Laws of gyroscopes / 3-axis gyroscope

Transcription

1 Laws of gyroscoes / 3-axis gyroscoe Princile The momentum of inertia of the gyroscoe is investigated by measuring the angular acceleration caused by torques of different known values. In this exeriment, two of the axes of the gyroscoe are fixed. The relationshi between the recession frequency and the gyro-frequency of the gyroscoe with 3 free axes is examined for torques of different values alied to the axis of rotation. If the axis of rotation of the force-free gyroscoe is slightly dislaced, a nutation is induced. The nutation frequency will be investigated as a function of gyro-frequency. Related toics Momentum of inertia, torque, angular momentum, recession, nutation. Tasks 1. Determination of the momentum of inertia of the gyroscoe by measurement of the angular acceleration. 2. Determination of the momentum of inertia by measurement of the gyro-frequency and recession frequency. Investigation of the relationshi between recession and gyrofrequency and its deendence on torque. 3. Investigation of the relationshi between nutation frequency and gyro-frequency. ig. 1: 3-axis gyroscoe. P PHYWE Systeme GmbH & Co. KG All rights reserved 1

2 Laws of gyroscoes / 3-axis gyroscoe Equiment 1 Gyroscoe with 3 axes Light barrier with counter Power suly 5 V DC/2.4 A Additional gyro-disc w. counter-weight Stowatch, digital, 1/100 sec Barrel base -PASS Slotted weight, 10 g, black Slotted weight, 50 g, black Holder for slotted weights Set-u and rocedure The exerimental set-u is shown in ig 1: 1. To start with, the olar momentum of inertia I of the gyroscoe disk must be determined. or this, the gyroscoe is fixed with its axis directed horizontally and ositioned on the exerimenting table in such a way that the thread drum rojects over the edge of the table (ig. 2). The thread is wound around the drum and the accelerating mass m (m = 60 g; late with 5 slotted weights) is fastened to the free end of the thread. Several exeriments are carried out for different dro heights h of the accelerating mass, from which the corresonding average falling time t from the moment the gyroscoe disk is released until the mass touches the floor is determined. The diagram of t 2 versus h is lotted and the moment of inertia of the gyroscoe disk is determined from the sloe of the straight line (ig. 3). 2. The gyroscoe, on which no forces act, and which can move freely around its 3 axes, is wound u and the duration t R of one revolution (rotation frequency) is determined by means of the forked light barrier, with the axis of the gyroscoe lying horizontally. Immediately after this, a mass m* = 30 g is hung at a distance r* = 27 cm into the groove at the longer end of the gyroscoe axis. The duration of half a recession rotation t /2 must now be determined with a manual stowatch (this value must be multilied by two for the evaluation). The mass is then removed, so the gyroscoe axis can regain immobility, and t R can be determined again. The inverse of the average value from both measurements of t R is entered into a diagram above recession time t P. In the same way, the other measurement oints are recorded for decreasing number of gyroscoe rotations. The sloe of the resulting straight line allows to calculate the momentum of inertia of the gyroscoe disk (ig. 6). 3. If a slight lateral blow is given against the axis of the rotating gyroscoe on which no forces are acting, the gyroscoe starts describing a nutation movement. The duration of one nutation t N is determined with the manual stowatch and this is lotted against the duration of one revolution t R, which is again determined by means of the forked light barrier (ig. 7). 2 PHYWE Systeme GmbH & Co. KG All rights reserved P

3 Laws of gyroscoes / 3-axis gyroscoe Theory and evaluation 1. Determination of the momentum of inertia of the gyroscoe disk If the gyroscoe disk is set to rotate by means of a falling mass m (ig. 2), the following relation is valid for the angular acceleration: d ω R dt = α = M I P (1) (ω R = angular velocity; α = angular acceleration; I =olar momentum of inertia; M = r = torque) According to the law of action and reaction, the force which causes the torque is given by the following relation: = m ( g a) (2) (g = terrestrial gravitational acceleration; a = trajectory acceleration) The following relations are true for the trajectory acceleration a and the angular acceleration α: a = 2 h t 2 ; α = a r (3) P PHYWE Systeme GmbH & Co. KG All rights reserved 3

4 Laws of gyroscoes / 3-axis gyroscoe (h = droing height of the accelerating mass, t = falling time; r = radius of the thread drum). Introducing (2) and (3) into (1), one obtains: t 2 = 2 I +2mr 2 (4) h mgr 2 rom the sloe of the straight line t 2 = f(h) from ig. 3, one obtains the following value for the momentum of inertia of the gyroscoe disk: I = (8,83 ± 0,15) 10-3 kg m² In general, the following is valid for the momentum of inertia of a disk: I P = 1 2 M R 2 = π 2 R4 d ρ (5) Taking the corresonding values for the radius R and the thickness d of the circular disk, and the secific weight of lastic ρ =0,9 g/cm 3, one obtains from (4): I = 8, kg m² 4 PHYWE Systeme GmbH & Co. KG All rights reserved P

5 Laws of gyroscoes / 3-axis gyroscoe 2. Determination of the recession frequency Let the symmetrical gyroscoe G in ig. 4, which is susended so as to be able to rotate around the 3 main axes, be in equilibrium in horizontal osition with counterweight C. If the gyroscoe is set to rotate around the x-axis, with an angular velocity ω the following is valid for the angular momentum L, which is constant in sace and in time: L = I P ω R (6) Adding a sulementary mass m* at the distance r* from the suort oint induces a sulementary torque M*, which is equal to the variation in time of the angular momentum and arallel to it. M * = m * g r * = d L d t (7) Due to the influence of the sulementary torque (which acts erendicularly in this articular case), after a lase of time dt, the angular momentum L will rotate by an angle dφ from its initial osition (ig. 5). d L = L dϕ (8) The gyroscoe does not tole under the influence of the sulementary torque, but reacts erendicularly to the force generated by this torque. The gyroscoe, which now is submitted to gravitation, describes a so-called recession movement. The angular velocity φ P of the recession fulfills the relation: ω P = dϕ d t = 1 d L L dt = 1 d L I P ω R d t = m* g r * I P ω R (9) P PHYWE Systeme GmbH & Co. KG All rights reserved 5

6 Laws of gyroscoes / 3-axis gyroscoe Taking ω P = 2π / t P and ω R = 2π / t R one obtains: 1 = m* g r * 1 t t R 4π 2 I P (10) P According to (10), ig. 6 shows the linear relation between the inverse of the duration of a revolution t R of the gyroscoe disk and the duration of a recession revolution t P for two different masses m* The sloes of the straight lines allow calculating the values of the momentum of inertia, for which one obtains: I = (8,89 ± 0,15) kg m 2 for m* = 0,03 kg and I = (9,29 ± 0,17) kg m 2 for m* = 0,06 kg 6 PHYWE Systeme GmbH & Co. KG All rights reserved P

7 Laws of gyroscoes / 3-axis gyroscoe The double value of the torque (double value of m*) causes the doubling of the recession frequency. If m* is hung into the forward groove of the gyroscoe axis, or if the direction of rotation of the disk is inverted, the direction of rotation of the recession is also inverted. If the sulementary disk identical to the gyroscoe disk is used too, and both are caused to rotate in oosite directions, no recession will occur when a torque is alied. 3. Determination of the nutation frequency ig. 7 reresents the relation ω N = k ω R ; t R = kt N (11) between the nutation frequency ω N and rotation frequency ω R The constant k deends on the different moments of inertia relative to the rincial axes of rotation. P PHYWE Systeme GmbH & Co. KG All rights reserved 7