Rigid-body rotation, the gyroscope, Euler angles

Equipment

• Rotatable 3D top canvas (drag or arrow keys)
• “Spin rate” slider (rad/s readout)
• “Pause”/“Resume” button to freeze the picture at an instant
• Red trace of the axis tip and gray vertical reference axis

Procedure

1. Identify the axis and precession orbit

   Drag the canvas to rotate the view. The top's blue symmetry axis is tilted from the gray vertical; the red trace is the locus of the axis tip — a circle around the vertical. This pictures a fixed tilt angle $\theta$ and a steadily increasing precession angle $\varphi$.

2. Vary the spin rate

   Move the “Spin rate” slider between 0.20 and 1.20 rad/s and read the value beside it. Faster spin $\psi$ means larger angular momentum $L \approx I_3\omega_3$. Remember the Euler angle $\psi$ is an orientation coordinate while $\omega_3$ is a rotation rate — do not conflate them.

3. Freeze the frame and read the axis position

   Press “Pause” to hold the top at an instant, then rotate the canvas to gauge the axis tilt against the vertical and how far along the red circle the tip has moved. Record the pair (instant, tip position on the precession circle). Press “Resume” to continue.

4. Compare with the slow-precession relation

   The gravitational torque, perpendicular to $\mathbf L$, rotates the angular-momentum direction and produces precession $\Omega_p \approx \tau/L$. Since $L \approx I_3\omega_3$, a faster-spinning top precesses more slowly. Compare the two spin settings you tried and state your qualitative prediction; note the model is a qualitative illustration and does not solve Euler's equations or capture nutation.

Simulation

Experiment history

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