Newton's Second Law for Rotation

Relate torque, moment of inertia and angular acceleration, and connect rotational dynamics with Newton's second law for translation.

General idea of the topic

The second law of rotation states that the net moment of the forces about an axis equals the moment of inertia times the angular acceleration. It is the rotational version of F = ma: the moment plays the role of force, the moment of inertia that of mass and the angular acceleration that of acceleration.

The topic develops that analogy and derives the fundamental equation by adding Newton's laws for all the particles in the body, where the internal moments cancel in pairs.

What you will truly understand

By the end you will be able to apply τ = Iα to a body with a fixed axis, combine it with the second law of translation in systems with massive pulleys and recognize which bodies resist rotation most: those with their mass far from the axis.

You will also know when the moment of inertia can be treated as constant.

Recommended learning route

1. Analogy with Translational Dynamics: term-by-term correspondence between F, m, a and τ, I, α.
2. Fundamental Equation of Rotational Dynamics: derivation of the fundamental equation, its validity and its assumptions.

Physical criterion you must acquire

The essential criterion is that τ and I refer to the same axis. The equation holds about a fixed axis or one through the center of mass; in other cases extra terms must be added. A zero net moment implies zero angular acceleration, that is, constant angular velocity or rest.

Connection with real problems

• massive pulleys in Atwood machines.
• starting and braking of motors and flywheels.
• accelerating wheels and disks.
• mechanisms with shafts and gears.

Final goal of the topic

The final goal is to apply the fundamental equation of rotation fluently and translate any translation problem into the language of rotation.