🚀 Work and Energy: The Universe’s Power Currency
The Big Idea: Energy is Like Money for the Universe
Imagine you have a piggy bank. You put coins in, you take coins out. The coins never disappear—they just move around.
Energy works exactly the same way!
When you push a ball, you’re spending your energy to give the ball speed. When a ball rolls up a hill, its speed energy transforms into height energy. Nothing is lost. It just changes form.
This is the story of how physicists discovered this beautiful truth.
🔧 What is Work?
The Simple Answer
Work = Using force to move something
But here’s the catch: If something doesn’t move, no work is done!
The Pushing Test 🧱
| Action | Force Applied? | Movement? | Work Done? |
|---|---|---|---|
| Push wall for 1 hour | ✅ Yes | ❌ No | ❌ No Work! |
| Push toy car across floor | ✅ Yes | ✅ Yes | ✅ Work Done! |
| Hold heavy backpack standing still | ✅ Yes | ❌ No | ❌ No Work! |
Mind-blowing, right? Your muscles get tired pushing a wall, but physics says you did zero work!
The Magic Formula
Work = Force × Distance × cos(angle)
W = F × d × cos(θ)
- W = Work (measured in Joules, J)
- F = Force (measured in Newtons, N)
- d = Distance moved (measured in meters, m)
- θ = Angle between force and movement direction
Why Does Angle Matter?
Think about pulling a wagon:
graph TD A["Pulling Straight"] --> B["θ = 0°"] B --> C["cos 0° = 1"] C --> D["Maximum Work!"] E["Pulling at Angle"] --> F["θ = 45°"] F --> G["cos 45° = 0.71"] G --> H["Less Work"] I["Pulling Sideways"] --> J["θ = 90°"] J --> K["cos 90° = 0"] K --> L["Zero Work!"]
📏 Work Done by Constant Force
When Force Stays the Same
Imagine pushing a shopping cart with steady force across the store.
Example: You push a 10 kg box with 50 N force for 3 meters.
W = F × d
W = 50 N × 3 m
W = 150 Joules
That’s 150 Joules of energy transferred to the box!
Real-Life Example: Lifting a Book
You lift a 2 kg book from floor to table (1 meter high).
Force needed = Weight = m × g
Force = 2 kg × 10 m/s²
Force = 20 N
Work = 20 N × 1 m = 20 Joules
🌊 Work Done by Variable Force
When Force Changes
Real life isn’t always simple! Sometimes the force you apply changes as you go.
The Spring Example 🎯
When you stretch a spring:
- At first: Easy! Little force needed
- As you stretch more: Harder! More force needed
The force increases as you stretch!
Spring Force: F = kx
(k = spring constant, x = stretch distance)
Calculating Variable Work
For changing forces, we use area under the force-distance graph!
graph TD A["Force-Distance Graph"] --> B["Calculate Area"] B --> C["Area = Work Done"] C --> D["For springs: W = ½kx²"]
Spring Work Formula
Work on spring = ½ × k × x²
Example: Stretching a spring (k = 200 N/m) by 0.1 m:
W = ½ × 200 × (0.1)²
W = ½ × 200 × 0.01
W = 1 Joule
⚡ The Work-Energy Theorem
The Most Powerful Idea
All the work done on an object equals the change in its kinetic energy!
Work = Change in Kinetic Energy
W = ΔKE = KE_final - KE_initial
Think of It Like This
Your energy “payment” (work) goes directly into the object’s “speed account” (kinetic energy).
graph LR A["You Do Work"] --> B["Energy Transfer"] B --> C["Object Speeds Up"] C --> D["Kinetic Energy Increases"]
Example: Pushing a Skateboard
A 5 kg skateboard starts at rest. You push it, doing 40 J of work.
W = KE_final - KE_initial
40 = KE_final - 0
KE_final = 40 J
Since KE = ½mv²:
40 = ½ × 5 × v²
v² = 16
v = 4 m/s
The skateboard now moves at 4 m/s!
🏃 Kinetic Energy: Energy of Motion
The Faster You Go, The More You Have!
Kinetic Energy (KE) = Energy an object has because it’s moving
KE = ½ × mass × velocity²
KE = ½mv²
Why Velocity is Squared (So Important!)
| Speed | KE Factor |
|---|---|
| 1× speed | 1× energy |
| 2× speed | 4× energy |
| 3× speed | 9× energy |
| 4× speed | 16× energy |
Double your speed = FOUR times the energy!
This is why car crashes at high speed are so dangerous.
Example: Running Child vs. Walking Adult
| Person | Mass | Speed | KE |
|---|---|---|---|
| Child running | 25 kg | 6 m/s | ½ × 25 × 36 = 450 J |
| Adult walking | 70 kg | 1.5 m/s | ½ × 70 × 2.25 = 79 J |
The small running child has MORE kinetic energy!
📦 Potential Energy: Stored Energy Waiting to Act
Energy’s Secret Bank Account
Potential energy is stored energy based on position or condition.
Think of it as a stretched rubber band or a ball held high—energy waiting to be released!
graph TD A["Potential Energy Types"] A --> B["Gravitational PE"] A --> C["Elastic PE"] B --> D["Height-based storage"] C --> E["Stretch/compression storage"]
🌍 Gravitational Potential Energy
Higher = More Stored Energy
The higher you lift something, the more energy you store in it.
Gravitational PE = mass × gravity × height
GPE = mgh
The Roller Coaster Secret 🎢
At the top of a roller coaster:
- Maximum height → Maximum GPE
- Minimum speed → Minimum KE
At the bottom:
- Minimum height → Minimum GPE
- Maximum speed → Maximum KE
Energy just converts back and forth!
Example: Book on a Shelf
A 1 kg book sits on a shelf 2 m high.
GPE = mgh
GPE = 1 kg × 10 m/s² × 2 m
GPE = 20 Joules
If it falls, those 20 J become kinetic energy!
The Reference Point Trick
GPE is always measured from somewhere. Usually the ground, but you choose!
| Reference Point | Book at 2m | Book on floor |
|---|---|---|
| Ground (0m) | GPE = 20 J | GPE = 0 J |
| Table (1m) | GPE = 10 J | GPE = -10 J |
The difference is always the same: 20 J!
🎯 Elastic Potential Energy
Springs and Stretchy Things
When you stretch or compress a spring, you store energy in it!
Elastic PE = ½ × spring constant × stretch²
EPE = ½kx²
The Bow and Arrow 🏹
- Pull back the bow → Store elastic PE
- Release → EPE converts to arrow’s KE
- Arrow flies! → KE in motion
Why x² Matters
| Stretch | EPE Factor |
|---|---|
| 1 cm | 1× |
| 2 cm | 4× |
| 3 cm | 9× |
Double the stretch = FOUR times the energy!
Example: Toy Cannon
A spring (k = 500 N/m) is compressed 0.05 m.
EPE = ½ × k × x²
EPE = ½ × 500 × (0.05)²
EPE = ½ × 500 × 0.0025
EPE = 0.625 Joules
When released, this energy launches the toy!
🔄 The Grand Connection: Conservation of Energy
Energy Cannot Be Created or Destroyed!
It only transforms from one type to another.
graph LR A["GPE at top"] --> B["Ball falls"] B --> C["KE at bottom"] C --> D["Ball bounces up"] D --> E["GPE again"]
The Pendulum Story
A swinging pendulum perfectly shows energy transformation:
| Position | Height | Speed | GPE | KE |
|---|---|---|---|---|
| Left (highest) | Max | Zero | Max | Zero |
| Middle (lowest) | Zero | Max | Zero | Max |
| Right (highest) | Max | Zero | Max | Zero |
Total energy stays constant!
🧮 Quick Reference Formulas
| Concept | Formula | Units |
|---|---|---|
| Work (constant F) | W = Fd cos(θ) | Joules (J) |
| Work (variable F) | W = Area under F-d graph | J |
| Work-Energy Theorem | W = ΔKE | J |
| Kinetic Energy | KE = ½mv² | J |
| Gravitational PE | GPE = mgh | J |
| Elastic PE | EPE = ½kx² | J |
🎓 Key Takeaways
- Work requires both force AND movement in the same direction
- Variable force work = area under force-distance curve
- Work-Energy Theorem connects force to speed change
- Kinetic energy increases with velocity squared
- Gravitational PE depends on height from reference
- Elastic PE increases with stretch squared
- Energy is conserved—it only transforms, never vanishes!
🌟 You’ve Got This!
Remember: Energy is the universe’s currency. Now you know how to count it, store it, and watch it transform. Every ball you throw, every spring you stretch, every hill you climb—you’re playing with the same physics that powers stars!
Keep exploring. Keep questioning. You’re already thinking like a physicist! 🚀