I. Introduction

The experiment is aim to:

• Measure the angular velocity of the flywheel ((s) and the precession of the gyroscope system((p).

• Using Ws and (p above to calculate Izz (mass moment of inertia of the gyroscope)

• Compare this experimental Izz to the theoretical Izz.

A gyroscope is a mechanism consisting of a spinning flywheel, mounted on a base so that its axis can turn freely in any direction. The gyroscopic motion occurs whenever the axis about which the body is spinning is itself rotation about another axis.

Gyroscope has a wide application in our daily life. The bicycle, for instance, is a good example of gyroscopic motion. The spinning wheel on bicycle act as a gyroscope, when we attempt to rotate the wheel about its axis, due to the procession of a gyroscope, the wheel will attempt to rotate about an axis perpendicular to the wheel axis. This procession keeps the wheel rotating either left or right so that the bicycle can stay upright. A gyrocompass is another example of the application of a gyroscope. The rotation of our earth about its axis gives it the properties of a huge gyroscope. The gyrocompass combines the action of the gyro rotor inside and the earth’s rotation to perform as a direction reference.

II. Method

1) Apparatus used:

• Gyroscope system(including a flywheel)

• Electrical motor

• Stroboscope

• Stopwatch

• Measuring ruler, pencil

2) Measuring angular velocity of the flywheel((s)

• Rev up angular velocity of the flywheel with an electrical motor.

• Hold the flywheel right in front of the stroboscope.

• Adjust the flashing rate of the stroboscope so that the frequency of the flashing matches up with the angular velocity of the flywheel, at this instant, the mark on the flywheel will appear stationary.

• Read the angular velocity from the stroboscope. This is the angular velocity((s) of the flywheel.

• Record this angular velocity((s) every 10 seconds.

3) Measuring the procession of the gyroscope((p)

• Find the centre of mass of the gyroscope, mark as 0, and then label the Length, L= -0.05m, -0.04m, -0.03m, -0.02m, -0.01m, 0.01m, 0.02m, 0.03m, 0.04m, 0.05m.

• Rev up the angular velocity of the flywheel with the electrical motor.

• Set the pivot point at 0, release the gyroscope gently from a near horizontal position so that we can minimise any discernible nutation. Allow the gyroscope to spin.

• Record the time spends for one complete revolution.

• Repeat above steps for 10 different Length L.

4) Theory

Izz = WL/((s *(p)

Where Izz = mass moment of inertia about z axis.

W = weight of the gyroscope

L = length from the mass centre to rotating axis

(s = angular velocity of the flywheel

(p = angular velocity of the gyroscope

III. Result

1) Angular velocity of the flywheel((s)

|Time(s) |Angular Velocity , (s,(RPM) |Angular Velocity, (s,(rad/s) |
| |Trial1 |Trial2 |Trial3 |Average | |
|0 |3000 |3000 |3000 |3000 |314.2 |
|10 |2900 |2600 |2900 |2800 |293.2 |
|20 |2800 |2600 |2700 |2700 |282.7 |
|30 |2700 |2700 |2600 |2667 |279.3 |
|40 |2600 |2500 |2500 |2533 |265.3 |
|50 |2500 |2300 |2400 |2400 |251.3 |
|60 |2400 |2250 |2300 |2317 |242.6 |

Table 1. Angular velocity of the flywheel ((s) during 60 seconds.

Plot the angular velocity (s (rad/s) against time (s), we can clearly see a decay of the angular velocity over the 60 seconds period. The rate of decay is fairly a constant, approximately 10 rad/s.

[pic]

Figure 1. Angular velocity ((s) agianst Time (s)

2) Procession of the gyroscope((p)

|Length,L(m) |Time for 1 rev |Procession, (p,(RPS) |Procession, (p,(rad/s) |
| |procession(s) | | |
|-0.05 |5.07 |0.197 |1.24 |
|-0.04 |6.55 |0.153 |0.96 |
|-0.03 |9.16 |0.109 |0.69 |
|-0.02 |13.31 |0.075 |0.47 |
|-0.01 |23.29 |0.043 |0.27 |
|0 |no rotation observed |
|0.01 |26.4 |0.038 |0.24 |
|0.02 |13.6 |0.074 |0.46 |
|0.03 |10.0 |0.100 |0.63 |
|0.04 |6.8 |0.147 |0.92 |
|0.05 |6.0 |0.167 |1.05 |

Table 2. procession of the gyroscope for different Length, L.

IV. Discussion

1) Calculation for theoretical Izz:

[pic]

Figure 2. Dimensions of the flywheel in mm.

For Area 1,

Volume 1 = π * [(0.14/2)2- (0.122/2)2] * 0.025 = 9.26*10-5 m3

Mass 1 = ρ*V1 = 7850 kg/m3 * 9.26*10-5 m3 = 0.727 kg

r = [(0.14/2) + (0.122/2) ]/2 = 0.0655 m

Izz 1 = mr2 = 0.727kg * (0.0655m)2 = 3.12*10-3 kg*m2

For Area 2,

Volume 2 = π * [(0.122/2)2- (0.039/2)2] * 0.004 = 4.20*10-5 m3

Mass 2 = ρ*V2 = 7850 kg/m3 * 4.20*10-5 m3 = 0.330 kg

r = [(0.122/2) + (0.039/2) ]/2 = 0.0403 m

Izz 2 = mr2 = 0.330kg * (0.0403m)2 = 5.36*10-4 kg*m2

For Area 3,

Volume 3 = π * [(0.039/2)2- (0.019/2)2] * (0.0105+0.004) = 1.32*10-5 m3

Mass 3 = ρ*V2 = 7850 kg/m3 * 1.32*10-5 m3 = 0.104 kg

r = [(0.039/2) + (0.019/2) ]/2 = 0.0145 m

Izz 3 = mr2 = 0.104kg * (0.0145m)2 = 2.19*10-5 kg*m2

For Area 4,

Volume 4 = π * [(0.026/2)2- (0.019/2)2] * (0.026-0.0105-0.004) = 2.85*10-6 m3

Mass 4 = ρ*V2 = 7850 kg/m3 * 2.85*10-6 m3 = 0.0224 kg

r = [(0.026/2) + (0.019/2) ]/2 = 0.0113 m

Izz 4 = mr2 = 0.0224kg * (0.0113m)2 = 2.86*10-6 kg*m2

( Izz = Izz 1 + Izz 2 + Izz 3 + Izz4

= 3.12*10-3 + 5.36*10-4 + 2.19*10-5 + 2.86*10-6

= 3.78 *10-3 kg*m2

Slightly different from the given theoretical value Izz = 3.85*10-3 kg*m2 , this may caused by roundings during the calculation.

2) Calculation for experimental Izz:

• Izz = WL/((s* (p)

• Mass of gyroscope =2.5 kg

Weight W = mg=2.5kg * 9.81 m/s2 =24.525 N

• Theoretical Izz = 3.85*10-3 kg*m2

• Error(%) = (Theoretical Izz – Experimental Izz) / Theoretical Izz

• If the time of one revolution is within 10s, then we assume the angular velocity of the flywheel is 314.2 rad/s.

For example:

Length L= -0.05 m, t=5.07s

(p = 1.24 rad/s

(s = 314.2 rad/s

( Izz = WL/(s (p = 24.525*0.05/(314.2*1.24) = 3.15*10-3 kg*m2

Error = (3.85*10-3 - 3.15*10-3)/ 3.85*10-3 =18.18%

• If the time of one revolution is more than 10s, for the purpose of accuracy, we use table 1 to calculate an average angular velocity of the flywheel.

For example:

Length L=0.01 m , t=26.4s

(p = 0.24 rad/s

From Table 1, we know that within 20 seconds, (s decays from 314.2 rad/s to 282.7 rad/s.

→average (s = (314.2+282.7)/2 = 298.5 rad/s

( Izz = WL/(s (p = 24.525*0.01/(298.5*0.24) = 3.42*10-3 kg*m2

Error = (3.85*10-3 - 3.42*10-3)/ 3.85*10-3 = 11.17%

• The result experimental Izz for each Length and its respective error are listed in the following table:

|Length,L,(m) |Experimental Izz (kg*m^2) |Error(%) |
|-0.05 |3.15*10^(-3) |18.18% |
|-0.04 |3.25*10^(-3) |15.58% |
|-0.03 |3.39*10^(-3) |11.95% |
|-0.02 |3.44*10^(-3) |10.65% |
|-0.01 |3.04*10^(-3) |21.04% |
| | | |
|0.01 |3.42*10^(-3) |11.17% |
|0.02 |3.51*10^(-3) |8.83% |
|0.03 |3.85*10^(-3) |0.00% |
|0.04 |3.39*10^(-3) |11.90% |
|0.05 |3.72*10^(-3) |3.38% |

Table 3. Experimental Izz and errors for each individual Length.

Average Experimental Izz = 3.42*10-3 kg*m2

Average Error = (3.85*10-3 - 3.42*10-3)/ 3.85*10-3 =11.17%

The greatest error in the set of test is 21.04%

3) Error analysis

The error appears in this experiment is quite large (11.17%), this could be caused by a number of reasons:

• From the first experiment, we see that the angular velocity of the flywheel decays over time. Therefore, for the second experiment, the assumption of a constant angular velocity is obviously not appropriate for producing an accurate result.

Possible improvement :

For the measurement that takes more than 10 seconds to complete a full revolution, instead of timing the full revolution, we can time half revolution so that the decreasing of angular velocity will not have such a great impact on the results.

• During the second experiment, some unavoidable nutations must be produced. This will cause some momentum lost in vertical movement and therefore affect the accuracy of the result.

Possible improvement :

The nutation of the gyroscope is very hard to avoid. The best thing we can do is releasing the gyroscope as gently as possible in a nearly horizontal position which we have already done in this experiment.

• The friction induced between the flywheel and its rotating axis as well as the air resistance are the major sources contribute to the decay of angular velocity of the flywheel. Friction presents almost in every experiment and is hard to avoid.

Possible improvement:

If we can set the experiment in a vacuum, in that way, we can reduce the air friction enormously and therefore produce a more accurate result.

V. Conclusion

In the first part of the experiment, we showed that the angular velocity of the flywheel decays from 314.2 rad/s to 242.6 rad/s over 60 seconds period. The rate of decay is approximately a constant with a value of 10 rad/s.

In the second part of the experiment, we obtained the angular velocity of the gyroscope for each length, L, through measuring the time the gyroscope spends for a complete revolution.

In conclusion, the aim of the experiment has been well achieved. We obtained a reasonable value for the mass moment of inertia of the flywheel (Izz = 3.42*10-3 kg*m2). Compare this value to the given theoretical Izz value (3.85*10-3 kg*m2), we calculated an error of 11.17% which is quite large. This may caused by several reasons. Some potential errors in the experiment were analysed in the report and accuracy may be improved in future experiments.

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L

flywheel

y

Rotate about y axis

Z