swing at exactly the same rate. And for both the spring and the pendulum, the period doesn't depend on amplitude either: swing wide or narrow, the timing stays the same. That amplitude-independence is what made pendulum clocks reliable timekeepers in the first place.

Check your understanding

1. What condition defines simple harmonic motion?
SHM is defined by F = -kx: the restoring force grows with displacement and always points back toward equilibrium. That specific proportional relationship — not just 'a force that pushes back' — is what produces smooth, repeating, sinusoidal motion.
2. A 0.50 kg mass on a spring with k = 50 N/m completes one oscillation in about how much time?
T = 2π√(m/k) = 2π√(0.50/50) = 2π√(0.010) = 2π(0.10) ≈ 0.63 s.
3. If you double the length of a pendulum, its period...
Since T = 2π√(L/g), the period scales with the square root of length. Doubling L multiplies T by √2 ≈ 1.41, not by 2 — a common trap since it feels like it should scale directly.
4. At the instant a mass on a spring passes through the equilibrium position (zero displacement), where has all its mechanical energy gone?
At equilibrium, x = 0, so spring potential energy (½kx²) is zero — every bit of the system's energy is kinetic at that instant, which is also why speed is at its maximum right there.
✅ Key takeaways
  • SHM arises whenever the restoring force is proportional to displacement and points opposite to it: F = -kx.
  • A mass-spring system oscillates with period T = 2π√(m/k); a pendulum (small angles) with T = 2π√(L/g) — neither depends on amplitude.
  • Total mechanical energy in SHM stays constant, continuously trading between kinetic energy (maximum at equilibrium) and potential energy (maximum at the extremes).
  • Pendulum period doesn't depend on the bob's mass, and the small-angle approximation is what keeps the formula accurate.