Dynamics

Torque Makes Angular Acceleration, ΣM = Iα

Torque M=F·d, moment of inertia I=Σmr², rotational Newton ΣM=Iα, shapes βMR² (hoop 1, disk ½, sphere ⅖), parallel axis I=I_cm+Md²

Just as force makes acceleration in straight-line motion, torque makes angular acceleration in rotation. The constant of proportionality is the moment of inertia I — a measure of how far the mass is spread from the axis, the "mass" for rotation.

What turns things is not just the force but where you apply it. Torque M = F × d — the longer the lever arm d from the axis, the more the same force turns it. Drag r to see.

The "mass" for rotation is the moment of inertia I = Σ m r². Even for the same mass, the farther from the axis the larger I, growing with r squared — which is why a skater spins faster with arms pulled in. Drag r.

Put the two together and you get Newton's second law for rotation — ΣM = Iα. Net torque makes angular acceleration, with I as the constant. It is the same shape as ΣF = ma with the letters swapped. Drag M and α = MI follows.

For the same mass and radius the factor β in I = βMR² depends on shape — hoop(1), disk(½), sphere(⅖). Rolling down a ramp, the small-β sphere descends fastest (acceleration ∝ 1(1+β)). Pick a shape to compare.

For the same body, moving the axis changes I. The parallel axis theorem I = I_cm + M d² — shifting a distance d from the center-of-mass axis adds M d². Drag d and I grows with d squared.

In PracticeTo sum up, the heart of rotational kinetics is the single line ΣM = Iα. Torque M = F × d is force times lever arm, and the moment of inertia I = Σ m r² is the rotational mass measuring how far the mass spreads from the axis. So for the same torque a larger I gives a smaller α. I varies with shape as βMR² (hoop 1, disk ½, sphere ⅖) and grows when the axis is shifted from the center of mass by the parallel axis theorem I = I_cm + M d². When a rigid body rolls, you set up ΣF = ma_cm together with ΣM = I_cm α and tie them with the rolling condition a = rα. The straight-line trio (F, m, a) has simply moved over to (M, I, α) in rotation. In the next lesson, D2, we view this rotation through energy — the rotational kinetic energy ½Iω², the sum of translation ½mv² and rotation ½Iω² in rolling, and the work–energy theorem that sums up the work done by a torque at a glance.

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