Unit 6 · Topic 06

Rotational dynamics

About an appropriate fixed or principal axis, net axial torque determines angular acceleration.

Unit 6Rotational dynamicsOpen navigation

Concept 01

Net torque and α

Essential The minimum you should retain

Under the model conditions, Στ_z=I_zα \sum\tau_z=I_z\alpha .

UnderstandInterpret and connect

At equal torque, larger I produces smaller |α|.

DeepenFormulation and conditions

The sign of α follows net torque under the stated convention.

ExploreConnections for further study

This scalar form is not a universal 3D vector equation.

Worked example

Angular acceleration of a disk

A solid 4.0 kg, 0.50 m disk receives a 10 N tangential rim force.

Given
  • M=4.0 kg
  • R=0.50 m
  • F=10 N
Target

Find α.

  1. Inertia

    I=(1/2)MR²=0.50 kg·m².

  2. Torque

    τ=FR=5.0 N·m.

  3. Dynamics

    α=τ/I=5/0.50.

  4. Result

    α=10 rad/s².

Conclusion

α=10 rad/s².

Mathematical relation

Axial rotational dynamics

Στz=Izα \sum\tau_z=I_z\alpha
Represents

Rotational second law about an appropriate axis.

Physical interpretation

This is not a universal 3D vector equality.

DeepenVariables, conditions, and checks

Variables

τ_z
axial torque component; usual unit: N·m
I_z
moment of inertia about z; usual unit: kg·m²

Conditions of application

  • Rigid body about an appropriate fixed or principal axis.

Dimensional check

N·m=kg·m²/s².

Two tangential forces on opposite sides produce oppositely signed axial torques and an out-of-plane resultant.diskF₁F₂τ₁ ⊙τ₂ ⊗τ_net, α ⊙

The torques from F₁ and F₂ have opposite signs; because |τ₁|>|τ₂|, τ_net and α point out of the plane.

Concept 02

The resultant governs

Essential The minimum you should retain

Opposing torques add algebraically.

UnderstandInterpret and connect

One large torque may be partly offset by another.

DeepenFormulation and conditions

The sum includes every external force about the axis.

ExploreConnections for further study

A zero-torque force may still enter other equations.

Concept 03

Translation coupled to rotation

Essential The minimum you should retain

Masses and pulleys require translational equations, rotational equations, and constraints.

UnderstandInterpret and connect

A no-slip cord relates linear acceleration to αR.

DeepenFormulation and conditions

Tensions are not set equal when the pulley requires net torque.

ExploreConnections for further study

Solving all unknowns together preserves the coupling physics.

Concept 04

A well-defined axis

Essential The minimum you should retain

I_z must correspond to the axis used for torques.

UnderstandInterpret and connect

A moving axis may require separating CM translation and rotation.

DeepenFormulation and conditions

In three dimensions, L and ω are not always parallel.

ExploreConnections for further study

The axial form remains powerful when its assumptions are stated.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Presenting Στ=Iα as a universal 3D vector equation.

Use the appropriate axial form for the axis and model.

Setting tensions equal for a pulley with nonzero I and α.

Their difference may supply net torque.