4. Steady precession!
If you had a little trouble going through the above concepts, here is your reward: you can now understand how a gyroscope works!
As mentioned the gyroscope initially dips slightly, wobbles up and down, and, in a short time, starts preceeding. The explanation of the dipping is somewhat subtle, so we shall deal with it later.
What we will do know is account for the fact that a gyro, after "dip-wobbling", preceeds instead of falling. Remember the equation of rotational motion (II)? If we rewrite it, taking the dt to the other side, it reads:
t dt= dL
In our case t is the torque due to the only two forces acting on the gyro: gravity (which can be thought as acting on the centre of mass of the gyro), and the reaction force of the ground [we suppose that the frictional forces and consequent torques are so small as to be neglectable].
The above equation, equates the directions of the involved vectors: we see that t produces a change in L parallel to it, and as time goes by, the angular momentum is constantly changing direction. So, as long as the torque is applied and as long as the gyro is spinning, the latter will not fall, but preceed. So the gyro posesses two angular velocities: w about its axis, and W about an axis perpendicular to the ground.
It's also pretty straightforward to work out the angular velocity of precession.
Fig.5: Angular momentum and related changes during precession.
It is clear that as the angular momentum changes in magnitude by dL, L turns through dq (Fig.5, above; and Fig.6, below). And the latter is dq=dL/Lsin(p). Thus :
W = dq/dt = (dL/dt)/Lsin(p) = t /L sin(p) (III)
Fig.6: Angular momentum and related changes during precession.
But t= Fg r sin(p) = Mgr sin(p), where Fg is the weight of the gyro, M is the mass of the gyro, and r is the length of the position vector of the centre of mass wrt to the point of contact with the ground. So W = Mg / r, but L = Iw, hence:
W = Mgr / Iw
We, furthermore, can find a vectorial relation for t. From (III) we see that t = W sin(p) so, introducing the vector W, along the precession axis:
t = W x L
This treatement is, by all means, not complete. I should add some complements in the future... In the mean time, if you wish, you can learn about angular momentum and gyroscopes from the following references.
-"An elementary treatment of the theory of Spinning Tops and Gyroscopic motion" by H.Crabtree (Longmans, 1903)
-"The Feynman lectures on Physics" by R.P. Feynman, R. Leighton and M.Sands vol. I, 20-5 (Addison-Wesley, 1989)
-"Fisica Generale I" by Marcello Cresti (Cleup, 1988)
-"Introduction to Classical Mechanics" by A.P.French and M.Ebison (Van Nostrand Reinhold International, 1986)
-"Fundamental University Physics" by M.Alonso and R.J.Finn (Addison-Wesley, 1968)
Please mail welcome criticisms and comments to: oti@geocities.com
