⚖️ Lesson 7: Proportions & Variation

How quantities scale — the secret to quick physics estimates

🎯 Learning Objectives

• Identify and solve direct, inverse, and joint variation problems
• Use proportions for unit conversions and scaling
• Apply variation to physics laws: F ∝ a, F ∝ 1/r², P ∝ V, etc.
• Make quick "back-of-envelope" estimates using scaling

🔑 Key Concepts

Direct Variation: y = kx

"y is directly proportional to x" or "y ∝ x". Double x → double y. Graph: straight line through origin.

Physics examples: F = ma (F ∝ a at constant m), p = mv (p ∝ v), KE = ½mv² (KE ∝ v² — not direct!), v = fλ (v ∝ f, v ∝ λ)

Inverse Variation: y = k/x

"y is inversely proportional to x" or "y ∝ 1/x". Double x → halve y. Graph: hyperbola.

Physics examples: F = Gm₁m₂/r² (F ∝ 1/r²), P = F/A (P ∝ 1/A), f = 1/T (f ∝ 1/T), R = V/I (R ∝ 1/I)

Joint Variation: y = kxz

y varies directly with multiple variables. F = ma (F ∝ m AND F ∝ a).

Power Laws: y = kxⁿ

Many physics relationships are power laws. n = 1 (direct), n = -1 (inverse), n = 2 (square), n = -2 (inverse square).

Law	Relationship	Type
Newton's 2nd	F = ma	F ∝ a, F ∝ m
Gravity	F = Gm₁m₂/r²	F ∝ 1/r² (inverse square)
Coulomb	F = kq₁q₂/r²	F ∝ 1/r²
Kinetic Energy	KE = ½mv²	KE ∝ v²
Centripetal	F = mv²/r	F ∝ v², F ∝ 1/r
Period (spring)	T = 2π√(m/k)	T ∝ √m, T ∝ 1/√k
Period (pendulum)	T = 2π√(L/g)	T ∝ √L, T ∝ 1/√g
Intensity	I = P/(4πr²)	I ∝ 1/r²

📐 Worked Examples

Example 1: Direct Variation — Force and Acceleration

A force of 10 N produces 2 m/s² acceleration. What force produces 5 m/s²?

F ∝ a  →  F₁/a₁ = F₂/a₂
10/2 = F₂/5
F₂ = 25 N
            

Example 2: Inverse Square — Gravitational Force

Two masses attract with 100 N at distance r. What force at distance 3r?

F ∝ 1/r²  →  F₁r₁² = F₂r₂²
100·r² = F₂·(3r)²
100 = F₂·9
F₂ = 11.1 N
            

Example 3: Square Root — Pendulum Period

A pendulum of length 1m has period 2s. What length gives 3s period?

T ∝ √L  →  T₁/√L₁ = T₂/√L₂
2/√1 = 3/√L₂
2 = 3/√L₂
√L₂ = 1.5
L₂ = 2.25 m
            

Example 4: Combined — Scaling a Projectile

Range R = v₀²sin(2θ)/g. At 45°, sin(90°)=1, so R = v₀²/g. v₀ doubles → what happens to R?

R ∝ v₀²
If v₀ → 2v₀, then R → (2v₀)² = 4v₀²
Range quadruples!
            

🏋️ Practice Problems

1. F = ma. If m doubles and a triples, F changes by what factor?
2. F = Gm₁m₂/r². If both masses double and r halves, F changes by what factor?
3. KE = ½mv². If v increases by 50%, KE increases by what percent?
4. T = 2π√(L/g). On Moon (g = 1.6 m/s²), a 1m pendulum has what period? (Earth g=9.8, T_earth=2s)
5. P = F/A. If force triples and area doubles, pressure changes by what factor?
6. I = P/(4πr²). At distance 2r, intensity is what fraction of intensity at r?
7. v = fλ. If f doubles and λ halves, v changes by what factor?
8. Centripetal: F = mv²/r. Design a curve where v doubles but F stays same — what must r do?
9. Hooke's Law: F = kx. Two identical springs in parallel: effective k? In series: effective k?

✅ Answers

Click to reveal answers

1) 6× 2) 16× 3) 125% increase (2.25×) 4) T_moon ≈ 5s 5) 1.5× 6) ¼ 7) no change

8) r must quadruple 9) Parallel: 2k; Series: k/2

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