# I Explorer-1 tipped over, but Euler equation doesn't agree

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1. Mar 9, 2016

### harmyder

$$\dot{\boldsymbol{H}} = \dot{\omega} + \boldsymbol{\omega}\times\boldsymbol H$$
Suppose body's got an impulse and 's started to rotate about its principle axis, say z, No more external moment from this time.
So, Euler equations become
$$0 = I_{xx}\dot{\omega}_x − (I_{yy} − I_{zz}) \omega_y \omega_z$$
$$0 = I_{xx}\dot{\omega}_y − (I_{yy} − I_{zz}) \omega_z \omega_x$$
$$0 = I_{xx}\dot{\omega}_z − (I_{yy} − I_{zz}) \omega_x \omega_y$$

Suppose small rotation was imparted along another axis z or y in such a way that $\omega_z >> \omega_x, \omega_z >> \omega_y.$

From differential equation we can get, we conclude that a body rotating about an axis where the moment of inertia is intermediate between the other two inertias, is unstable.

But, Explorer-1 was rotation about its longest axis - the one with smallest inertia, and tipped over due to internal forces, so rotation about axis with smallest inertia moment is unstable. Doesn't that contradict to the previous result?

Last edited: Mar 9, 2016
2. Mar 14, 2016