# PHY6200 W07

## Chapter 10: Rotational Motion of Rigid Bodies

### Reading:

In Taylor, read sections 10.3 to 10.4 for today, and 10.4 to 10.6 for Monday.

### Recall

### Rotation about Any Axis; the Inertia Tensor

**L** = **I** **ω**

where L and ω are (1×3) column vectors, and I is the (3×3) matrix

**I** =

I_{xx} | I_{xy} | I_{xz} |

I_{yx} | I_{yy} | I_{yz} |

I_{zx} | I_{zy} | I_{zz} |

An individual element of the matrix, ij, where i and j = x, y, or z, is

I_{ij} = &sum_{a} m_{a}(δ_{ij} r_{a}² - r_{ai}r_{aj})

#### Example: Inertia Tensor for Lamina

Derive relations among the elements of the inertia tensor for a lamina.

A lamina is a planar object.
Being flat, we can orient it to lie in the x-y plane so that all points have z=0.
Then we see immediately that I_{xz} = I_{yz} = 0.
Additionally there's a relation between the diagonal elements.
To see this, notice that, since z=0 for all points,

I_{xx} = &sum_{a} m_{a}y_{a}²

I_{yy} = &sum_{a} m_{a}x_{a}²

I_{zz} = &sum_{a} m_{a}(x_{a}² + y_{a}²) = I_{xx} + I_{yy}

The remaining element, I_{xy} = &sum_{a} m_{a}x_{a}y_{a}

#### Angular Momentum for an Arbitrary Angular Velocity

**ω**_{31} = **ω**_{32} + **ω**_{21}

### Principal Axes of Inertia

As discussed in the cone example, a diagonal inertia tensor means that a rotation about one of the axes has angular momentum parallel to angular velocity.
This is nice, when we happen to have an object that produces a diagonal inertia tensor.

But wait!
There's a theorem from linear algebra that says that any symmetric matrix can be rotated to a different basis in which it is diagonal.
Every inertia tensor is symmetric, therefore we can always diagonalize the inertia tensor, for any object no matter how oddly shaped.

Physically, this means that every object has directions that they can be spun around such that the angular momentum is parallel to the angular velocity vector.
We write this mathematically as saying that the angular momentum equals a constant times the angular velocity or

**L** = **I****ω** = λ**ω**

The directions where this is true are called the principal axes of the body.
For the cone example, the x, y, and z axes defined in the calculation are principal axes.
The constants of proportionality are the diagonal elements of the diagonalized inertia tensor (**L** = λ**ω**), and are called the principal moments.

#### Kinetic Energy of a Rotating Body

We can easily calculate rotational kinetic energy by exploiting the principle axes and principal moments.

T = ½∑_{i} λ_{i}ω_{i}²

### Finding the Principal Axes; Eigenvalue Equations

© 2007 Robert Harr