Tool · Simulator
Spin Lab
A rigid-body simulator that integrates Euler's equations live. Choose a block or a cylinder, launch it spinning in free space or as a top on a pivot, and watch \(\wvec\), \(\Hvec\) and the energy as it tumbles, wobbles, precesses and nods.
Mode
Body
Initial angular velocity (body axes)
Top on a pivot
Angular velocity in body axes
The plot shows \(\colX{\omega_x}\), \(\colY{\omega_y}\), \(\colZ{\omega_z}\) over the last 10 s of simulated time. For free motion \(|\Hvec_G|\) and \(T\) stay constant; the relative drift is a check that the equations are integrated accurately.
The plot shows the tilt \(\theta\) of the axis over the last 2 s, scaled about its mean so that small nodding shows. A flat line is steady precession. The precession rate is the average rate at which the axis circles the vertical.
How to use the Spin Lab
In free mode the body floats in space with no forces, turning about its center of mass: \(\Hvec_G\) is fixed in space and the kinetic energy is constant (Lesson 7). Set the initial angular velocity in body axes, where \(x\), \(y\), \(z\) are the body's principal axes, and press Launch.
In top mode a cylinder stands on a pivot below its center of mass, and gravity supplies a moment. Choose its tilt and spin, and release it from rest (it nutates) or launch it at the steady-precession rate of Lesson 8 (it glides).
The simulator integrates Euler's equations \(I_x\dot\omega_x = M_x + (I_y - I_z)\omega_y\omega_z\) (and cyclically) with a fourth-order Runge–Kutta method, and tracks the body's orientation with a quaternion. The readouts show how well \(|\Hvec|\) and the energy are conserved.
Ideas to try
Each idea loads a preset and launches it.
The tennis-racket flip
A book-shaped block spun about its intermediate axis flips over and over, although \(\Hvec_G\) never moves (Lesson 7). Compare with the maximum and minimum axes.
A wobbling coin
A flat cylinder (\(I_z > I\)) spun with a small wobble precesses about \(\Hvec_G\) in the retrograde sense: its spin \(\dot\psi\) relative to the circling axis is opposite to the precession \(\dot\phi = H_G/I\) (Lesson 8).
A football spiral
A long cylinder (\(I > I_z\)) with the same kind of wobble precesses in the direct sense. Compare the precession rate with \(\dot\phi = H_G/I\).
A nodding top
Released from rest at a tilt, a top drops a little and then precesses while nodding: the axis tip traces a scalloped path with a cusp at each high point. Lower the spin and the nutation grows (Lesson 8).
Steady precession
Launch the same top at the slow steady-precession rate from the quadratic of Lesson 8, and it glides round at constant tilt.
A sleeping top
Spun fast and nearly upright, a top barely moves: its large angular momentum makes gravity's moment a slow sideways drift.