Hasan Unlu

Intermediate Axis Theorem

One of the interesting physics phenomena is intermediate axis theorem, if a rigid object has increasing moment of inertia for each orthogonal axis and you spin it axis of intermediate moment of inertia, it oscillates back and forth in that axis.

Here is the video:

The analysis of system using rigid body equations:

In that case $M_1$ , $M_2$ and $M_3$ are zero. Because no external torque is applied. $I_1<I_2<I_3$ , $w_1\approx 0$ and $w_3\approx 0$ , in order to solve higher order differential equation MATLAB can be used. Here is func.m file:

function dydt = func(t, y)
I1 = 1;
I2 = 2;
I3 = 3;
dydt = zeros(3, 1);
dydt(1) = (I2 - I3) * y(2) * y(3) / I1;
dydt(2) = (I3 - I1) * y(1) * y(3) / I2;
dydt(3) = (I1 - I2) * y(1) * y(2) / I3;

from command line or other .m file run the following lines.

clear all;
[t, y] = ode45(@func, [0:0.1:100], [.01 4 .01]);
plot(t, y(:,1), t, y(:,2) ,t , y(:, 3))

Numerical solution of differential equations shows $w_2$ is oscillating between -4 and 4. While changing rotation, very short amount of time $w_1$ and $w_3$ are non zero.
data1 -> $w_1$
data2 -> $w_2$
data3 -> $w_3$