Euler's laws of motion for a distributed mass are:
$$F = \frac{d}{dt} MV_{cm},\ N = \frac{d}{dt} L$$
$F$ are the sum of the external forces, $M$ the total mass, $V_{cm}$ the velocity of the centre of mass. $N$ are the sum of the moments of the external forces about some given point, L the total angular momentum about the same point.
If a gyroscope is supported at its base with its axis horizontal, it precesses at a constant angular velocity. Using the above equations, how does one show this?
Best Answer
Start off ignoring gravity. The spin axis is horizontal? Well then, you have an L vector. Very simple, nothing happens.
Now turn on gravity. This pulls down on the centre of mass, which is elsewhere from the pivot. That gives you an N vector - cross product of force with location relative to the pivot. This is perpendicular to the axis - so perpendicular to L. This is the change in L, according to N=dL/dt. In an arbitrarily small change in time, dt, we find dL will not change the length of L but will change its directions. Repeat indefinitely for every dt in a finite time interval t_1 to t_2.