Simple harmonic motion
A bouncing spring and a swinging pendulum look nothing alike — but they run on the exact same cosine curve, because both feel a restoring force pulling straight back towards the middle. So build it and watch. Switch between a mass-spring system and a simple pendulum, drag the sliders, and every reading — period, top speed, energy — updates live from the same equations as your NCERT chapter.
The rules you're seeing
What makes it SHM
The restoring force is always proportional to displacement and pointed back the other way: F = −kx for the spring, and the equivalent for a small-angle pendulum. Different physics, same cos/sin maths.
Period is amplitude-independent
T doesn't depend on A (spring) or θmax (pendulum, for small angles). A big swing and a small swing of the same pendulum take exactly the same time per cycle — which is why pendulums make good clocks.
Energy trade-off
In the mass-spring system, kinetic and potential energy are constantly converting into each other — but KE + PE stays exactly equal to the total energy at every single instant.
Spring vs pendulum
Same underlying maths — ω = √(k/m) vs ω = √(g/L) — but different quantities drive the restoring force: mass and stiffness for a spring, gravity and length for a pendulum.
Part of the Oscillations chapter — read the notes, grab the formula sheet and take the quiz. One of Priodemy for School, free with every EduSuite school.
What makes an oscillation 'simple harmonic'
The defining condition
Not every back-and-forth motion is simple harmonic. SHM requires one specific relationship: the restoring force must be proportional to the displacement and directed opposite to it, written F = −kx. The minus sign is the physics — the force always points back towards the equilibrium position, and it grows in proportion to how far you have strayed.
Because F = ma, this gives a = −ω²x, and that equation is what produces a sinusoidal solution. Displacement varies as x = A sin(ωt + φ), which is why every SHM graph is a sine or cosine curve. If the force law is anything other than proportional, the motion may still oscillate, but it is not SHM and none of the standard formulas apply.
Period does not depend on amplitude
This is the result students find hardest to believe, so it is worth testing directly: change the amplitude here and watch the period stay fixed. Pull the mass twice as far and it has twice the distance to cover, but the restoring force is also twice as large, so it moves proportionally faster. The two effects cancel exactly.
This property, called isochronism, is what makes oscillators useful for timekeeping. A pendulum clock keeps time even as the swing decays, and the same principle underlies the quartz crystal in a wristwatch. For a spring the period is T = 2π√(m/k), and for a simple pendulum T = 2π√(L/g) — note that the pendulum's period does not involve the mass of the bob at all.
Energy sloshing between two forms
At the extremes the object is momentarily at rest, so kinetic energy is zero and all the energy is potential. At the centre the displacement is zero, so potential energy is zero and the speed — and kinetic energy — is maximum. In between, the two exchange continuously while the total stays constant at E = ½kA². Watch the energy bars trade height as the motion proceeds; the sum never changes.
Mistakes that cost marks
Assuming the pendulum formula always holds. T = 2π√(L/g) is an approximation valid only for small angles, where sin θ ≈ θ. At large amplitudes the motion is still periodic but no longer simple harmonic, and the period grows.
Confusing frequency with angular frequency. ω is measured in radians per second and f in hertz, related by ω = 2πf. Substituting one for the other introduces a factor of 2π that is easy to miss and hard to spot afterwards.
Thinking acceleration is greatest at the centre. Speed is greatest at the centre; acceleration is greatest at the extremes, where displacement — and therefore restoring force — is largest. At the centre the acceleration is exactly zero.