1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 20
G. A. Krafft
Jefferson Lab

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

3. Precession of Top (Simplified)
Because torque normal magnitude
cannot change. If theta constant
Top precesses about the z-axis

4. Dynamics of Rigid Bodies
Repeating arguments from before, main adjustment is to allow the angular velocity vector to depend on time
Identify (use )

5. Euler Equations The rigid body equations of motion are
where the primes indicate that you use body frame coordinates
Because the mass locations constant in the body frame
For body coordinates along the three principal axes

6. Free Precession
When there is no torque, can solve Euler equations approximately as follows. Assume angular frequency in one principal direction is much bigger than the other two
Stable when I3 the biggest or smallest MOI principal direction; then the coefficient positive
Unstable if I3 is the intermediate value!

7. Free Precession of a Symmetrical Body
When I1 = I2 , can solve Euler equations exactly when no torque

8. Motion in Space and Body Frames
Earth has
Chandler wobble!

9. Euler Angles
General description of rotation matrices in terms of three angles, including the two usual polar coordinates specifying the main rotation axis
Rotation 1
Rotation 2
Rotation 3
Defines ϕ
Defines θ
Defines ψ

10. Angular Velocity and Momentum
Remember relative angular velocities add
In the body frame for I1 = I2 (Taylor trick is a principal axis normal to
Angular momentum is

11. Kinetic Energy Recall rotational kinetic energy is
Evaluate in body frame
Lagrangian for Spinning Top

12. Motion of Spinning Top Constants of the motion
Third equation of motion

13. Steady Precession in θ
Assume θ constant. Then

14. Effective Potential for θ Motion

15. Nutation

16. Sleeping Tops Condition for motion θ = 0 to be stable
For stability coefficient must be positive
For a coin spinning along a radius on a table

17. Top Precession More Accurately
Using our “driven oscillator” ideas, can derive the top precession more accurately. The torque is
When projected onto the body axes (Taylor trick: define psi as angle body x-axis makes with the torque)