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

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

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

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

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

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!

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

Motion in Space and Body Frames Earth has Chandler wobble!

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

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

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

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

Steady Precession in θ Assume θ constant. Then

Effective Potential for θ Motion

Nutation

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

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)