simple harmonic motion — HSC mathematics extension 2, live

Simple Harmonic Motion

Simple harmonic motion is one equation wearing a hundred costumes - a pendulum, a spring, a bobbing buoy, an alternating current. What makes them the same thing is not the shape of the motion but its cause: acceleration always points back towards the centre, and grows in exact proportion to how far you have strayed from it. So x-double-dot = -n squared (x - c) is not a description of the wobble, it IS the definition of it, and everything else - the sine solution, the v-squared formula, the period - is that single line pushed through calculus. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

mathematics extension 2 · mechanics · simple harmonic motiondrag it · it is yours

Mechanics works like a crowd on the dancefloor.

Mechanics is Newton's second law and a choice. Every problem here is F = mẍ; what decides whether it is a two-line answer or an unsolvable mess is which form of acceleration you write down. Use dv/dt when the force depends on time, and v dv/dx when it depends on position. Read what the force actually depends on, choose accordingly, and the calculus does the rest.

The floor stands still, everyone at rest before the track starts.

The beat pulls the crowd, driving them further the harder it pushes.

They swing back and forth, oscillating evenly around the centre.

what examiners catch — Students forget that SHM requires proving both the proportionality AND the negative sign in the acceleration equation a = -n²x, or they incorrectly identify damped motion as simple harmonic.

five things, not forty.

The equation IS the definition: proving motion is simple harmonic means producing the form x-double-dot = -n squared (x - c) from whatever you were given, whether that is displacement, velocity or acceleration

The minus sign carries the physics - F(x) = m x-double-dot is a RESTORING force, always aimed back at the centre c, never away from it, which is what makes the motion repeat instead of run away

x = A sin(nt + alpha) + c and x = A cos(nt + beta) + c are the same solution in different clothes - verify one by differentiating it twice and watching the original equation reappear, never by memorising it

v squared = n squared [A squared - (x - c) squared] falls straight out of writing acceleration as v dv/dx - it says speed is greatest at the centre and exactly zero at both ends of the swing

The period 2 pi / n depends on n alone, so amplitude does NOT change how long a swing takes - which is precisely why a pendulum clock keeps time as it winds down

nothing here is a standalone fact.

knowhere maps every concept to what it rests on and what rests on it — 1 underneath this one, 0 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptsimple harmonic motionSimple harmonic motion occurs when acceleration is always proportional to displacement from equilibrium and directed toward it, producing sinusoidal position functions regardless of the physical system.

sits under itkinematics with variable accelerationbecause SHM is the case where acceleration is a function of displacement

4 more, same treatment.

Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.

calculus applied to motionin the appnewton's laws in vector formin the appkinematics with variable accelerationin the appresisted motion and terminal velocityin the app

they don't read it. they run it.

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