← Mechanical Systems

5 Rotational dynamics

Torque τ = F r (perpendicular) · τ = I α The rotational analogue of F = ma. α = angular acceleration. Rotational kinematics ω = ω₀ + α t · θ = ω₀t + ½αt² · ω² = ω₀² + 2αθ Same shape as linear SUVAT. Angular momentum L = I ω Conserved when no external torque acts. Rotational KE E<sub>k(rot)</sub> = ½ I ω²

From the rim to the axle

Linear ↔ angular d = r θ · v = r ω · a = r α For a string unwinding from a wheel, or a wheel rolling without slipping: the distance, speed and acceleration at radius r are r times the angle (in radians), the angular velocity and the angular acceleration.

Angular momentum of something moving in a line

Point mass passing an axis L = m v r r is the distance from the axis, with v at right angles to it. It is the same as Iω for a point mass, since I = mr² and ω = v/r. Use it when a person or object moving in a straight line joins something that turns: L before = L after.

Conservation of angular momentum

  • With no external torque, I₁ω₁ = I₂ω₂.
  • Ice skater pulls arms in → I decreases → ω increases (spins faster). Classic exam scenario.
  • A rolling object shares energy between translation (½mv²) and rotation (½Iω²).
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Momentum is always conserved in a collision — kinetic energy is not

Momentum (p = mv, or L = Iω for rotation) is conserved in EVERY collision with no external force or torque. Kinetic energy is conserved only in an ELASTIC collision. Solve momentum FIRST to find the unknown velocity, then compute kinetic energies afterward to classify the collision — assuming elasticity before checking is the most common route to a wrong answer. The same trap appears with angular momentum: a skater pulling their arms in conserves L, but their rotational KE actually INCREASES, because they do work against the rotation pulling their arms in. Conservation of one quantity never implies conservation of the other.

Practise just this