Foundations — Energy in SHM — KE + PE = ½kA² (constant)

This page assumes you have seen none of the symbols before. We build each one from a picture, then hand you the finished toolbox so the parent page reads like plain English.

0 · The scene we are describing

Everything in this chapter happens to a block tied to a spring on a smooth table. Look at the picture and hold it in your head — every symbol below is a label on some part of it.

• The block can only slide left–right.
• There is one special spot where the spring is neither stretched nor squeezed — the equilibrium point. We will call that spot "zero".
• Pull the block out, let go, and it swings back and forth forever (in the ideal, frictionless world).

Now we name the parts.

1 · $x$ — displacement (where the block is right now)

The picture: $x$ is the little arrow from the centre line out to the block. It is a signed number — the sign is the direction. Look back at figure s01: the labelled arrow marked $x$ is exactly this.

Why the topic needs it: the spring's push and its stored energy both depend on how far you are from centre. Without a number for "how far", we cannot write any formula.

2 · $A$ — amplitude (the furthest the block ever gets)

The picture: the block swings between $x=+A$ (far right) and $x=-A$ (far left). Those two extreme spots are the turning points: the block stops there for a split second and reverses.

Why the topic needs it: $A$ is set once, at the start, by how far you pulled. Because it never changes during the motion, it is the perfect thing to build a constant total energy out of — which is why $\frac{1}{2}k{A}^2$ works.

3 · $k$ — spring constant (how stiff the spring is)

The picture: imagine pulling the block a distance $x$. The spring pulls back with a force. Double the stretch → double the pull. The proportionality number in "pull $=$ (number) $\times$ stretch" is $k$.

This is Hooke's Law: $F=-kx.$

• The $k$ is the stiffness we just described.
• The $x$ is the displacement from section 1.
• The minus sign means the force points opposite to the displacement: pull right ($x>0$) and the spring pulls left ($F<0$). This "always aimed back toward centre" behaviour is what makes it a restoring force.

Why the topic needs it: $k$ links position to force, and force is what stores energy. The whole energy formula is built on $k$.

4 · $m$ — mass (how heavy the block is)

The picture: a heavy block responds sluggishly to the same spring; a light block snaps back and forth quickly. Same spring, different weights → different rhythms.

Why the topic needs it: kinetic energy (motion energy) depends on mass. Also, $m$ and $k$ together decide how fast the oscillation is (next symbol).

5 · $\omega$ — angular frequency (how fast it cycles)

It is built from the two things we already have: $\omega =\sqrt{\frac{k}{m}}.$

Reading this formula in words: stiffer spring (bigger $k$) → faster wobble; heavier block (bigger $m$) → slower wobble. The square root is there because ${\omega }^2=k/m$ pops out naturally when you solve the motion equation (done in the SHM definition note).

Why the topic needs it: $\omega$ is the bridge that lets the parent write energy two equivalent ways, $\frac{1}{2}k{A}^2$ and $\frac{1}{2}m{\omega }^2{A}^2$. They are the same thing because $m{\omega }^2=k$.

6 · $t$, $\cos$, $\varphi$ — the motion as a function of time

The parent starts from: $x(t)=A\cos (\omega t+\varphi ).$

Let us earn every piece.

• $t$ = time in seconds. "$x(t)$" just means "the value of $x$ at time $t$" — position is a function of time.
• $\cos$ (cosine) = a smooth wave that rocks between $+1$ and $-1$ forever, never getting bigger. Multiply it by $A$ and you get a smooth rock between $+A$ and $-A$ — exactly the block's motion. That is why cosine and not some other shape: it repeats, it is bounded, and its wiggle matches the spring's.
• $\varphi$ (phi) = the phase, a starting-offset. It just says "where in the cycle the block was at $t=0$." If you release from the far right, $\varphi =0$; release from elsewhere and $\varphi$ shifts the wave sideways.

Why the topic needs it: to prove the energy is constant, the parent plugs this $x(t)$ into the energy formulas. The magic identity ${\sin }^2+{\cos }^2=1$ then wipes out the time — that is the whole point.

7 · $v$ — velocity (how fast, and which way, the block moves)

The picture: near the ends the block crawls, stops, reverses ($v\to 0$). Through the middle it whizzes ($v$ biggest). This is velocity in SHM.

The parent writes: $v(t)=\frac{dx}{dt}=-A\omega \sin (\omega t+\varphi ).$

The symbol $\frac{dx}{dt}$ is the derivative — read "the rate at which $x$ changes as time ticks." We use a derivative here, and not just "distance over time", because the block's speed is different at every instant; the derivative is the tool that gives the instantaneous speed.

Why the topic needs it: kinetic energy is $\frac{1}{2}m{v}^2$, so we need $v$ before we can talk about motion-energy at all.

8 · Energy, $\frac{1}{2}$, and "$\propto$"

• Energy = a single number measuring "capacity to do work." Measured in joules (J). Two flavours here: kinetic (because moving) and potential (because stretched).
• The $\frac{1}{2}$ in $\frac{1}{2}m{v}^2$ and $\frac{1}{2}k{x}^2$ is not a guess — it falls out of adding up (integrating) the force as the stretch builds from $0$ to $x$. Half of the maximum, because the force grew linearly from zero. (The parent shows this integral in Step 1.)
• $\propto$ means "proportional to" — "grows in step with." Writing $E\propto A^2$ says: if you double $A$, $E$ grows by $2^2=4$. No equals sign needed because we only care about the scaling, not the exact constant.

Why the topic needs it: these two energies, added, give the constant total — the headline result. The idea that "no coins are lost" is conservation of mechanical energy; drop that assumption and you get damped oscillations where the total slowly leaks away.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, reread its section above before opening the parent page.