The Gyroscope "A large acquaintance with particulars often makes us wiser than the possession of abstract formulas, however deep." W. James OBJECTIVES To qualitatively observe some of the motions of a gyroscope. To quantitatively verify a simple motion. THEORY A gyroscope is defined as a rigid rotating object, symmetric about one axis. Generations of children, back at least to Greek antiquity, have found fascination in the behavior of tops, to give the gyroscope its common name. A number of eminent physicists have also found the complex behavior of spinning objects a matter of interest and a fit subject for detailed analysis. More recently, very carefully engineered gyroscopes were used for navigation because the axis of spin points in a nearly fixed direction when external torques are small. This makes the gyroscope a good replacement for a magnetic compass, particularly in regions where magnetic compasses are unreliable. As with any mechanical system, the motion of a gyroscope can be understood completely by a systematic application of ! F = m! a to all the particles of which the rigid body is made. It is much more efficient, however, to exploit the fact that most of the forces act between the particles of the body, and simply have the effect of making it rigid. The overall motion is then described by ! ! =d! L dt (1) with ! ! =! r "! F the torque due to external forces ! F . Although an apparently simple equation, analysis of the resulting motions can become very complicated. We will simplify matters by considering the somewhat idealized example sketched in Fig. 1. Our toy is spinning about its axis with an angular speed !, supported at one end on a frictionless bearing. Choosing the origin at the pivot, gravity will produce a torque about the origin because the center of mass is not necessarily above the pivot point, but there are no other external forces that can produce a torque because the bearing is assumed frictionless. This implies that both Lz and ! must be constant. Further, the total mechanical energy, includingPHYS 111 The Gyroscope 2 gravitational potential, must also be constant. The motion will still be interesting, but these conditions let us understand some qualitative features. First, consider the case where the top is spinning rapidly with its axis more or less horizontal. The external force is vertically downwards, so the torque is horizontal, perpendicular to the axis of rotation. Since the spin angular momentum ! L s is parallel to the axis of rotation, Eq. 1 tells us that d! L is perpendicular to ! L s, and to a first approximation ! L s+ d! L is the same length as ! L s but pointing in a different direction. In fact, as the motion proceeds the tip of ! L s traces out a circle at a constant angular speed !, as indicated in Fig. 1. This motion is called precession, and we will demonstrate below that it follows from the equations of motion. To start the gyroscope, we will hold the axis fixed and set the rate of spin to the desired value. If we then move the axis at the precession speed and release it, the motion will be a smooth precession. If, instead, the axis is released from rest the tip will trace out small 'scallop' or looping motions, superimposed on the overall precession. This is called nutation, and arises from conservation of mechanical energy. The precessional motion represents additional kinetic energy, relative to the state with the axis fixed. Since ! is constant (frictionless bearing), the additional kinetic energy must come from a loss of gravitational potential. In other words, the center of mass must fall a little bit, tipping the axis of rotation, in order for the top to precess. If the spin is rapid, the drop is small, and the precession is affected only slightly. Overall, the tip of the axis bounces up and down a little, and the precessional speed varies a little. If the spin is not fast enough the character of the motion changes drastically, but that is a complicated story. For a quantitative comparison with Eq. 1, we can derive the angular frequency ! of the steady precession, assuming there is no nutation. In that situation we can write the total angular momentum as the sum of the spin and precession angular momenta ! L =! L s+! L z= I!cos"sin#ˆ i + sin"sin#ˆ j + cos#ˆ k ()+ Iz$ˆ k (2) r!"!mg!dL!L!s Fig. 1 A simple gyroscope, showing precession.PHYS 111 The Gyroscope 3 where Iz is the moment of inertia about the vertical, and the angles are shown in Fig. 2. For steady precession, ! is constant and d!/ dt = ", so d! L dt= I!"#sin$sin%ˆ i + # cos$sin%ˆ j (). (3) The vector from the pivot to the center of mass is ! r = r cos!sin"ˆ i + r sin!sin"ˆ j + r cos"ˆ k (4) so the gravitational torque is ! ! =! r " m! g = #rmg sin$sin%ˆ i + rmg cos$sin%ˆ j . (5) These will satisfy Eq. 1 if ! =rmgI", (6) demonstrating that uniform precession is a possible solution to the equations of motion. It is also amusing to note that ! is independent of the tilt angle, perhaps an unexpected result. !"xyzr#mg#Ls#center of mass Fig. 2 Coordinates for analyzing gyroscopic motion.PHYS 111 The Gyroscope 4 EXPERIMENTAL PROCEDURE 1. Physical arrangement The gyroscope we will use is a solid metal sphere supported on a cushion of air, as sketched in Fig. 3. The air cushion effectively supports the sphere under its geometric center, so if the sphere were perfect there would be no torques acting at all. The sphere actually has a rod fastened to it, which serves to displace the center of mass away from the geometric center of the sphere and allow gravity to exert a small torque on the system. A sliding weight can be fastened to the rod to adjust the torque as desired. The rod also serves as a convenient marker for the axis of rotation, and a handle for manipulating the gyroscope. A secondary jet of air is available to spin the sphere very rapidly. There are a number of precautions which you must be aware of to avoid problems for yourself or the apparatus. The rod is strong enough to support the heavy sphere only in the vertical position, so avoid picking up the sphere by the rod if at all possible. The sphere itself is rather soft metal