Rotation and torque: study guide

Key ideas

Rotation mirrors straight-line motion. Torque plays the role of force, moment of inertia the role of mass, and angular velocity and acceleration replace velocity and acceleration.

• Torque is τ = rF sin θ, the force times the perpendicular distance from the axis. A net torque gives an angular acceleration: τ = Iα.
• Moment of inertia depends on how the mass is spread out: the farther from the axis, the larger I. A solid disk has ½MR², a hoop MR² and a solid sphere ⅖MR².
• Angular momentum L = Iω is conserved when no outside torque acts, which is why a skater spins faster when the arms are pulled in.
• A rolling object without slipping has v = ωR, and its kinetic energy is the sum of ½mv² and ½Iω².

Key equations

• τ = rF sin θ
• τ = Iα
• L = Iω
• KE = ½Iω²
• I = ½MR² (solid disk)
• v = ωR (rolling without slipping)

Worked example

A solid disk of mass 2.0 kg and radius 0.50 m spins at 10 rad/s. I = ½MR² = 0.5 × 2.0 × 0.25 = 0.25 kg·m². Angular momentum L = Iω = 2.5 kg·m²/s. Kinetic energy = ½Iω² = 0.5 × 0.25 × 100 = 12.5 J. To reach 10 rad/s from rest in 5.0 s needs α = 2.0 rad/s², so τ = Iα = 0.50 N·m.

Common mistakes

• Using the full distance in τ = rF without the sin θ factor, so the force is not taken perpendicular to the lever arm.
• Using the wrong moment of inertia. Check which shape and which axis the formula belongs to.
• Putting angles in degrees into formulas that need radians.

Practice

Test yourself with practice questions on this topic: Rotation and torque