🌀 OSCILLATORS & SPRINGS
PAGE 1 OF 5, THE MOTION BATTERY
STRETCH AND SNAP
A SPRING IS A BATTERY FOR MOTION
Pull a coil spring and it fights back. Release it and the stored stretch turns into motion. Compress it and the same thing happens in reverse. A spring stores elastic potential energy in its coils, then hands that energy to a mass as kinetic energy when it snaps toward its natural length. That back-and-forth swap is the heart of oscillation: a repeating cycle of stretch, speed, stretch the other way, and speed again. Clocks, car suspensions, trampolines, and guitar strings all rely on this timed dance between tension and motion.
🔋 ENERGY STORE
Elastic potential energy in an ideal spring is (1/2)kx², where k is the spring constant and x is how far the spring is stretched or compressed from its rest length.
BOUNCE!
STRETCH
🧲 Coils store elastic potential energy
📏 Farther stretch means more stored energy
✋ You do work on the spring to load it
📏 Farther stretch means more stored energy
✋ You do work on the spring to load it
RELEASE
🚀 Potential energy becomes kinetic energy
🌀 Mass races through the equilibrium point
🔁 Then overshoots and stores energy again
🌀 Mass races through the equilibrium point
🔁 Then overshoots and stores energy again
PAGE 2 OF 5, HOOKE'S LAW
UT TENSIO, SIC VIS
FORCE PROPORTIONAL TO STRETCH
In 1678, Robert Hooke published a law that still anchors spring physics: the restoring force is proportional to how far the spring is displaced from its natural length, and it always points back toward that length. Written as F = −kx, the minus sign means the force opposes the displacement. The constant k is the spring constant, measured in newtons per metre. A stiff spring has a large k and needs a big force for a small stretch. A soft spring has a small k and stretches easily. Hooke's law holds only within the elastic limit. Stretch too far and the metal permanently deforms, so the simple linear rule no longer applies.
📐 F = −kx
F is the restoring force, k is the spring constant, and x is displacement from equilibrium. The force always tries to return the spring to its rest length.
HOOK!
STIFF VS SOFT
🏋️ Large k: hard to stretch
🪶 Small k: easy to stretch
📊 Units of k are N/m
🪶 Small k: easy to stretch
📊 Units of k are N/m
RESTORING FORCE
⬅️ Pull right, force points left
➡️ Push left, force points right
🎯 Always aims at equilibrium
➡️ Push left, force points right
🎯 Always aims at equilibrium
ELASTIC LIMIT
✅ Within limit: spring returns fully
❌ Beyond limit: permanent bend
⚠️ Hooke's law then fails
❌ Beyond limit: permanent bend
⚠️ Hooke's law then fails
PAGE 3 OF 5, SIMPLE HARMONIC MOTION
AMPLITUDE
📏 Max displacement from equilibrium
🔊 Larger amplitude: more energy stored
⏱️ Period stays the same for ideal SHM
🔊 Larger amplitude: more energy stored
⏱️ Period stays the same for ideal SHM
PERIOD
⏳ T = 2π√(m/k) for a mass-spring
🏋️ Heavier mass: longer period
🧲 Stiffer spring: shorter period
🏋️ Heavier mass: longer period
🧲 Stiffer spring: shorter period
THE PERFECT CYCLE
SIMPLE HARMONIC MOTION
When Hooke's law holds and friction is negligible, a mass on a spring performs simple harmonic motion, or SHM. Position, velocity, and acceleration all vary smoothly with time like sine and cosine waves. At maximum stretch, speed is zero and all energy is potential. At the equilibrium point, stretch is zero, speed is maximum, and all energy is kinetic. Then the mass overshoots, compresses or stretches the other way, and the cycle repeats. For an ideal mass-spring system, the period depends only on mass and spring constant: T = 2π√(m/k). Pull the mass farther and it still takes the same time per cycle, because the stronger restoring force speeds it up just enough to cover the longer path in the same period.
⚡ DID YOU KNOW?
In ideal SHM, period does not depend on amplitude. A gentle pull and a hard pull on the same mass-spring system complete one cycle in the same time.
CYCLE!
PAGE 4 OF 5, ENERGY SWAP
PE TO KE TO PE
THE ENERGY DANCE
Oscillators are energy traders. At the turning points, elastic potential energy is at its peak and kinetic energy is zero. Midway through, kinetic energy peaks and potential energy bottoms out. If you ignore friction and air resistance, the total mechanical energy stays constant: E = (1/2)kx² + (1/2)mv². Real springs lose a little energy each cycle to heat and sound, so the amplitude slowly shrinks. That is damping. Engineers sometimes add dampers on purpose, as in car shock absorbers, so the bounce dies quickly instead of rocking forever. Other systems, like a driven clock or a radio circuit, feed in just enough energy to keep the oscillation going at a steady amplitude.
🔄 CONSERVATION
In an ideal oscillator, total energy is constant. Potential and kinetic forms trade places every quarter cycle, but their sum does not change.
SWAP!
TURNING POINT
🛑 Speed is zero for an instant
🔋 All energy is potential
↩️ Force is strongest here
🔋 All energy is potential
↩️ Force is strongest here
MIDPOINT
⚡ Speed is maximum
🏃 All energy is kinetic
🎯 Net spring force is zero
🏃 All energy is kinetic
🎯 Net spring force is zero
DAMPING
📉 Friction drains energy each cycle
🚗 Shock absorbers damp on purpose
🔇 Amplitude slowly fades away
🚗 Shock absorbers damp on purpose
🔇 Amplitude slowly fades away
PAGE 5 OF 5, SPRINGS EVERYWHERE
FROM CLOCKS TO CARS
OSCILLATORS IN EVERY MACHINE
Once you see springs as motion batteries, machines make more sense. Mechanical clocks use a balance wheel and hairspring as a timed oscillator. Car suspensions use coil springs to store the jolt of a bump, then release it gently while dampers stop endless bouncing. Musical instruments turn vibrating strings and air columns into sound waves at precise frequencies. Even atoms in a solid vibrate like tiny masses on springs, which is why solids have natural frequencies and can resonate. Material science chooses alloys and coil shapes so k, strength, and fatigue life match the job. Mathematics and metalwork meet in every reliable oscillator.
TICK!
REAL WORLD
🕰️ Clocks keep time with oscillators
🎸 Strings vibrate at set frequencies
🚗 Suspensions cushion road bumps
🎸 Strings vibrate at set frequencies
🚗 Suspensions cushion road bumps
REMEMBER
🌀 KEY FACTS
A spring stores elastic potential energy and releases it as motion. Hooke's law: F = −kx within the elastic limit. Ideal mass-spring SHM has period T = 2π√(m/k), independent of amplitude. Energy swaps between PE and KE each cycle. Real systems damp unless energy is added back.
✅ Spring = motion battery
✅ F = −kx (restoring force)
✅ PE and KE trade places
✅ Period depends on m and k
✅ F = −kx (restoring force)
✅ PE and KE trade places
✅ Period depends on m and k
🧠 QUIZ TIME!
OSCILLATORS & SPRINGS · 5 QUESTIONS
QUESTION 01
Hooke's law says the restoring force of an ideal spring is:
QUESTION 02
At the equilibrium point of an ideal mass-spring oscillator, which is true?
QUESTION 03
For an ideal mass-spring system, the period T equals:
QUESTION 04
Elastic potential energy stored in an ideal spring is:
QUESTION 05
In ideal simple harmonic motion, if you increase the amplitude:
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