370(7): Motion of the Gyroscope, General Theory

This general theory is developed in terms of Euler angles and spherical polar coordinates and can be used for any situation. It is exemplified by the motion of a gyro with one point fixed. In general the complete information about a gyroscope should be obtained by using both the Euler angles and the spherical polar coordinate system, related by Eq. (8). The latter can be used as a check on the correctness of the numerical solution. The torque and force calculations are exemplified by working out the lab frame torque and force which give the potential energy U = mghcos theta of the gyroscope with one point fixed. In the spherical polar system the torque is Eq. (29), adn in the Cartesian system the lab frame torque is Eq. (34). The lab frame force is the force of gravitation on the centre of mass of the gyroscope, Eq. (36), in the -k direction. The torque balance equation is Eq. (38) which shows that the torque due to gravitation is balanced by the torques of the Euler equations. This is the reason why the gyro does not fall over. We are now ready to evaluate the Laithwaite experiment and Shipov experiment by adding a torque in the lab frame. Any kind of torque can be used in Eq. (38).


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