Foundations — Spring-mass system — horizontal, vertical

Before you can read the parent note comfortably, you need to earn every symbol it throws at you. Below, each item gives: plain words → the picture → why the topic needs it. They are ordered so each one leans only on the ones above it — no symbol appears until the block that defines it.

The building blocks, one at a time

1. Natural length and the coordinate axis

Picture the spring just sitting there, coils evenly spaced, neither stretched nor squished. Every "how far have we strayed" measurement starts from thinking about this relaxed shape.

• The axis convention (fix it once, for good): we point one axis along the spring's line of action and call that direction positive.

  • Horizontal case: a spring on a table. We point the axis to the right and call rightward positive.
  • Vertical case: a hanging spring. We point the axis straight down and call downward positive. In each case there is only one axis, chosen along the motion — so "positive" is never ambiguous, it just means "the way the axis points."

• Why the topic needs it: the natural length is the reference for how much a spring is stretched, and the single-axis convention lets us write one clean equation instead of juggling left/right/up/down by hand.

2. Position and the coordinate $x$

Picture a mass sitting on a smooth table with a ruler laid underneath along the axis. The zero mark of the ruler is placed exactly where the mass rests when nothing disturbs it. Then $x$ is simply what the ruler reads under the mass.

• Why the topic needs it: every law on the page — the force, the energy, the wave-like motion — is written as a rule about how $x$ behaves as time passes. Without a clean zero point, none of the formulas simplify.

3. Equilibrium — "home base"

Picture the mass perfectly still, the spring neither pulling nor pushing harder than the other forces. That balance point is home.

• Why the topic needs it: oscillation is swinging around this point. In the horizontal case home is the spring's natural length; in the vertical case gravity stretches the spring first, so home moves down — but it is still just "the balance point."

4. The spring constant $k$

Picture two springs: a flimsy slinky ($k$ small) and a stiff car spring ($k$ large). Pull each by the same distance — the stiff one yanks back much harder. That "harder per metre" is exactly $k$.

• Why the topic needs it: $k$ is the strength of the "pull-back" that causes the oscillation. It appears in every period and frequency formula. This comes straight from Hooke's Law.

5. Force $F$ and the minus sign in $F=-kx$

Why the minus? Look at the picture: stretch the spring to the right ($x>0$) and it pulls left (force negative). Push it left ($x<0$) and it pushes right (force positive). The sign of $F$ is always opposite the sign of $x$ — that opposite-ness is what the "$-$" encodes.

• Why the topic needs it: this minus sign is the entire reason motion oscillates instead of running away. See Hooke's Law.

6. Mass $m$ and inertia

Picture flicking a marble versus flicking a bowling ball with the same finger-push: the marble leaps, the bowling ball barely stirs. That reluctance is inertia, measured by $m$.

• Why the topic needs it: inertia is why the mass overshoots home instead of stopping there. Spring pulls it home → inertia carries it past → spring pulls it back the other way. Pull-back plus overshoot equals oscillation.

7. Velocity $v$ and the dot notation $\dot{x}$

Picture the mass mid-swing: at that instant $v$ is the speed shown on an imaginary speedometer glued to it, with a sign for direction. Where the position graph rises steeply, $v$ is large.

• Why we use a "dot" and not just "$v$": the dot is a compact way to say "time-derivative of whatever is under it." It lets us stack: one dot = velocity, two dots = acceleration. This keeps the equation of motion short.

8. Acceleration $\ddot{x}$ (double dot)

Picture the mass at the far end of its swing, momentarily stopped ($v=0$) but being yanked hardest by the spring. It is not moving yet, but its motion is about to change fastest — that is large acceleration.

• Why the topic needs it: Newton's second law $F=m\ddot{x}$ links the spring's pull to how the motion changes. Combine it with $F=-kx$ and you get the master equation.

9. Amplitude $A$

Picture the two turning points of the swing, left and right of home. Their distance from home is $A$ on each side (equal, because the spring is symmetric).

• Why the topic needs it: $A$ sets the total energy budget and the maximum speed ($v_{max}=A\omega$), as the energy block below makes precise.

10. The cosine function and the general solution

Picture a wheel turning at a steady rate with a dot on its rim. The shadow of that dot cast onto a wall moves out, slows, reverses, and returns — one full turn of the wheel is one full back-and-forth of the shadow. That shadow's motion is a cosine. This is why cosine is the natural language of oscillation: it is what steady circular turning looks like from the side.

11. The Greek letter $\varphi$ (phase)

Picture the same wiggle slid left or right along the time axis. $\varphi$ measures that slide. If you release from the far end, $\varphi =0$; release from home moving, and $\varphi$ shifts.

• Why the topic needs it: the general solution $x(t)=A\cos (\omega t+\varphi )$ must fit any starting condition, and $\varphi$ is the dial that adjusts the start.

12. Period $T$, frequency $f$, angular frequency $\omega$

These three all describe how the motion is spaced out in time. They are three views of one clock.

Picture a full cosine wiggle of the position-versus-time graph: $T$ is the horizontal width of one complete hump-and-dip. $f$ counts how many humps fit in one second. $\omega$ measures the same rhythm using the angle of a spinning wheel that would trace this exact wiggle.

• Why $\omega$ and not just $f$? The equation of motion naturally produces a $\cos (\omega t)$, whose argument must be an angle. $\omega$ is the tool that converts elapsed time into that angle — that is precisely why it carries the $2\pi$. This links to Simple Harmonic Motion.

13. Energy words: KE, PE, and $E$

Picture a bank account: at the ends of the swing the money is all in the "spring" account (PE full, KE zero); at home it's all in the "motion" account (KE full, PE zero). The total never changes — it just sloshes between the two. See Energy in SHM.

• Why the topic needs it: conservation of $E$ gives a second, independent road to the same $\omega$, and instantly hands you the max speed.

How these foundations feed the topic

Read the map top-down as a story: the coordinate $x$ (measured from equilibrium, itself referenced to the natural length) and the stiffness $k$ together produce Hooke's law. Hooke's law plus Newton's law — which needs the mass $m$ and the dot notation for rates — assemble into the equation of motion. That single equation, solved with the cosine, delivers the angular frequency $\omega =\sqrt{k/m}$, which sets the timing words $T=2\pi \sqrt{m/k}$ and $f$. Separately, the amplitude $A$ together with $\omega$ fixes the energy picture. Both the equation of motion and the energy view are two faces of the same thing: Simple Harmonic Motion.

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, reread that block above.

Connections

• Spring-mass system — horizontal, vertical — the parent topic these foundations unlock.
• Hooke's Law — source of $F=-kx$ and the meaning of $k$.
• Simple Harmonic Motion — where $\omega$, $T$, $\varphi$ all come together.
• Energy in SHM — the KE/PE/total-energy trio in action.
• Springs in Series and Parallel — combining stiffnesses $k$.
• Simple Pendulum — a cousin oscillator to compare timing words against.