Torque and Rotational Dynamics
Core ideas
Torque τ = rF sinθ measures a force's tendency to rotate an object about an axis; it depends on the lever arm and the angle between r and F. Static equilibrium requires both ΣF = 0 and Στ = 0. Rotational inertia I depends on mass distribution; larger I means more resistance to angular acceleration. Newton's second law for rotation is Στ = Iα.
Candidates must locate the axis, compute lever arms, and apply equilibrium to beams, ladders, and hanging signs. Choosing the pivot at an unknown force eliminates that force from the torque equation - a powerful shortcut.
Angular kinematics mirror linear kinematics (ω = ω₀ + αt, etc.), with the same graphical relationships.
Traps
- Using the full distance r instead of the perpendicular lever arm.
- Forgetting that a force through the pivot produces zero torque.
- Assuming more mass always means more I - distribution matters (a hoop beats a disk of the same mass).
- Omitting one of the two equilibrium conditions.
Rule of thumb
For static equilibrium, place the pivot at the point where the most unknown forces act, then write Στ = 0 about that pivot. Check ΣF = 0 afterward.
Worked example
A uniform 4 m beam of weight 200 N is supported at one end and by a cable at the other, with a 100 N load 1 m from the pivot. About the pivot: torque from load = 100·1 = 100 N·m; from beam weight = 200·2 = 400 N·m; total = 500 N·m clockwise. Cable torque = T·4, so T = 125 N.