Gyroscope

Gyroscope
THE GYROSCOPE
REFERENCES
The Feynman Lectures on Physics, Chapter 20 (a very nice, intuitive description of the operation of the
gyroscope). Copy available at the Resource Centre.
Most Introductory Physics texts (e.g. A.Halliday and R.Resnick, Physics; M.Sternheim and J.Kane,
General Physics) discuss Angular Motion, Moments of Inertia, Simple Harmonic Motion, etc.
Operation and Instruction Manual for the MITAC Gyroscope (available at the Resource Centre)
INTRODUCTION
One of the most interesting areas in the science of rotational dynamics is the study of spinning solid objects,
tops, hoops, wheels, etc. From the gyrocompass (which indicates true North, rather than Magnetic North)
to an understanding of how a cyclist turns corners, the applications of this field of study are both practical
and fascinating. This experiment is designed to
introduce you to some of these interesting and
often counter-intuitive properties of rotating
bodies.
The apparatus consists of a 2.7 kg cylindrical
rotor that is spun at a constant rate by an
electric motor. The rotor is mounted in a
double gimbal arrangement which allows it to
assume any orientation.
THEORY
Angular Motion. The basic equations for
angular motion can often be obtained simply
from those for linear motion by making the
following substitutions (bold quantities are
vectors):
Linear variables
Force, F
Mass, m
Velocity, v
Momentum, p
Acceleration, a
Y
Y
Y
Y
Y
Y
Angular variables
r
Torque, τ
Moment of Inertia, I
r
Angular velocity, ω
Angular Momentum, L
Angular acceleration, a
NB. The analogy needs to be treated with caution; e.g. I is not a constant property of the body, as is mass, since its
value depends the axis around which it is measured.
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r d rp d ( mvr )
r
Thus Newton’s Law F =
=
= ma
dt
dt
r
r
r
r dL d(Iω)
r
, becomes τ =
=
= Iα
dt
dt
.
So we see that a torque τ applied to a body rotating with angular momentum L produces a change in that
angular momentum according to the relationship:
r
r
r dL
τ=
dt
(1)
r
r
r
L
=
I
ω
where τ is the applied torque;
is the angular momentum, ω is the angular velocity of
rotation of the body and I is the moment of inertia of the body about the axis of rotation (i.e., about the
r
ω axis).
If this torque
r
τ
r
is applied perpendicular to the direction of L (i.e., perpendicular to the direction of ω )
then from equation (1), L does not change in magnitude but does change in direction (if you do not
understand why this is so, check with a demonstrator before going any further). This change in
r
direction of L (and thus also of ω ) is called precession and appears as a rotation of the direction of the
r
L vector in space with a precession angular velocity of Ω p . Then it can be shown that:
r
r
τ = Ωp × L
EXPERIMENTS
1. Introductory experiment
Study the gyroscope (gyro) and ensure that you
can identify the outer frame, the outer gimbal
(which has a vertical rotation axis), the inner
gimbal (with horizontal rotation axis), and the
caging screw which locks the outer gimbal to the
frame, along with the associated equipment:
weights, springs, supports and the scale.
Instrument alignment
- Set the gyro on a level table.
- Lock the outer vertical gimbal to the frame
with the caging knob.
- Start the motor and allow a minute or so for
the rotor to attain a constant operating speed.
- Adjust the arrow head and/or tail weights by
turning them until the arrow remains in a
horizontal plane, which indicates a balance
about the inner gimbal axis.
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(2)
Free (two degrees of freedom) gyro
Unlock the outer gimbal by releasing the caging screw, then start the motor. Observe the orientation of the
spin vector arrow. Carefully lift the gyro by its base. What happens to the direction of the spin vector
arrow?
After returning the gyro on the table, carefully push down on the arrow or the head to produce a torque
vector. Notice the opposing torque. Observe the resulting tendency to rotate (precess) about the vertical
axis. Determine the directions of the spin, torque and precession vectors and verify that the spin vector
precesses toward the torque vector.
Captive (one degree of freedom) gyro
Lock the outer gimbal to the base. Except for small frictional and inertial torques, there is now no directly
opposing torque when the arrow is depressed as before. This can be explained by the basic gyro equation
(2):
r
r
τ = Ωp × L
(2)
r
With the outer gimbal locked, the gyro is not free to precess and the precession angular velocity Ω p must
be zero. Then, by Eq. (2), the angular momentum L must be zero, which means no torque is generated to
oppose the torque caused by pushing on the arrow. The gyro seems to have lost its inertial properties.
2 - Determination of rotor angular momentum
You need: 300g and 150g torquer weights; stop watch.
The gyro has to be balanced.
Unlock the outer gimbal by releasing the caging knob. Attach one of the torquer weights on the shaft of the
spin vector arrow and secure the detent in one of the notches.
Orient the spin vector arrow in the horizontal plane and then release the arrow, allowing the gyro to precess
about the vertical axis. Oscillations which may occur can be damped by applying a very slight torque to the
outer gimbal about the precession axis.
Determine the time the outer gimbal takes to rotate five complete revolutions. Calculate the precession
r
frequency f and the precession angular velocity Ω p :
Ω p = 2 πf
(3)
Measure the lever arm: the distance (along the spin vector arrow) between the inner gimbal rotational axis
and the notch on which the torquer weight was placed.
Express the applied torque (in N m) by multiplying the force due to the torquer weight (in N) by the lever
arm (in m).
Use (2) to determine the angular momentum L of the gyro.
Repeat the experiment for three or four other combinations of torquer and lever arm. Two weights are
supplied; each weight can be placed in any of the different positions on the shaft; more than one weight can
be used at a time.
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3 - Measurement of rotor moment of inertia. Analytical determination of rotor moment
of inertia.
The moment of inertia of a body measures the distribution of its mass about the axis of rotation. The angular
momentum L of a rotating body is dependent upon the moment of inertia I:
r
r
L = Iω
(4)
r
L is the angular momentum, ω is the angular (spin) velocity of rotation of the body and I is the moment
r
of inertia of the body about the axis of rotation which is the ω (vector) axis.
The objective of this experiment is to calculate the rotor moment of inertia from: the angular momentum as
measured in Experiment 2, and the spin angular velocity to be measured in this experiment.
Balance the gyro, apply power to the rotor and allow approximately one minute for the rotor to come up
to speed. Measure the time required to complete ~50 revolutions, calculate the rotor spin velocity in rad/s.
Use (4) and the previously determined L to calculate I.
In order to calculate the rotor moment of inertia we shall consider the solid to be made of two bodies
having moments of inertia I1 and I2. The moment of inertia I of the rotor can be calculated by using the
method of superposition. To simplify the calculation, we neglect the small hub at the centre and the rounding
of all sharp edges. The rotor is a composite solid made up as the difference between two right circular
cylinders. To get the solid, we have to subtract I2 from I1:
1
1
I1 = M1r12 = V1ρ r12
2
2
(5)
I1 (r1 )
I2
=
(r2 )
I
I2 =
1
1
M 2 r22 = V2 ρr22
2
2
M1,2 , r1,2 , V1,2 are masses, radii and volumes of the two cylinders; ρ is the density of the cast iron
material (7 × 103 kg/m3).
Calculate the value for the moment of inertia of the gyro rotor. Compare the value to that measured before.
Discuss the sources of error.
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4 - Measurement of system moments of inertia
In this experiment two composite moments of inertia of the system will be measured:
a) the inner gimbal plus rotor moment of inertia about the inner gimbal axis of freedom;
b) the inner gimbal and outer gimbal moment of inertia about the vertical axis.
When the gimbals are restrained by springs (see Figures 1, 2a, 2b and 3) the gyroscope would oscillate
with a natural frequency
ω =
K
I
(6), where K is the torque constant and
I is the composite moment of inertia of the inner gimbal plus rotor. The torque constant is equal to the ratio
of the output torque to the output angular displacement:
K=
τ
(7)
θ
In the situations shown in Figures 1, 2a, 2b and 3, the torque applied to the rotor-plus-gimbal is ,
r
r r
r
r
τ = Σ F × l by definition of torque. Here Σ F is the sum of the forces acting at a distance l from
the axis of rotation. These forces are provided by the two springs. If k 1 and k 2 are the spring constants
of the two springs, d1 and d2 the extensions from the unstressed length when the rotor is at rest, and ∆
the extension of one spring and the concomitant compression of the other as the rotor oscillates (vertically
in Figures 2a and 2b, horizontally in Figure 3), we have:
τ = Σ F × l = [ k 1 ( d1 + ∆ ) − k 2 (d 2 − ∆ )]
(8)
which, if k 1 . k 2 . k and d1 . d2 , becomes :
τ = 2k l ∆
(9)
Now ∆ = lθ where θ is the angular displacement of the rotor-plus-gimbal (see Figure 1).
Equation (7) becomes:
τ
2 kl 2θ
K=
=
= 2 kl 2
(10)
θ
θ
and the angular frequency:
2 kl 2
ω=
I
You will assume k 1 . k 2 . k ~ 76.5 N/m
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(11)
Figure 1.
Figure 2a.
Figure 3.
Figure 2b.
Procedure
A. Moment of Inertia about Inner Gimbal Axis
Balance the gyro (see Instrument alignment, above). Attach the pointer and spring lever to the inner
gimbal then apply power to the gyro rotor. Balance the outer gimbal by means of the adjustment weights
on the shaft of the spin vector arrow. Attach the springs as shown in Figure 2a and 2b.
Deflect the spin vector arrow a small amount (5 to 10 degrees) and then release it.
Time ~ 20 oscillations with a stopwatch. Compute the natural frequency in oscillations/second. Calculate
K from Eq. (10) and then use Equation (11) to compute the “inner” moment of inertia Ii.
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B. Moment of Inertia about the Outer Gimbal Axis
Use the gyro configuration shown in Figure 3 without rotor excitation.
Orient the spin vector arrow in the horizontal plane.
Deflect the outer gimbal ~ 5 to 10 degrees about the vertical axis and then release it.
Time ~20 oscillations with a stopwatch; compute the natural oscillation frequency in oscillations/second.
Use equations (10) and (11) to compute the “outer” moment of inertia Io of the rotor, inner gimbal and
outer gimbal about the vertical axis.
5. Analytical study of nutation
Nutation is the oscillation of the spin arrow as it precesses. This may be observed by giving a downward
tap to the head of the spin arrow as it precesses; a damped oscillatory motion will be observed. Another
exercise would be to clip one of the weights to the spin arrow and observe the precession.
The angular frequency of the nutation depends on the inertial properties of the spinning rotor. If Iow is the
moment of inertia of the outer gimbal plus rotor plus weight about its vertical axis, and Iiw the moment
of inertia of the inner gimbal plus rotor plus weight about its horizontal axis, it can be shown that the angular
frequency of nutation of the rotor is given by:
ω n = 2 πf n =
L
I iw I ow
(12)
Io and Ii determined in Experiment 4 become now Iow and Iiw. They have a term added to take account
of the extra contribution of the torquer mass.
Balance the gyro and unlock the outer gimbal. Apply power to the rotor and align the spin vector arrow
in the horizontal plane. Clip a weight (use the maximum torque available) and time ~50 cycles of nutation.
Calculate the nutation frequency in oscillations/second. Compute the nutation frequency using Equation (12)
and compare the result with the measured value.
Observations
- A limitation of the apparatus comes from friction in the bearings of the gimbals. This is particularly
noticeable with a large weight hung on the rotor axis. If friction did not exist, the arrow would not drop with
time. You should figure out which bearing is causing this and you might work out how to move the base
to eliminate this torque while you perform your experiment.
- One of the most useful features of a gyroscope is the stability of its direction of spin in space. Gyroscopic
stability can be understood by rewriting Equation (1):
r Iω f − Iω i
τ =
∆t
where
ω f and ω i
are the final and initial angular speed (spin) of the object. If
r
τ
is small or applied
for a short time, it will have only a small effect in changing the angular momentum of the object and the spin
direction will be constant.
RM Serbanescu - Feb. 2003. Previous versions of this guide sheet were written by D. Harrison (1974),
J. Vise (1988) and T. Key (1995)
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