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Gyroscopic Motion Experiment: Understanding Precession and Rotational Mechanics, Study notes of Engineering Physics

University of Nottingham Engineering Physics

An experimental script for studying gyroscopic motion at keele university's physics/astrophysics laboratory. The script outlines the theory behind the experiment, the use of a spinning ball in an air bearing, and the investigation of precessional motion. The document also details the experimental procedure, including the measurement of angular velocity and precessional frequency using a pick-up coil and stop-clock.

Typology: Study notes

2010/2011

Uploaded on 09/08/2011

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EXPERIMENT E

Keele University Physics/Astrophysics Laboratory

School of Physical and Geographical Sciences Experimental Scripts

43

Gyroscopic Motion

1. Introduction and Theory

The principal item of apparatus in this experiment is a massi ve ball mounted in an air bearin g

connected to a compressed air line. The air bearing fixes the position of the centre of the ball but allows it

to freely rotate, so that the phenomena of rotational mechanics c an be stud ied und er condition s of very

low friction. The ball is fitted with an aluminium rod to which can be added discs which serve to apply a

torque to the ball (see f ig. 1). Thi s Ball and its bear ing are eng ineered to a hi gh precision so tre at them

both with respect. Do nothing to damage the bearing surfaces and do not switch off the air supply while

the ball is still spinning.

Fig. 1 Steel Ball and air bearing

Rotations about vertical and horizontal axes are illustrated

The response of rotating bod ies to ext ernal tor que’ s i s very counter int uit ive. Demonstrate this

for yourself by apply ing forces to the spinn ing ball. Set the ball rotatin g about a verti cal axi s by hand

spinning the rod (without any d isc s). Now u se a r u le r or penci l to gent ly appl y a horizontal force to the

rod. Feel how the inertia of the sp inning body appears to be enhanced comp ared to its non spinnin g

state and observe that the motion of the rod is at right angle s to that expected. These disconcerting

properties are characteri stic of rotat ing bodies . Rapid ly rotating bod ies in sp ecial low torque mountings

are generally ref erred to a s g yroscope s and they have important appl icat ions in inertia l gu idance sy stems.

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Keele University Physics/Astrophysics Laboratory 43

Gyroscopic Motion

Introduction and Theory

The principal item of apparatus in this experiment is a massive ball mounted in an air bearing connected to a compressed air line. The air bearing fixes the position of the centre of the ball but allows it to freely rotate, so that the phenomena of rotational mechanics can be studied under conditions of very low friction. The ball is fitted with an aluminium rod to which can be added discs which serve to apply a torque to the ball (see fig. 1). This Ball and its bearing are engineered to a high precision so treat them both with respect. Do nothing to damage the bearing surfaces and do not switch off the air supply while the ball is still spinning.

Fig. 1 Steel Ball and air bearing Rotations about vertical and horizontal axes are illustrated The response of rotating bodies to external torque’s is very counter intuitive. Demonstrate this for yourself by applying forces to the spinning ball. Set the ball rotating about a vertical axis by hand spinning the rod (without any discs). Now use a ruler or pencil to gently apply a horizontal force to the rod. Feel how the inertia of the spinning body appears to be enhanced compared to its non spinning state and observe that the motion of the rod is at right angles to that expected. These disconcerting properties are characteristic of rotating bodies. Rapidly rotating bodies in special low torque mountings are generally referred to as gyroscopes and they have important applications in inertial guidance systems.

Keele University Physics/Astrophysics Laboratory 44

Disconcerting though these effects undoubtedly are, classical mechanics gives a perfectly accurate account of them. For a body, like the ball in this experiment, subject to an applied torque (but no net force) Newton's laws of motion give

Γ = d Ldt = I ddt^ ω^ (1)

Γ is the applied torque vector, L is the angular momentum vector of the body ω is the angular

velocity vector ( ω = 2 π f )and I is the moment of inertia of the body about the axis of rotation. For a

sphere of radius R and mass M

I = (^25) MR 2 (2) A particular solution of equation(1) is that of uniform precession where both Γ and L maintain constant magnitude but rotate steadily at the precessional frequency Ω, as illustrated in fig. 2. It can be seen from fig.2 that d L dt =^ Ω ×^ L (3) therefore equation(1) will be satisfied if

Γ = Ω×ω I (4)

In the case where ω and L are in a horizontal plane this becomes

Γ = Γ D + Γ o = Ω ω I (5)

Here ΓD is the torque due to the discs and Γo is the fixed torque due to the combined effects of the rod and the hole in the ball which accommodates the rod.

Fig. 2 Plan view illustrating precessional motion, Γ is horizontal and Ω is vertical

Keele University Physics/Astrophysics Laboratory 46

tolerable proportions by inserting the point of a pencil or ball pen into the small recess at the tip of the rod to steady the motion. (b) Use the pick-up coil (as in fig. 3) and the stop-clock to measure ω and Ω. (c) By a combination of varying the position of the disc on the rod (without going too near the end) and adding more discs, further measurements should be made to check the validity of (5) over a range of values of ΓD (a factor of three variation or more). (d) Plot a graph of ω × Ω against ΓD. From this graph determine values for I and Γo by re-arranging equation (5). Compare this value of I with the theoretical value of equation(2) (values of M and R are marked on the ball) and comment on the value of Γo. (e) As a further check on equation(5) measurements could be made at a fixed value of ΓD for a range of values of ω. This could be done by spinning the ball at a high frequency and making measurements as the ball slows down naturally (or by applying gentle braking if necessary). Plot a graph of ω against 1/Ω (or vice-versa, which would be better ?) for a fixed value of ΓD.