A2. Gyroscope. We shall study the motion of the rigid body about a fixed point (free axis gyroscope), measure precession speeds and observe nutation

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1 A. Gyroscope. BJETVE F THE EXPERMENT We shall study the motion of the rigid body about a fixed point (free axis gyroscope), measure precession speeds and observe nutation. BAKGRUND THERY A gyroscope is a solid body able to move freely about a fixed point (fig. 1). n this case the motion consists of a rotation about the A axis called instantaneous free axis. f is the instantaneous rotation vector, the speed of a point P of the solid is given by the equation: v P r (1) Fig. 1: A random body rotating about a fixed point. The axes a1,a,a 3 are attached to the body. The behavior of the gyroscope is derived from the theorem of angular momentum: dl M () that states that the instantaneous variation of the angular momentum L (with respect to ) is equal to the total moment of force applied M, with respect to. f we call the components of the vector : 1,, 3 in the referential a1a a 3, then: L (3)

2 This matrix made up of nine integrals represents the inertial tensor with respect to the point in the referential a1aa3. The diagonal terms are the respective inertial momenta with respect to the axes a1,a,a3. The other terms are called inertial products or centrifugal moments. L isn t parallel to. n general The expression (3) of the angular momentum can be simplified by choosing an orthonormal basis attached to the solid (there exists a possible choice for every point ) for which the inertial 1 3 matrix is diagonal. For these principal axes of inertia, (discovered by Euler in 176), we can write: L ii i1 e i i (4) i being the components of along the axes 1,, 3, e i being the corresponding unit vectors. The motion of such a body is rather complex and the general solution of equation () is complicated. n the following experiment we will study homogeneous solids with a cylindrical symmetry, for instance a symmetrical top (fig. ) for which the axes e,e r, e are principle axes of R x 3 G mg e x e r e inertia. n this basis, we have: Rotation vector: And thus A B 11 cos sin B A 33 L (5) Where the vector equation of motion is dl B M e, G mg G mg sin( ) e x 1 => The horizontal component of L vanishes. Fig. : Diagram of a gyroscope dl dl And e L mgl sin( ) e (6) Ri R with G l and e cos sin (7) After a few calculations, we get : B sin B A cos A B A B sin cos A sin mgl sin 3 equations to describe the 3 degrees of freedom of our system Depending on the initial conditions three particular cases are possible. (8) (9)

3 3 M and 1) f G, then L const This property is the basis of numerous applications (e.g. autopilot, indication of the horizon, gyroscopic compass, boat stabilization, etc.) The direction of the axis of rotation remains fixed with respect to the x, y, z basis, regardless of the motion of the fixture (Fig. 3). A G,L fixes z y x Fig. 3 : Free gyroscope ) f the solid rotates quickly about a principal axis of inertia, for instance about the axis e r, the rotational energy is large with respect to the potential energy 1 (1) A mgl We can therefore consider only the inertial moment A with respect to the axis, allowing us to write: L A e r (11) This rotation generates a slow precession: the vector L rotates about a cone (fig. 4) at a constant angular velocity const. The nutation is negligible, and e. With const, equation (6) becomes: dl Ri dl e L A er A sin( ) e mgl sin( ) e (1) R Which leads to the simple expression: mgl A (13) The pure precession motion (without nutation) occurs for minimum E of the function L A cos cos U A cos mgl cos (14) eff Asin 3 sin with L A cos cos B 3) For E > E, a nutation motion is added to the precession (periodic variation of the angle between and ). tow extreme values 1 référentiel de fixation (sous-marin, bateau, avio n etc.)

4 4 Fig. 4 : General motion of a symmetric top spinning about a fixed point, subject to its own weight. Precession and nutation of the three possible cases. with loops with cusps sinusoidal. EXPERMNETAL SETUP The gyroscope is made of a steel ball attached to a metal rod forming the axis of the gyroscope. A weight can be moved along the rod to modify its center of gravity, thus changing the torque applied to the gyroscope. The angular velocity of the gyroscope is measured using a colored pattern on a mass that is lit with a stroboscope. A small ball bearing the end of the rod allows us to modify direction of the axis while the gyroscope is rotating. n order to get quantitative results it is necessary to conserve the angular velocity of the gyroscope. This can be achieved within comfortable limits using air cushions and pressurized air to make the gyroscope hover. The stroboscope is made of a mercury vapor light whose frequency can be varied between scales ranging from 1 15 Hz, 3-5 Hz and 1 15 Hz. The angular velocity of the precession s measured using a manual stopwatch. An image of the setup can be found in figure 5.

5 5 V. SUGGESTED EXPERMENTS AUTN! Handle the gyroscope with care. Avoid any falls or impacts. rregularities on the ball or the cavity can greatly increase the friction. The rod of the gyroscope is fragile. Never go over rpm! 1) Make sure the rod is tightly screw into the ball, and press the sliding weight towards to bottom (zero displacement along the rod). ) Turn on the compressor (valve on the wall). 3) f necessary, clean the ball and the cavity. Delicately place the gyroscope in the cavity. 4) Turn on the stroboscope, at maximal frequency. 5) Using two fingers, hold the ball bearing at the tip of the rod, and bring it to its horizontal position. Then, start rotating the ball using the faucet of the tangential air stream. 6) Measure the angular velocity by following these steps: bserve the colored pattern, and progressively reduce the frequency of the stroboscope, until the colors aren t mixed, and seem static. At this point, the frequency of the stroboscope is equal to that of the gyroscope (it generally starts around 15 rpm). Measure the reduction of the rotation frequency over time due to friction (for instance every 3 seconds for 3 minutes). Plot the graph t. Repeat steps 5 and 6 five times 7) Study the precession. n order to do so, move the weight to the origin, start the gyroscope, and slowly move the axis away from its vertical position. Then measure the precession speed ver the time interval where the rod s angle and the angular velocity can be supposed constant, determine the dependency of with respect to d, the distance separating the weight to the ball. Plot ( d). 8) Fix d and, and study the dependency of with respect to by determining and every and minute, for 6 minutes. Repeat three times, in order to estimate the error. Plot the graph compare with equation (13). 9) Determine the dependency for and d constant. Plot Fig. 5 : mage of the experimental setup

6 6 V. BBLGRAPHY All theories and explanations found in the experiment can be found in any general physics class, for instance: - H. GRUBER ET W. BENT, «Mécanique Générale», PPUR, G. BRUHAT : "mécanique. Masson & ie Editeurs, Paris, Pages 11 à R. FEYNMAN : "mécanique 1".nterEditions, Paris, Pages -7 à KTTEL, W. KNGHT, M. RUDERMAN: cours de Berkeley "mécanique" volume 1.Librairie Armand olin, Paris 197. Pages 38 à H. GLDSTEN : "mécanique classique", Presses Universitaires de France, Paris Pages