5.5 Newton's Second Law in Rotational Form

Rotational analog of F = ma

Newton's second law for rotation states net torque = Iα, where τ_net is the sum of torques, I is rotational inertia, and α is angular acceleration. This mirrors F = ma, with torque replacing force, moment of inertia replacing mass, and angular acceleration replacing linear acceleration. It applies about a fixed axis or the center of mass.

Applying τ = Iα to systems

For rigid bodies (pulleys, rods, disks), sum all torques about the rotation axis, set equal to Iα, and combine with Newton's second law for any attached masses using a = rα (for strings that don't slip). This lets you solve for tensions, accelerations, and angular accelerations in systems like Atwood machines with massive pulleys.

Rotational kinetic energy & work-energy tie-in

Torque doing work on a rotating object relates to the rotational kinetic energy, KE = ½Iω². The work-energy theorem extends to rotation: net work by torque equals the change in rotational KE. This connects τ = Iα problems to energy methods when angular velocities are needed instead of accelerations.

Key terms

Newton's second law (rotational form)

τ_net = Iα — net torque equals moment of inertia times angular acceleration.

Moment of inertia (I)

Rotational analog of mass; measures resistance to angular acceleration, depends on mass distribution relative to axis.

Angular acceleration (α)

Rate of change of angular velocity; α = τ_net/I.

No-slip condition

For a string or rolling object without slipping, linear acceleration a = rα (r = radius).

Rotational kinetic energy

KE_rot = ½Iω²; work done by net torque changes this rotational kinetic energy.

Try an AP-style question

A net torque of 8 N·m acts on a disk with I = 2 kg·m². What is α?

1. 4 rad/s²
2. 16 rad/s²
3. 0.25 rad/s²
4. 10 rad/s²