Gyroscope Physics

7/28/2019 Gyroscope Physics

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Gyroscope physics

One of the evergreens of classical mechanics demonstrations is the behavior that canbe elicited from a gyroscope.

The word 'gyroscope' was coined by the french physicist Foucault. Foucault was activein optics, in the manufacturing and testing of lenses and mirrors, in the chemistry ofphotography, and he did research in electromagnetism. Today he is mainly known forthe pendulum setup that is called 'Foucault pendulum'.

Free spinning gyroscope


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direction of motion a spinning gyroscope wheel will remain pointing in the samedirection.

An object left to free motion will move in a straight line. The spinning of the gyroscopewheel can be thought of as combining two oscillations, perpendicular to each other.

Each of these oscillations remains on the same line, making the plane that is defined bythe two perpendicular lines a stationary plane. Hence the spin axis, perpendicular to theplane, keeps pointing in the same direction.

A gyroscope subject to torque

I will use the following naming convention: I will take the word 'gyroscope' to refer to theassembly of gyroscope wheel and all of the suspension mount together. I will call thespinning mass - usually a disk-shaped object - the 'gyroscope wheel'.

Bicycle wheel

The image shows a demonstration from a lecture by professor Walter Lewin. Using anelectric motor he spins up a bicycle wheel to a hair raising velocity, and then he hooksup the wheel to a rope suspended from the ceiling. Initially, walking up to the rope, hesupports both ends of the axle. When the rope takes the weight the wheel starts

precessing.

Picture 2, picture 3, image.Source:MITopencourseware physics 8.01, Youtube video 35:30

Gyroscope
http://www.youtube.com/watch?v=zLy0IQT8sskhttp://www.youtube.com/watch?v=zLy0IQT8sskhttp://www.youtube.com/watch?v=zLy0IQT8sskhttp://www.youtube.com/watch?v=zLy0IQT8ssk


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Pictures 4 and 5 show a gyroscope in a multi-axed gimbal mounting. The yellowhousing enables swivel, the red housing enables pitch. The wheel's bearings rest on afixed axle that extends out of the red housing.

Notice especially the instant at 47:10, when professor Lewin happens to manipulate the

yellowhousing. The turningof the yellow housing is transmitted to the gyroscope wheel,and just for a moment you can see how the gyroscope wheel responds to that.

Picture 4, picture 5, image.Source:MITopencourseware physics 8.01, Youtube video 46:00

The demonstrations by professor Lewin are so vivid because he spins the wheels sofast. (You definitely shouldn't try that at home.)

The purpose of this article is to show how to understand the behavior of the gyroscopein terms of the laws of motion.

Naming conventions
http://www.youtube.com/watch?v=zLy0IQT8sskhttp://www.youtube.com/watch?v=zLy0IQT8sskhttp://www.youtube.com/watch?v=zLy0IQT8sskhttp://www.youtube.com/watch?v=zLy0IQT8ssk


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Picture 9. Image48:00 into the video, only seconds away from adding a weight.

Image 9 is at 48:00 into the video.

Let me go step by step over what happens at the exact instant that the weight is added.

When the weight is positioned onto the axle rod the force that it exerts starts topitch the gyroscope wheel.

The pitching motion causes swiveling motion: precession. The precessing motion adds a tendency to pitch up, counteracting the

downpitching tendency from the brown weight. The gyroscope settles into a sustained dynamic configuration, neither pitching up

nor pitching down.

Settling into the precessing motion happens very quickly; you don't actually see ithappening. It may look as if the wheel's motion has changed directly to the final

precessing motion, but in fact it has gone through the above described process.

Self-adjusting

Picture 10. Image38:20 into the video. Faster precession when extra torque has caused further pitching down.


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Appendix - Derivation of F=-2msvr

Picture 13. Image

Repeat of image 8

The derivation below deals with what is illustrated with image 13. The rolling moves theparts in the shown quadrant closer to the swivel axis. And the question is: how large isthe tendency to pull ahead of the general swivel? To obtain the answer to that I gothrough two steps:

1. First I answer the question: if circumnavigating mass is pulled closer to the axis ofrotation with a particular radial velocity, how large will it's tangential acceleration be?

2. Then I turn that around: the amount of force that is necessary to preventangular

acceleration is proportional to the tendency to accelerate.

Step 1. is the case where angular acceleration is not prevented (that is: no torque),which means that angular momentum is conserved.

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Differentiating the expression for the angular momentum:

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Using the chain rule to obtain an expression in terms of a factor dr/dt .

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Dividing by r, and rearranging

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r(d/dt) = at = the tangential acceleration.

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This expression gives the tangential acceleration that occurs if there is no torquepresent.

Multiplying both sides with m gives the corresponding force:

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The above completes step 1. of the two steps. Step 2. is to see that if at every pointprecisely that force is exerted (in the opposite direction) then change of angular velocityis prevented, and the part will remain moving in radial direction with the same velocity.

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Gyroscope Physics