1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 19
G. A. Krafft
Jefferson Lab

2. General Rigid Bodies
Setup: let the unprimed coordinate system be those in an inertial frame and the primed coordinates be a coordinate set tied to the rigid body.
Terminology/Jargon
Unprimed Coordinate System: Space-fixed Frame
Primed Coordinate System: Body-fixed Frame
For simplicity, the origin of the primed coordinate system is almost always the center of mass of the rigid body

3. Motion of Center of Mass
Use Space-fixed coordinate system. Newton’s Second Law applies
Integrate equations of motion in the space-fixed frame
In many cases the motion of the Center of Mass is simply solved, e.g. Coin flip or motion of a baton in gravity. Center of mass follows usual motion in a gravitational field.

4. Body-Fixed Coordinates
Discuss rotational part of the problem using the primed, body-fixed frame
Need general method for transforming vectors and vector coordinates between coordinate systems

5. Transformation of Unit Vectors
Because scalar product symmetric
Going unprime to prime back to unprime leaves you where you started

6. Orthogonal Matrices Yield Rotations
In matrix form
Such matrices are called orthogonal. Any transformation between right-handed sets of orthonormal unit vectors will lead to an orthogonal matrix with determinate 1. They preserve the scalar product when (matrix) multiplied by coordinate vectors.
Euler’s theorem: in three dimensions any orthogonal matrix with unit determinate is of the form
a rotation through ψ in 3 dimensions about a unit vector

7. Rotation Matrices
Expression for the rotation matrix (for your future reference!)
Euler’s theorem follows by showing any unit determinate orthogonal matrix has an eigenvector with eigenvalue 1

8. Inertia Tensor
In 1 dimension when an object rotates there is a linear relationship between angular momentum and angular velocity
In 3 dimensions both l and ω are vectors. M is a new type of object called a (Cartesian) tensor.
Thought experiment: rotate a point mass
Need to provide a torque!

9. Dynamic Spin Balancing
How can we make ? Add a balancing mass!
No torque needed to have this motion. Angular momentum conserved and directed along angular velocity vector. E.g. a baton will rotate with constant angular momentum around its CM
Definition of Principal Axis: Rotation direction of a rigid body such that angular velocity and angular momentum align

10. General Inertia Tensor
Because angular momentum is additive, simply sum over all masses in the body. Integrate for continuous mass distributions
As above, angular momentum will rotate along with the mass of the rigid body, unless the angular momentum is aligned with the angular velocity, i.e., the rotation is around a principal axis

11. Terminology
Diagonal elements of the inertia tensor are called the moments of inertia. Same concept/expression as from 1 D case in elementary physics
The non-diagonal matrix elements are called the products of inertia.
In body-fixed frame the inertia tensor is constant. It is usually (not always!) evaluated with the origin at the center of mass

12. Finding Principal Axes
For alignment of the angular momentum and angular velocity
This is only possible if there are solutions to the problem
As mentioned previously, if there are solutions to this problem, λ is called an eigenvalue for the matrix M. The vector direction associated with the solution of the problem with eigenvalue λ generally depends on the eigenvalue. This eigenvector direction will provide a principal axis direction.
Linear algebra teaches us when this is possible

13. Some Math
Note that M is real and symmertic. This means the matrix is Hermitian. Hermitian matrices have a very important property. The eigenvalues must be real (not true for rotation matrices!) and the eigenvectors corresponding to distinct eigenvalues must be perpendicular.
Subtract and consider cases and

14. Body Frame Coordinate Choice
Align the coordinate axes for body frame along the principal axes directions. This is the same as performing a similarity transformation with the matrix of eigenvectors; M’=E-1ME becomes diagonal. The diagonal elements are called the principal moments of inertia
The inertial tensor in this frame is
For axially symmetrical masses I2 is equal to I3 and the principal axes can be any pair of orthogonal vectors in the plane normal to the eigenvector for I1 (the symmetry axis)

15. Baton Point Mass Point masses of mass m at x ± L/2
No products of inertia because y and z coordinates are all zero
What does M11= 0 mean?

16. Baton Spherical Masses
Moment of inertia for sphere about its center is 2mR2/5. Use parallel axis theorem (a HW problem generalizing the 1 D result!) to get
for each sphere.

17. Thin Coin
Let the z-axis be perpendicular to the coin through the center, with the origin on the coin. All z-values are 0.
The M12 is zero by symmetry

18. Thick Coin
Need the additional integral
To be added to the I2 integral

19. Cube Origin at corner. Use integrals like
Origin at center of mass. Use integrals like