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Studying gyroscope in a steady Precession

Name of Author

Institution Affiliation

Aim

The aim of the experiment is to verify the association between the spin velocity, the applied moment, the precession angular velocity and the rotational mass moment of inertia on the example of a gyroscope in a steady precession.

Background theory

A gyroscope is a mechanical device with a wheel or disk mounted so that it can rotate rapidly about an axis that is itself free to alter in direction. The figure below is a classical construction of mechanical gyroscope with the rotor mounted on gimbals –pivoted supports allowing rotation around different axes (Rana, 2001). The major characteristic of a gyroscope is a probability to respond onto change of rotation axis direction which enables the determination of dir

The equation of motion for a gyroscope is given by: Where denotes angular momentum of the gyroscope and denotes torque of the external forces. Since , where is the vector of angular velocity and I represents the moment of inertia of the gyroscope. When M=0, no moments of the external forces is experienced, ω= 0

Considering that torque due to gravity, G, causing increased change dL in the angular momentum of the gyroscope and causing increase in time too, denoted by dt then:

Meaning that the direction of torque is the same as that of angular momentum. Therefore, the direction of L changes in the direction of torque. L must always point along the axle, hence, it is always perpendicular to torque due to gravity, G, and therefore L vector follows G

In a circular manner (Rana, 2001). The rate of precession, Ω, can be found when a gyroscope undergoes steady precession, this is when the axle is horizontal so that |m(r ⋀ g)| = mrg = |G. From the above equations we conclude that the equation due to increased change in the angle of precession, dФ in given by:

hence,

Apparatus

Significant disk can easily rotate about a horizontal axis. Two countertop weights can be employed for handling gyroscope (Rana, 2001). Disk movement is the consequence of thread that has been prematurely signed up onto axis. Stroboscope is needed to evaluate the content spinning frequency, that this precession volume is calculated with stopwatch. Counterweights large is provided: 900 grms. And 25 gr and disc momentum: 0. 0139 kg/m2.The diagram below shows the view of the experimental block.

Procedure

1. An observable mark is put on the disc

2. The counterweights are moved on a horizontal position in order to balance the gyroscope

3. The pin is caused with a sharp push and the thread enrolled onto a horizontal axis

4. The disc spinning frequency is measured using a stroboscope whose frequency is varied for it to synchronize with mark spinning frequency.

5. On the gyroscope frequency add-on is placed.

6. The precision frequency is measured that is, when is measured with a stop watch since it is the period of precession, and the stopwatch measurement is taken of 10 spins completion and an angle of rotation taken.

7. The procedures 2-5 are repeated numerous times for accuracy

Measurements and observations

Thus, the gyroscope motion is defined by equation

as.

: rpm rad/s

The angle:

Accuracy of :

Torque

Thus we use the equation:

No.

, rpm

, rad/s

, Nms

, rad/s

1

300

31.5

0.44

0.112

2

545

38

0.52

0.095

3

545

44

0.61

0.080

4

550

50

0.70

0.072

5

555

58

0.8

0.059

Results and discussion

It is observed that most point fit on a straight line of best fit but there are some anomalous points in the first graph. In the second graph, there are three anomalous points removed and the best line of fit is drawn. Thus, from the graph the relationship between angular velocity and the inverted precision frequency and linear dependency will be given by the following equation:

Where from the values of and b are 2.71 and 3.31 respectively. Theory explains that the dependency should be linear as shown from the graph above. That is,

Coefficient is not zero due to imprecision of data recorded.

Sources of errors

1. Errors in observing the accuracy of the time measurement

2. Errors in observing the accuracy in the angle of measurement

3. Errors in observing the accuracy of rotational frequency measurement

To correct the errors, more exact values of and should be taken.

Conclusions

The experiment results and observations taken clearly shows a relationship of with the theoretical values. Since the linear dependency obtained is proved to be linear.

References

Daniel Kleppner, Robert Kolenkow, An Introduction to Mechanics, Cambridge University Press, 2013 - P. 572.

P K Srivastava, Mechanics, New Age International, 1997 - P. 412.

Joag, Rana, Classical Mechanics, Tata McGraw-Hill Education, 2001.

http://physics.kenyon.edu/EarlyApparatus/Mechanics/Gyroscope/Gyroscope.html

Name of Author

Institution Affiliation

Aim

The aim of the experiment is to verify the association between the spin velocity, the applied moment, the precession angular velocity and the rotational mass moment of inertia on the example of a gyroscope in a steady precession.

Background theory

A gyroscope is a mechanical device with a wheel or disk mounted so that it can rotate rapidly about an axis that is itself free to alter in direction. The figure below is a classical construction of mechanical gyroscope with the rotor mounted on gimbals –pivoted supports allowing rotation around different axes (Rana, 2001). The major characteristic of a gyroscope is a probability to respond onto change of rotation axis direction which enables the determination of dir

The equation of motion for a gyroscope is given by: Where denotes angular momentum of the gyroscope and denotes torque of the external forces. Since , where is the vector of angular velocity and I represents the moment of inertia of the gyroscope. When M=0, no moments of the external forces is experienced, ω= 0

Considering that torque due to gravity, G, causing increased change dL in the angular momentum of the gyroscope and causing increase in time too, denoted by dt then:

Meaning that the direction of torque is the same as that of angular momentum. Therefore, the direction of L changes in the direction of torque. L must always point along the axle, hence, it is always perpendicular to torque due to gravity, G, and therefore L vector follows G

In a circular manner (Rana, 2001). The rate of precession, Ω, can be found when a gyroscope undergoes steady precession, this is when the axle is horizontal so that |m(r ⋀ g)| = mrg = |G. From the above equations we conclude that the equation due to increased change in the angle of precession, dФ in given by:

hence,

Apparatus

Significant disk can easily rotate about a horizontal axis. Two countertop weights can be employed for handling gyroscope (Rana, 2001). Disk movement is the consequence of thread that has been prematurely signed up onto axis. Stroboscope is needed to evaluate the content spinning frequency, that this precession volume is calculated with stopwatch. Counterweights large is provided: 900 grms. And 25 gr and disc momentum: 0. 0139 kg/m2.The diagram below shows the view of the experimental block.

Procedure

1. An observable mark is put on the disc

2. The counterweights are moved on a horizontal position in order to balance the gyroscope

3. The pin is caused with a sharp push and the thread enrolled onto a horizontal axis

4. The disc spinning frequency is measured using a stroboscope whose frequency is varied for it to synchronize with mark spinning frequency.

5. On the gyroscope frequency add-on is placed.

6. The precision frequency is measured that is, when is measured with a stop watch since it is the period of precession, and the stopwatch measurement is taken of 10 spins completion and an angle of rotation taken.

7. The procedures 2-5 are repeated numerous times for accuracy

Measurements and observations

Thus, the gyroscope motion is defined by equation

as.

: rpm rad/s

The angle:

Accuracy of :

Torque

Thus we use the equation:

No.

, rpm

, rad/s

, Nms

, rad/s

1

300

31.5

0.44

0.112

2

545

38

0.52

0.095

3

545

44

0.61

0.080

4

550

50

0.70

0.072

5

555

58

0.8

0.059

Results and discussion

It is observed that most point fit on a straight line of best fit but there are some anomalous points in the first graph. In the second graph, there are three anomalous points removed and the best line of fit is drawn. Thus, from the graph the relationship between angular velocity and the inverted precision frequency and linear dependency will be given by the following equation:

Where from the values of and b are 2.71 and 3.31 respectively. Theory explains that the dependency should be linear as shown from the graph above. That is,

Coefficient is not zero due to imprecision of data recorded.

Sources of errors

1. Errors in observing the accuracy of the time measurement

2. Errors in observing the accuracy in the angle of measurement

3. Errors in observing the accuracy of rotational frequency measurement

To correct the errors, more exact values of and should be taken.

Conclusions

The experiment results and observations taken clearly shows a relationship of with the theoretical values. Since the linear dependency obtained is proved to be linear.

References

Daniel Kleppner, Robert Kolenkow, An Introduction to Mechanics, Cambridge University Press, 2013 - P. 572.

P K Srivastava, Mechanics, New Age International, 1997 - P. 412.

Joag, Rana, Classical Mechanics, Tata McGraw-Hill Education, 2001.

http://physics.kenyon.edu/EarlyApparatus/Mechanics/Gyroscope/Gyroscope.html