Revise: Rotational Dynamics
Scalar and vector products, torque, the relation between torque and moment of inertia, equilibrium, and conservation of angular momentum.
τ = rF sinθ; maximum turning effect when force is perpendicular to the lever arm.
A longer wrench needs less force for the same torque on a bolt.
I = mr² for a point mass — mass farther from the axis matters much more.
Pulling arms in reduces r and sharply cuts a skater's moment of inertia.
Complete equilibrium needs BOTH ΣF = 0 and Στ = 0.
A 300 N child 1.5 m out balances a 450 N adult 1 m out on a seesaw.
L = Iω is the rotational equivalent of linear momentum p = mv.
A 4 kg·m² wheel at 5 rad/s has L = 20 kg·m²/s.
With zero external torque, I₁ω₁ = I₂ω₂ — angular momentum stays constant.
A skater pulling in their arms spins up from 2 to 8 rad/s.
A · B = AB cosθ gives a scalar; maximum when vectors are aligned.
10 N at 60° to a 5 m displacement gives a dot product of 25.
|A × B| = AB sinθ gives a vector, perpendicular to both; maximum when perpendicular.
12 N perpendicular to a 0.4 m arm gives a cross product of 4.8.
τ = Iα is the rotational version of F = ma.
12 N·m on a 3 kg·m² wheel gives α = 4 rad/s².