Foundations — Simple harmonic motion — definition, restoring force F = −kx

This page assumes you have seen none of the notation. We build each symbol from a picture, in an order where every new idea only leans on ones already made.

0. The stage: a mass, a spring, a home position

Picture a block sitting on a smooth table, tied to a wall by a spring. Left alone, it sits still at one spot. Nudge it, and it slides back and forth. Everything in this topic is spoken about this picture.

Why do we need a "home"? Because every other symbol measures something relative to home. Without agreeing where zero is, "displacement" and "restoring force" would be meaningless.

1. $x$ — displacement (a signed distance from home)

The picture: lay a ruler along the table with $0$ at home. The block's position on that ruler is $x$. Look at the red dot in the figure above — its label $x$ is read straight off the ruler.

Why the sign matters: the whole topic is about a force that opposes $x$. If $x$ had no sign, we could never say "opposite." A signed $x$ lets one formula handle both sides at once.

2. $A$ — amplitude (the farthest $x$ ever gets)

The picture: two dashed walls at $-A$ and $+A$ in the figure below. The block bounces between them, touching each wall for an instant then turning around.

Why we need it: $A$ sets the size of the motion. A gentle nudge gives small $A$; a hard pull gives big $A$. Remarkably (we'll see) the rhythm does not depend on $A$ — but we still need the symbol to describe how wide the swing is.

3. Force $F$ and the restoring idea

Look at the two coral arrows in the figure: on the right side the arrow points left, on the left side it points right. Both aim at home. That "aim at home" behaviour is what the word restoring captures — and it is why the block never runs away; it is always being dragged back.

4. $k$ — the force constant (how stiff the spring is)

The picture: in the figure above, pull the block twice as far and the coral arrow becomes twice as long. That "arrow length grows in step with distance" is exactly what $k$ measures — it is the proportionality between how far and how hard.

(This proportional-spring law has its own home: see Hooke's Law.)

5. Rate of change — velocity $v$ and acceleration $a$

We need language for how position changes over time. Two words:

The picture: watch the block. At home it is racing through (big $v$). At the walls $\pm A$ it has momentarily stopped ($v=0$) but is being yanked hardest (big $a$). So $v$ and $a$ peak at opposite moments.

Why we bring in the derivative here

To turn "how fast $x$ changes" into an exact symbol, we need one mathematical tool: the derivative. We choose it — and not simple division $\Delta x/\Delta t$ — because the velocity keeps changing at every instant, so we need the instantaneous rate, the slope at a single point, not an average over a chunk of time.

The picture: on a graph of $x$ against time, the slope of the curve at each point is $v$; how that slope itself bends is $a$. A straight line has zero acceleration; a curving line accelerates.

6. Newton's second law — the bridge $F=ma$

Why the topic needs it: we have a rule for the force ($F=-kx$, §4) and a symbol for acceleration ($a=d^{2}x/d{t}^2$, §5). Newton's law is the bridge that welds them into one statement about the motion: $ma=-kx.$ Without this bridge, the spring law and the motion would be two unconnected facts.

7. $\omega$ — angular frequency (the rhythm number)

Rearranging $ma=-kx$ gives $a=-\frac{k}{m}x$. The combination $k/m$ keeps appearing, so we give it a name.

The picture: think of a spot going around a circle at a steady rate; its shadow on a wall slides back and forth exactly like our block. $\omega$ is how many radians of that circle are swept each second — that's why it's called angular frequency even though our block moves in a straight line. This shadow connection is the whole story of Uniform Circular Motion.

8. The sine wave $A\cos (\omega t+\varphi )$ — the shape of the motion

The picture: plot the block's position against time and you get a smooth wave that rises and falls forever — a cosine curve, drawn below. Its height is $A$, its repeat-length in time is $T$, and $\varphi$ shifts it sideways.

Why sine/cosine and nothing else

We need a function whose second derivative equals minus itself (because §6 said $a=-{\omega }^{2}x$, i.e. the acceleration is $-{\omega }^2$ times the position). Ask: what shape, differentiated twice, comes back to itself but flipped in sign? Only the sine/cosine family does this — that is precisely why SHM is a sine wave and not some other curve. The full solving of this is the job of the parent note; here we just meet the symbols.

9. Energy symbols $KE$, $U$, $E$

The picture: a valley shape $U=\frac{1}{2}k{x}^2$. The block is like a marble rolling in this bowl: high on the sides = lots of stored energy, no speed; at the bottom = no stored energy, top speed. This "bowl" picture is the deep reason SHM is everywhere, and it's the subject of Energy in SHM and (for why every valley bottom is a parabola) Taylor Series.

Prerequisite map

Equipment checklist

Hide the right side and test yourself — you are ready for SHM once every line is instant.

Connections

• Parent topic — SHM definition & $F=-kx$
• Hooke's Law — where the linear spring force $F=-kx$ comes from
• Uniform Circular Motion — the circle whose shadow is our block (source of $\omega$)
• Energy in SHM — the bowl $U=\frac{1}{2}k{x}^2$ and energy sloshing
• Taylor Series — why every valley bottom is a parabola
• Simple Pendulum — another restoring force in disguise
• Damped Oscillations — what changes when we add friction