# Hooke's Law
*10 beats · narrated by Antoni · Canvas Tutor v0.5*
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## 1. Step 01 — Topic
> 🎙 *Here's a question: when you stretch a spring, how does it push back? Today we'll uncover Hooke's Law — the rule that tells us exactly how springs (and lots of other things) resist being deformed.*
> "When you stretch a spring, how hard does it push back — and does it depend on how far you stretch it?"

## 2. Step 02 — Definition
> 🎙 *Hooke's Law says the restoring force a spring exerts is proportional to how far you've stretched or compressed it from its natural length. We call that displacement x. The law is: F equals negative k x.*
The restoring force of a spring is proportional to its displacement x from the natural length:
$$F \;=\; -k\,x$$
k — spring constant (N/m)
x — displacement (m)
F — restoring force (N)

## 3. Step 03 — The Minus Sign
> 🎙 *That negative sign is everything. It means the spring always pushes back opposite to the direction you've displaced it. Stretch it right, it pulls left. Compress it left, it pushes right. That's why we call it a restoring force.*
> ✗ The minus sign means the force is always opposite to displacement — it restores, it doesn't amplify!
$$F \;=\; \underbrace{-k}_{\text{restoring}}\, x$$
> ✓ x > 0 (stretched) → F x  0 (pushes out)

## 4. Step 04 — Diagram — Rest
> 🎙 *Let's draw this out. Here's a spring attached to a wall on the left and a block on the right. In the natural state, the spring sits at its rest length and exerts no force at all.*
Natural length:spring relaxed,x = 0, F = 0.
*[Diagram: 8 elements — baseline, baseline, text, dashedLine, text, block 'm', +2 more]*

## 5. Step 05 — Diagram — Stretched
> 🎙 *Now we pull the block to the right by a distance x. The spring stretches, and immediately it pulls back to the left with force F equals negative k x. The bigger the stretch, the bigger the pull-back.*
Stretched:x > 0F = −kx (pulls left)
$$F = -kx$$
*[Diagram: 8 elements — text, dashedLine, text, block 'm', dashedLine 'x=0', dashedLine, +2 more]*

## 6. Step 06 — Spring Constant k
> 🎙 *The spring constant k tells you how stiff the spring is. A large k means a very stiff spring — it takes a big force to produce even a small stretch. A small k means a floppy spring — even a tiny force creates a large displacement.*
The spring constant k measures stiffness:
$$k \;=\; \dfrac{|F|}{|x|} \quad \text{(N/m)}$$
> ✗ Large k → very stiff(steel spring, bone)
> ✓ Small k → very soft(bungee cord, foam)

## 7. Step 07 — Worked Example
> 🎙 *Let's solve a worked example. A spring has k equal to 200 newtons per metre. We compress it by 5 centimetres — that's 0.05 metres. What force does it exert?*
Given:  k = 200 N/m,  x = −0.05 m
**Student work:**
- $$F = -k\,x$$
- $$F = -(200)(-0.05)$$
- $$F = +10 \text{ N}$$
> ✓ Positive F means the spring pushes outward — correct for compression!

## 8. Step 08 — Trap — Missing Sign
> 🎙 *Here's a classic trap. Students often drop the negative sign and write F equals k x. That gives the right magnitude but the wrong direction — your force points the same way as the stretch, which is physically impossible for a spring.*
Common mistake:
$$\color{red}{F = +kx} \quad \leftarrow \text{ WRONG}$$
Always write:
$$F = -kx \quad \leftarrow \text{ restoring}$$
> ✗ Dropping the minus sign means your spring amplifies displacement instead of fighting it — springs don't do that!

## 9. Step 09 — Verify
> 🎙 *Let's do a quick limit check to make sure the equation makes sense. If x equals zero — no displacement — then F equals zero. Perfect: a spring at rest length feels no force. And as x grows larger, F grows proportionally. Both limits match reality.*
**Verification:**
- $$x = 0 \Rightarrow F = 0$$ — no stretch, no force ✓
- $$x \uparrow \Rightarrow |F| \uparrow$$ — bigger stretch, bigger force ✓
- $$\text{sign}(F) = -\text{sign}(x)$$ — force opposes displacement ✓
> **Result:** $$F = -k\,x$$

## 10. Step 10 — Elastic Limit
> 🎙 *One last big idea: Hooke's Law only works within the elastic limit. Stretch a spring too far and it deforms permanently — the law breaks down and F is no longer proportional to x. Always check that you're in the linear region.*
Hooke's Law holds only within the elastic limit. Beyond it, the spring deforms permanently.
*[Diagram: 6 elements — axes, text, text, vectorArrow 'Hooke region', dashedLine 'elastic limit', vectorArrow 'permanent deformation']*
> — In the yellow region: F ∝ x (linear, Hooke's Law applies).In the red region: law breaks down.