Summary

General Principles

Solving Rotational Dynamics Problems

Model Model the object as a rigid body.

Visualize Draw a pictorial representation.

Solve Use Newton’s second law for rotational motion:

α = \frac{τ_{net}}{I}

Use rotational kinematics to find angles and angular velocities.

Review Is the result reasonable?

Conservation Laws

Energy is conserved for an isolated system.

Pure rotation $E_{mech} = K_{rot} + U_{G} = \frac{1}{2} I ω^{2} + M g y_{cm}$
Rolling $E_{mech} = K_{rot} + K_{cm} + U_{G} = \frac{1}{2} I ω^{2} + \frac{1}{2} M v_{cm}^{2} + M g y_{cm}$

Angular momentum is conserved if $\vec{τ}_{net} = \vec{0}$ .

Particle $\vec{L} = \vec{r} \times \vec{p}$
Rotation about a fixed axle $\vec{L} = I \vec{ω}$

Important Concepts

Torque is the rotational equivalent of force:

τ = r F sin ϕ = r F_{t} = d F

The vector description of torque is

\vec{τ} = \vec{r} \times \vec{F}

The figure illustrates a wrench tightening a bolt.

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The figure illustrates a vertically oriented disk whose circular faces face left and right.

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A system of particles on which there is no net force undergoes unconstrained rotation about the center of mass:

\begin{matrix} x_{cm} = \frac{1}{M} \int x d m & y_{cm} = \frac{1}{M} \int y d m \end{matrix}

The gravitational torque on a body can be found by treating the body as a particle with all the mass M concentrated at the center of mass.

Vector Description of Rotation

Angular velocity $\vec{ω}$ points along the rotation axis in the direction of the right-hand rule.

For a rigid body rotating about a fixed axle, the angular momentum is $\vec{L} = I \vec{ω}$ .

Newton’s second law is $\frac{d \vec{L}}{d t} = \vec{τ}_{net}$ .

The figure illustrates a horizontally rotating solid disc.

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The moment of inertia

I = \sum_{i} m_{i} r_{i}^{2} = \int r^{2} d m

is the rotational equivalent of mass. The moment of inertia depends on how the mass is distributed around the axis. If $I_{cm}$ is known, I about a parallel axis distance d away is given by the parallel-axis theorem: $I = I_{cm} + M d^{2}$ .

Applications

Rigid-Body Model

Size and shape do not change as the object moves.
The object is modeled as particle-like atoms connected by massless, rigid rods.

Rigid-Body Equilibrium

An object is in static equilibrium only if both $\vec{F}_{net} = \vec{0}$ and $\vec{τ}_{net} = \vec{0}$ .

The figure illustrates a ladder with no rotational or translational motion resting on a wall on the left.

Rolling Motion

For an object that rolls without slipping

\begin{array}{l} v_{cm} = R ω \\ K = K_{rot} + K_{cm} \end{array}

The figure illustrates a rotating circular object on a horizontal surface.

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