Moment of inertia and rotational kinematics

In a rotating rigid body, every point goes round in a circle about the axis, and all points share the same ω and α. A point at distance r has speed v = rω, so points further out move faster.

Add up ½mv² for every particle: K = ½(Σmᵢrᵢ²)ω². The bracket is the moment of inertia, I = Σmᵢrᵢ², so K = ½Iω². I plays the part of mass in rotation: the larger it is, the harder the body is to start spinning, or to stop.

I depends not just on the mass but on how far it sits from the axis, and so on the axis chosen. Mass far from the axis counts for much more (r²). Common results:

The radius of gyration k is defined by I = Mk²: the distance at which all the mass could be put to give the same I. For a ring k = R; for a disc k = R/√2.

Example: a 2 kg disc of radius 0.1 m spins at 50 rad/s. I = ½ × 2 × 0.1² = 0.01 kg m², and K = ½ × 0.01 × 50² = 12.5 J.

With constant angular acceleration, the Chapter 2 equations carry over, with θ for x, ω for v and α for a:

• ω = ω₀ + αt
• θ = ω₀t + ½αt²
• ω² = ω₀² + 2αθ

Watch the units: one revolution is 2π rad, and to change rpm into rad/s multiply by 2π/60. Example: a fan starts from rest and reaches 300 rpm = 10π rad/s in 5 s. Then α = 10π/5 = 2π rad/s², and θ = ½ × 2π × 5² = 25π rad, which is 12.5 turns.