M6e Gyroscope with three Axes

Transcription

1 Fakultät für hysik und Geowissenschaften hysikalisches Grundraktikum 6e Gyroscoe with three Axes Tasks 1. Determine the moment of inertia of the gyro disk from a measurement of the angular acceleration for a known torque as well as from a measurement of the rotation seed.. easure the deceleration of the gyro by friction.. easure the recession frequency as a function of the rotation frequency of the gyro disk for two different torque values. 4. easure the frequency of nutation as a function of the rotation frequency of the gyro disk. iterature. Alonso, E.J. Finn, hysics,. 4ff.A. Tiler, hysics (rd Edition),. 60ff Accessories Gyroscoe (B Scientific), rotational seed meter, stowatch Key Asects for rearation - Angular momentum, torque, angular velocity - oment of inertia, moment of inertia tensor - Rotation around fixed and free axes - Gyroscoe, recession, nutation - Euler s equations 1

2 Remarks Warning. The rotational seed meter works with a red laser ( nm) of laser class (maximum ower < 1 mw). The beam window is indicated by a triangular laser warning sign. Do not look directly into the laser beam or direct reflections; do not aim with the laser at other eole. aser radiation of class might lead to eye injury. To match the otimal measuring range of the rotation seed meter (R = rotations er minute) to the rotation seeds tyically achieved, the gyro disk is marked by 8 reflecting stries; take this into account in your evaluation. In order to obtain reliable readings, the distance between seed meter and gyro disk should be below 5 cm. Warning. The gyro disc can be accelerated by a string; when seeding the gyro u, take care of other ersons in the immediate surroundings; take further into account that the string sometimes acts like a whi. For all tasks calculate the measurement uncertainties. Task 1 The gyro axis is ositioned horizontally and is fixed to the triod stand by the additional stand rod. The string is wound on the bobbin and a weight is fixed at the string end. easure the time the weight needs to reach the floor from a certain rest osition; immediately afterwards measure the rotation seed of the gyro disk. Reeat the exeriment, average over ten values and calculate the standard deviation. During the rearation of the exeriment think of a way to determine the moment of inertia from the data. Derive the corresonding equations. Comare the measured value of the moment of inertia with the one calculated from mass and diameter. This is the comonent I of the moment of inertia tensor. Task Friction leads to a deceleration of the gyro and therefore influences the studies on recession and nutation. Estimate the influence of friction by measuring the daming constant of the gyro rotation. Using the string, set the gyro into rotation, such that it reaches a rotation seed of the order of 500 rm. Determine the decrease of the rotation seed over a eriod of 10 s by measuring and rotocolling the rotation seed every 10 s. Determine the daming constant from an aroriate grah of the rotation seed against time. Task The fixation of the gyro to the triod stand is removed and with the use of the counterweights the gyro axis is brought into equilibrium. Check that the axis is in indifferent equilibrium for arbitrary values of its inclination with resect to the horizontal. Wind the free gyro u while holding the gyro axis fixed. Hook the additional weight onto the screw at the end of the gyro axis oosite to the rotating disk and let the gyro go. easure the time for half a recession eriod with the stowatch; both at the start and at the end of the half recession eriod measure the rotation seed of the gyro disk. Use the average value of the rotation seeds as rotation frequency. erform the measurements for ten different rotation frequencies in the range between 00 and 600 rm and for additional weights of 47 g and 94 g.

3 lot the recession frequency in an aroriate way as a function of the rotation frequency and comare to theory. Task 4 The free gyro is brought into equilibrium by adjusting the counter weights and is wound u. Subsequently, a nutation is generated by a brief vertical imulse on the gyro axis. The time for three nutation eriods is measured with the stowatch and the rotation seed is recorded at the start and end. These measurements are erformed for ten different rotation frequencies in the range between 150 and 500 rm. lot the nutation frequency in an aroriate way as a function of the rotation frequency of the gyro. Determine the value of the comonent I 1 of the moment of inertia tensor. Theoretical Basics A rigid body that is freely rotating or that is suorted in a single oint is called gyro. For gyros often the directions of the angular momentum and the angular velocity are not arallel; it is just this what makes u for the aeal of their intricate motion. The rotational roerties of a rigid body are described by the moment of inertia tensor I. Angular momentum and angular velocity are then related by I (1) In the rincial axes system of the rigid body the moment of inertia tensor has diagonal form I I I I. () Often the rincial axes can be found by viewing the symmetry roerties of the body. In the following only symmetric gyros with I1 I will be considered. The motion of a gyro is described within the laboratory system which is assumed to be an inertial system by the fundamental equation for rotary motion: d, () where denotes the torque acting on the gyro. It is favorable to consider a second coordinate system, namely the rincial axes system that is rigidly fixed to the rigid body and rotates with resect to the laboratory system with the instantaneous angular velocity. In this system the angular momentum vector is denoted by. In analogy to the analysis of rotating frames of reference, one might show that the angular momentum vectors in the two reference frames are related by

4 d d. (4) This equation might be exressed in coordinate form with the coordinates referring to the rincial axes of the rigid body. This yields the following system of equations: which is known as Euler s equations. d1 1 I1 ( I I) d I ( I1 I) 1, (5) d I ( I I1) 1 1. Acceleration of a disk by a falling mass Consider a disk with mass m and radius R that rotates around a fixed axis through the center of mass and erendicular to the large disk face. The moment of inertia is then 1 I mr. (6) A massless bobbin with radius r is fixed to the disk. A massless thread is wound around the bobbin and a weight with mass m is fixed to the end of the thread. The weight falls under the action of gravity, starting from rest, and drives the disk. With the tension F transmitted by the thread one has: I rf, (7) m a m g F where denotes the angular acceleration of the disk, a the acceleration of the weight. Since the thread rolls of the bobbin one has a r. can be calculated from fall height h and fall time t. Alternatively, one might use conservation of energy: m gh I mv I mr (8) to determine the moment of inertia from a measurement of the rotation seed.. recession We consider a symmetric gyro consisting of gyro disk, axis and counterweights that is suorted in its center of mass. If the gyro disk rotates around the figure axis, angular velocity and angular momentum are arallel. Since no torque acts, the angular momentum is conserved, the figure axis of the gyro oints along the angular momentum direction that in this case is horizontal. A small mass m is fixed to the axis, at a distance z from the center of mass; this leads to a torque r G (9) 4

5 erendicular to the angular momentum. G m g denotes the force of gravity and r the vector ointing from the center of mass to the additional mass. The torque changes the direction of the angular momentum, but not its absolute value such that the angular momentum recesses with the angular velocity around the vertical. During the time interval the angular momentum turns by an angle d with such that the angular frequency of the recession is given by d d, (10) d d /. (11) In vector form this can be written as. (1) This motion of the angular momentum is called recession; it occurs with the recession angular velocity and is driven by a torque. The additional mass fixed at a distance z from the vertical rotation axis leads in case of a horizontally adjusted gyro to a torque m gz (1) The angular momentum is obtained as I I I, (14) 1 where is the comonent of the angular velocity along the figure axis. Since, the first term on the right hand side of (14) is neglected.. Nutation et us consider a force free symmetric gyro with no torques acting on it. In this case the angular momentum is conserved and oints along a fixed direction in the laboratory system. If the angular velocity is arallel to the figure axis, both angular velocity and angular momentum are arallel to the figure axis and the gyro disk rotates around the fixed angular momentum axis. In the interesting case, however, angular velocity and angular momentum are not arallel, after e.g. the figure axis was deflected by a temorary imulse. In this case Euler s equations read: 5

6 d1 I1 ( I I1) 0 d I1 ( I I1) 1 0 d I 0. (15) With I I 1 (16) I1 one obtains d1 0 d 1 0 const, (17) such that cos( t) sin( t) I1 cos( t) I1 sin( t) I, (18) i.e. the angular velocity axis (rotation axis) recesses on a cone around the figure axis. This cone is called olhode cone. The absolute value of the angular velocity is given by the recession frequency, however, is. 6, the value of On the other hand, the rojection of the angular velocity onto the direction of the angular momentum is given by: ( 1 ) I1 I I1 I. (19) This is constant, i.e. the angle between rotation axis and angular momentum direction is constant. Since the angular velocity changes with time, it has to recess on a cone around the angular momentum direction; this cone is called herolhode cone. In effect, the figure axis moves around the angular momentum direction, i.e. the olhode cone rolls u on the herolhode cone. Thereby, the figure axis recesses around the angular momentum direction. The joint motion of figure axis and instantaneous rotation axis is called nutation. The half values of the oening angles of olhode cone and nutation cone can be calculated from the angles enclosed by the angular velocity or angular momentum and the figure axis. The half oening angle of the nutation cone is given by

7 tan 1, (0) I I the half oening angle of the olhode cone by tan. (1) In the case of a rolate gyro I 1 > I, such that the oening angle of the nutation cone is larger than that of the olhode cone. This case is illustrated in the figure. arameters Diameter disk: ass disk: Distance disk-vertical axis: asses counterweights: ass ieces for additional mass: Distance additional mass-vertical axis: Diameter bobbin: R = 50 mm m = 1500 g z S = 165 mm m G1 = 1400 g, m G = 50 g m = 47 g each z = 75 mm r = 65 mm 7