R EPORT EPORT
FOR LAB
A11: THE GYROSCOPE
Nathaniel Kan 12/01/01 ABSTRACT:
In this lab we studied the relationships between the moment of inertia, angular momentum and torque using a gyroscope as our rotating system. Then we studied precession in a rotating system, using a weight to apply a torque to rotate the gyroscope. PURPOSE:
Calculate the moment of inertia of the gyroscope moment of inertia around a central axis. Use this and experimental data to determine the angular momentum of the gyroscope, and use this to calculate the precession under un der different circumstances. PRINCIPLES:
Part 1: Quantities representing rotational motion have analogs in translational motion. For velocity, the analogous rotational quantity is angular velocity: velocity: Omega = d(Theta)/dt We get the direction by using the right hand rule; looking at our right hand, h and, wrapping the fingers around in the direction of rotation and sticking your thumb straight out, the angular velocity is in the direction of the thumb. We find the angular momentum to be: L = I(Omega), where L is in the same direction as Omega The torque, the rotational analog of force, is equal to the force times the distance, or the derivative of angular momentum: Torque = dL/dt = I(dOmega/dt) Part 2: To find the precession, P precession, P , we use the formula, based on our observed speed of rotation, d(Phi)/dt: d(Phi)/dt = 2(pi)P
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We can also calculate the theoretical precession, using the formula: d(Phi)/dt = mgd/L PROCEDURE:
Part 1: After turning on the power to the gyroscope and ensuring that all gimbals are in working order, we verify that the angular momentum stays fixed when rotating the base of the gyroscope. Next we use a stroboscope to measure the frequency of rotations. Then we observe what happens to the gyroscope as we push up, down, left and right. We use these observations to relate torque and angular momentum (see Results section). We then push the on the gimbals and record those observations. Part 2: To measure precession, we perform several trials with varying weights. We attach a weight to the shaft underneath the arrowhead. This applies a torque to the rotor, which then rotates or precesses about the vertical axis. Then we measure the rate of precession, in revolutions per second. CALCULATIONS :
See Lab book page 42 and A11 Appendix A for calculation of the rotor’s moment of inertia (taken about axis through center). I = MR 2 / 2 I1 = 5346.36 g (20.0 cm / 2)2 See Lab book page 43 for calculation of the angular momentum. L = Omega * I = 26.60 ± 0.02 1/s * 0.0198862 kg m2 = 0.5290 ± 0.0004 kg m2 / s See Lab book page 45 for calculations of theoretical values of the precession. d(Phi)/dt = mgd/L = = (.4725 ± .0001 kg)(9.8 m/s2)(.0370 ± .0005 m)/(.52897 ± .00040 kg m2/s) = .3239 ± .0044 rad/s
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R ESULTS:
Final values: I = .0198862 kg m2 L = .52897 ± .00040 kg m2/s Omega = 26.6 ± .2 1/s Part 1: In Part 1 we familiarized ourselves with the gyroscope. When we push on the gyroscope’s arrowhead, the rotor does not move in the direction we might have expected. When applying a force downwards, the torque, equal to the change in the angular momentum, is to the left (according to the right hand rule). Because of this, the rotor begins precessing clockwise (looking down from the top). Likewise, if we push up, the rotor precesses counter-clockwise. If we push right, the rotor moves down, and if we push left, the rotor moves up. See lab book page 43 for diagrams. When holding the outer gimbal steady and pushing, the rotor reacts differently. When pushing up or down, the rotor responds by moving up or down, respectively, but applies a force which is counteracted by the hand holding the outer gimbal. When pushing on the rotor itself, the rotor will move up or down depending on whether the push is applied to the left or right side, respectively. This is predicted by the righthand rule (see diagram lab book page 44). Part 2: After performing three trials of measuring the precession with three different weights, we find the theoretical values do not quite match the actual values (see lab book pages 45 and 46 for exact values). We repeated the precession experiment with angle Phi at different angles other than 90 degrees, and found that the rotor precesses at the same rate. This is because when calculating the theoretical values of the rotor’s rate of precession, sin(Phi) cancels out (see lab book page 46 for proof).
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