Mouna Fattouh
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Lab: Gyroscope
Introduction:
A gyroscope is a device for measuring or maintaining orientation, based on the principles of
conservation of angular momentum. A mechanical gyroscope is essentially a spinning wheel
or disk whose axle is free to take any orientation.
Gyroscopes are used in modern aircrafts, where an inertial guidance system uses spinning
gyroscopes to monitor and control the orientation of the aircraft. Also, wheels on
motorbikes act as gyroscopes and make the bike easier to balance (stay up right) when
moving.
Aim:



To study a simple gyroscopic system consisting of a flywheel rotating at high at a
high angular velocity counter balanced by a weight
To obtain some insight into the two modes of motion of a gyroscope: precession and
nutation
To measure the pressing angular velocity and to compare with the theoretical
prediction.
Method
The Experimental gyroscopic assembly is shown in Figure 1. The assembly includes a metal
bar with a spinning flywheel on one end and a counterweight on the other end. The metal
bar is balanced on a stand by a sliding pivot.
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A.
Measure the decay of the angular velocity of the flywheel over time
1.
2.
4.
Place a spinning motor in contact with the flywheel to spin it up.
Using a stroboscope measure the angular velocity of the flywheel in revolutions per
minute. This is done by adjusting the frequency of the stroboscope until a stationary
X image is visible off the fly wheel.
Measure the angular velocity of the flywheel in increments of 10 seconds until 60
seconds have elapsed.
For reliability repeat steps 1 to 3.
B.
Calculating the Izz (mass moment of inertia of the gyroscope)
1.
Locate the centre of mass by sliding the pivot until the bar connecting the flywheel
to the counterweight is horizontal. Mark the center of mass position on the bar.
From the centre of mass position mark out ± 10mm, ± 20mm, ± 30mm, ± 40 mm
and ± 50mm.
Place a spinning motor in contact with the flywheel to spin it up to 3000rpm.
Detatch the spinning motor from the flywheel
Place the gyroscope gently onto the stand, balancing it on its centre of mass pivot
point, and once it starts to undergo precessional motion, measure the time it takes
to complete one revolution using a stop watch.
Repeat step 3 to 5 but this time balancing the gyroscope in turns on the positions
marked in step 2.
For reliability, repeat steps 1 to 6.
3.
2.
3.
4.
5.
6.
7.
Calculation of Izz Theoretical
We can obtain a theoretical value for the flywheel’s moment of inertia about the z-axis by
treating the flywheel as four concentric rings of different mass. These four areas are shown
in figure 2. Applying equation 1, the moment of inertia equation to each of them, and
adding these components, the total moment of inertia could be calculated.
1
4
4
𝐼𝑧𝑧,𝑇ℎ𝑒𝑜𝑟. = 2 𝜌𝜋𝑡(𝑟𝑜𝑢𝑡𝑒𝑟
− 𝑟𝑖𝑛𝑛𝑒𝑟
)
Where
 is the density of the material 7850 kg/m3
t is the thickness of the ring (m)
router and rinner are the outer and inner radii of the ring.
(1)
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1
Area 1: 𝐼𝑧𝑧,1 = 2 (7850)𝜋(0.025)(0.074 − 0.0614 ) = 3.133 × 10−3 𝑘𝑔/𝑚2
1
Area 2: 𝐼𝑧𝑧,1 = 2 (7850)𝜋(0.004)(0.0614 − 0.01954 ) = 6.7579 × 10−4 𝑘𝑔/𝑚2
1
2
1
(7850)𝜋(0.0135)(0.0134
2
Area 3: 𝐼𝑧𝑧,1 = (7850)𝜋(0.0145)(0.01954 − 0.00954 ) = 2.4396 × 10−5 𝑘𝑔/𝑚2
Area 4: 𝐼𝑧𝑧,1 =
− 0.00954 ) = 3.3985 × 10−6 𝑘𝑔/𝑚2
Total Moment of Inertia is
4
𝐼𝑧𝑧,𝑇ℎ𝑒𝑜𝑟. = ∑
𝐼𝑧𝑧,𝑛 = 3.837 × 10−3 𝑘𝑔/𝑚2
𝑛=1
≈ 3.85 x 10−3 𝑘𝑔/𝑚2
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Calculations
Experiment A
Table 1: Angular Velocity ( ωs ) of flywheel
Time (s)
ωs (rpm)
st
nd
0
10
1 trial
3000
3000
2 trial
3000
2950
20
30
40
50
60
2900
2910
2750
2600
2500
2900
2800
2750
2700
2600
Average ωs
(rpm)
Average ωs
(rad/s)
3000
314.159
311.541
2975
2900
2855
2750
2650
2550
303.687
298.975
287.979
277.507
267.035
Graph 1: Change in Angular Velocity in time
Angular Velocity (rad/s)
Angular Velocity
320
310
300
290
280
270
260
250
240
0
10
20
30
40
Time (sec)
50
60
Experiment B:
Table 2: The precession, ωp (angular velocity of the entire gyroscope)
Length
(mm)
from com
-50
-40
-30
-20
-10
0
+10
+20
Time (s) for one revolution about the y-axis
Trial 1
4.51
6.2
8.75
13
25
No Precession
23.75
12.48
Trial 2
5.28
6.98
10.49
13.67
28.26
No Precession
25.84
12.78
Average time
(s)
ωp (rpm)
ωp (rad/s)
4.895
6.590
9.620
13.335
26.630
No Precession
24.795
12.630
12.257
9.105
6.237
4.499
2.253
No Precession
2.420
4.751
1.284
0.953
0.653
0.471
0.236
No Precession
0.253
0.497
Mouna Fattouh
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+40
+50
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9.51
6.9
5.73
8.97
7.07
5.88
5
9.240
6.985
5.805
6.494
8.590
10.336
0.680
0.900
1.082
Table 3: Calculation of mass moment of inertia
Length (m)
from c.o.m
W (N.m)
ωs (rad/s)
ωp (rad/s)
Izzexp
% Error
-0.05
-0.04
-0.03
-0.02
-0.01
24.525
24.525
24.525
24.525
24.525
314.16
314.16
314.16
314.16
314.16
1.284
0.953
0.653
0.471
0.236
-3.040 x10−3
-3.277 x10−3
-3.586 x10−3
-3.315 x10−3
-3.308 x10−3
21.04
14.89
6.85
13.90
14.08
0.01
0.02
0.03
0.04
0.05
24.525
24.525
24.525
24.525
24.525
314.16
314.16
314.16
314.16
314.16
0.253
0.497
0.68
0.9
1.082
Average
3.086 x10−3
3.141 x10−3
3.444 x10−3
3.470 x10−3
3.607 x10−3
3.327 x10−3
19.85
18.40
10.54
9.88
6.30
13.57
Discussion/Error Analysis
From part A of the experiment, and angular velocity of the flywheel was calculated and how
it decays over time. From the results, it is clear that the decay is relatively constant over
time. Errors in experiment can be attributed to:
 Adjusting the stroboscope frequency in time to get a reading within 10 seconds
 Delay in time reading, between time reading off stopwatch and reading for
frequency
Such errors result in inaccuracies of angular velocity, hence carrying out the experiment
twice and taking an average to make such errors minimal in final results.
It was also seen that at the centre of mass, there was no precission due to the fact that it is
at equilibrium, demonstrating conservation of angular momentum
In part B of the experiment, the moment of inertia of the gyroscope about the z-axis
𝑊𝐿
was experimentally calculated using the formula 𝐼𝑧𝑧,𝐸𝑥𝑝. = 𝜔 𝜔 , and found to to be an
𝑠
𝑝
average of 3.327 x10−3. This value is seen to be lower than the theoretical value of 3.85
x10−3 by an average of 13.57%. the greatest error however was seen to be from -0.05m
away from com. The further away from the centre of mass, the greater room there is for
error.
Such error in percentage could be mainly due to human error in:
 Timing of 1 revolution
 Judgment of 1 revolution
 Measurement in length from centre of mass
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 Placement of pivot
 Orientation and parallax error

Several assumptions made could also contribute to such discrepencies. These include:
 Accepting the mass of the gyroscope to be 2.5 kg without actually measuring
it.
 Assuming the flywheel is still rotating at 3000 rpm from the time it moves
from the motor to the pivot.
Also, Nutational motion which is the bobbing up and down of the metal bar as it
precesses can also affect the results. When the bar is released from large offsets, such as
50mm, there is a greater chance of instability, hence nutation will occur. This error could be
reduced by placing the gyro on the pivot slowly to minimize the amount of nutation.
This experiment could be more reliable if the above errors are minimized. This could
be done by more careful measurements and by using data loggers for timing as well as
modification to the experimental apparatus (gyroscope) such that the flywheel did not
require to be detached and then reattached. Also if time had permitted, the experiment
needed to be repeated at least more than three times to get consistency.
If these corrections are made than it would be possible to see a greater agreement
between theory and experiment.
Conclusion
The three objectives of the experiment were successfully demonstrated, the first
two in experiment A, by which angular velocity of a flywheel was seen to uniformly decay
over time.
The third objective was to record the precession (the angular velocity of the entire
gyroscope) and to calculate the Izz (mass moment of inertia of the gyroscope) and comparing
this experimental Izz with the theoretical result. The comparison was somewhat successful
with Izz, experimentally calculated to be 3.327 x10−3 kg/m2; this value was lower than the
theoretical value of 3.85 x 10-3 kg/m2 by 13.57%. By minimizing the amount of errors in the
experiment, such a percentage could be reduced.