A.4 Rigid body mechanicsIB Physics HL: Revision notes

The torque τ of a force F about an axis is τ = Fr sin θ, where r is the distance from the axis to the point where the force acts and θ is the angle between the force and the line joining that point to the axis. Torque is greatest when the force is perpendicular (θ = 90°) and zero when the force passes through the axis (θ = 0). Unit: N m.

A body is in rotational equilibrium when the resultant torque is zero: it is either not rotating or rotating at constant angular velocity. An unbalanced torque on an extended rigid body causes angular acceleration. For full equilibrium, both the resultant force and the resultant torque must be zero.

Rotation is described by angular displacement Δθ (rad), angular velocity ω (rad s⁻¹) and angular acceleration α (rad s⁻²). For uniform angular acceleration:

• Δθ = ((ωf + ωi)/2)t
• ωf = ωi + αt
• Δθ = ωit + ½αt²
• ωf² = ωi² + 2αΔθ

These mirror the linear suvat equations. One revolution = 2π rad.

The moment of inertia I measures how hard it is to change a body's rotation. It depends on the mass and on how that mass is distributed about the axis. For a system of point masses, I = Σmr². Mass far from the axis contributes much more because of the r² term: a ring has a larger I than a solid disc of the same mass and radius. I is different for different axes through the same body.

τ = Iα, where τ is the (average) resultant torque.

A rotating body has angular momentum L = Iω (kg m² s⁻¹). Angular momentum is conserved unless a resultant external torque acts, so if I decreases (a skater pulls in her arms) ω increases.

A resultant torque acting for a time gives an angular impulse: ΔL = τΔt = Δ(Iω).

A rotating body has rotational kinetic energy Ek = ½Iω² = L²/2I. When I changes with L conserved, Ek changes: pulling mass inwards increases Ek (work is done by the body), and adding mass that must be brought up to speed decreases Ek (energy is dissipated). Angular momentum can be conserved while kinetic energy is not.