Rotational Inertia Race

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About This MicroSim

This simulation races four objects with different shapes (and therefore different rotational inertias) down an inclined ramp. All objects have the same mass and radius, but their different mass distributions affect how fast they accelerate.

The Physics

The acceleration of a rolling object down a ramp is:

a = g sin θ / (1 + I/mr²)

Where: - g = gravitational acceleration - θ = ramp angle - I = rotational inertia - m = mass - r = radius

Objects with smaller I/mr² ratios accelerate faster!

Race Rankings (Fastest to Slowest)

1. Solid Sphere (I = 0.4 mR²) - Most mass near center
2. Solid Cylinder (I = 0.5 mR²)
3. Hollow Sphere (I = 0.67 mR²)
4. Hoop/Ring (I = 1.0 mR²) - All mass at maximum radius

Key Insight

The race results are independent of mass and radius - shape alone determines the winner! This is because both gravitational force and rotational inertia scale with mass.

Lesson Plan

Discussion Questions

1. Why does the solid sphere always win?
2. What if we slid the objects instead of rolling them?
3. How does changing the ramp angle affect the race?