Foundations — Velocity and acceleration in SHM — v = ω√(A² − x²)

This page builds the topic's every symbol from nothing — plain meaning, then a picture, then the reason the topic can't live without it. We introduce no letter before its own section. Read top to bottom; each block uses only what came before, and by the end you will assemble the target relation yourself.

1. The oscillation itself — what are we even watching?

Picture a small bead sliding back and forth along a straight groove. There is one special spot — the middle — that the bead keeps returning to. Everything in this topic is measured from that middle.

Look at the figure: the groove is horizontal, the middle is marked with a small circle, and the bead is shown once to the left and once to the right. The topic is the story of that bead's journey.

2. $x$ — displacement, the "where am I" number

Why the topic needs it: the whole point is to find speed and acceleration at a given place. Without a labelled position axis there is no "place" to talk about. Notice the sign already tells us direction, which we'll reuse for acceleration.

3. $A$ — amplitude, the "edge of the world"

The bead does not fly off forever; it swings out, slows, stops, and comes back. The farthest it ever reaches has a name.

Look at the figure: the two turning points sit at $x=+A$ and $x=-A$. The shaded band between them is the entire territory the bead can occupy.

4. Turning space into a spinning circle — the reference circle

Here is the trick that makes SHM easy. Take a point moving steadily around a circle of radius $A$. Shine a light from the side and watch its shadow on the groove below. That shadow performs exactly SHM.

Look at the figure: the violet dot goes round the circle at constant speed. Its horizontal shadow (magenta) slides back and forth between $-A$ and $+A$ — the bead. The radius of the circle equals the amplitude $A$. This is the Reference circle (projection of uniform circular motion).

Why the topic needs it: it converts a back-and-forth motion (hard to describe) into a steady rotation (easy to describe with angles). Every symbol below is really an angle or a rate of turning on this circle.

5. Angle $\theta$, and the two functions $\sin \theta$ and $\cos \theta$

To say where on the circle the dot is, we need an angle — and a rule for reading off its coordinates.

Look again at the figure of Section 4: the orange arc is $\theta$, opening counter-clockwise from the right-pointing zero-line. Fixing this convention is what makes the next two definitions unambiguous.

6. $\omega$ — angular frequency, "how fast the circle spins"

The dot turns at a steady rate. That rate has a name.

Picture: a fast spinner (large $\omega$) makes the shadow rattle back and forth quickly; a slow spinner makes a lazy sway. $\omega$ sets the pace of the whole oscillation — see Angular frequency ω and time period T.

7. $\varphi$ — phase, "where the dot started"

Two beads can swing with the same $A$ and $\omega$ yet be out of step — one at the middle while the other is at the edge. What distinguishes them is their starting angle.

Picture: rotate the whole circle by $\varphi$ before starting the clock. This is the foundation of Phase and phase difference. If it confuses you now, set $\varphi =0$ everywhere — the bead simply starts at the middle moving outward.

8. $v$ — velocity, and the chain rule that gives it

Why the derivative? "Rate of change" is exactly what a derivative measures: how much $x$ moves in a whisker of time $dt$. No other tool gives instantaneous speed; average speed over a whole swing would hide the fact that the bead is fastest at the middle.

Here is where $\cos \theta$ from Section 5 does real work: the derivative of sine is cosine, so velocity naturally carries a cosine. This single line is what later lets us write $\cos \theta =\frac{v}{A\omega }$.

Picture on the circle: the shadow moves fastest when the dot crosses the top or bottom (its horizontal motion is greatest) — exactly when the shadow sits at the middle, $x=0$. So speed is maximum at the centre, zero at the edges.

9. $a$ — acceleration, derived by differentiating once more

Why a second derivative? Velocity itself is not constant in SHM — the bead speeds up toward the middle and slows toward the edges. To capture that changing-ness we differentiate the velocity of Section 8.

Why the minus sign matters: it says acceleration always points opposite to displacement — back toward the middle. That restoring behaviour is the definition of SHM. See Simple Harmonic Motion — definition a = −ω²x.

10. The Pythagorean identity — the bridge that removes time

We now have two equations that both hide the same angle $\theta =\omega t+\varphi$: $\sin \theta =\frac{x}{A}\text{(from Section 5)},\cos \theta =\frac{v}{A\omega }\text{(from Section 8)}.$ We want $v$ in terms of $x$ alone — with the hidden angle (and hence time) gone. The tool for that is the identity linking $\sin$ and $\cos$.

Picture: on the reference circle of radius $1$, a point has coordinates $(\cos \theta ,\sin \theta )$. Its distance from the centre is always $1$, and Pythagoras on that right triangle gives exactly ${\cos }^2\theta +{\sin }^2\theta =1$. This is Pythagorean identity sin² + cos² = 1.

11. Energy words — kinetic and potential (the see-saw)

The target relation is secretly energy book-keeping, so two more words:

Look at the figure: as $x$ grows the potential bar rises and the kinetic bar falls by the same amount — a see-saw whose total height never changes. That constant total is precisely why $v$ shrinks exactly as $x$ grows, reproducing $v=\pm \omega \sqrt{A^2-x^2}$.

How these foundations feed the topic

Each arrow below just says "is needed before." Read it as a checklist of order: equilibrium → displacement → amplitude → circle → angle & its two functions → spin rate → phase → the two derivatives → the identity → the final relation, with energy as the parallel route.

Equipment checklist

Cover the right side and test yourself. If any answer is shaky, reread that section before the main note.

Connections

• Simple Harmonic Motion — definition a = −ω²x
• Energy in SHM — kinetic and potential
• Phase and phase difference
• Reference circle (projection of uniform circular motion)
• Angular frequency ω and time period T
• Pythagorean identity sin² + cos² = 1