# Simple derivation of the Equations of a Gyroscope

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• alan123hk
In summary: I found it quite difficult to follow the derivation of the equations for a gyroscope but these videos helped me a lot in understanding the basic principle.In summary, the videos explained the principle of gyroscope in a very simple and easy to understand way.
alan123hk
Frankly, it is difficult for me to understand the derivation of mathematical equations that describe and explain the motion behavior of a gyroscope, particularly the reason why it spins in one direction, gravity tries to rotate it in a second direction, but it actually ends up turning in the third direction and dose not fall down.

I'm looking for an as simple and easy to understand as possible way for deriving the equations of gyroscope to help me better understanding the basic principle of it.

Any help would be appreciated.

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alan123hk said:
Frankly, it is difficult for me to understand the derivation of mathematical equations that describe and explain the motion behavior of a gyroscope, particularly the reason why it spins in one direction, gravity tries to rotate it in a second direction, but it actually ends up turning in the third direction and dose not fall down.

I'm looking for an as simple and easy to understand as possible way for deriving the equations of gyroscope to help me better understanding the basic principle of it.

Any help would be appreciated.
View attachment 236714

That's an interesting way to look at it. The gyroscope is supported on its stand. If there were no stand it would fall down! The vertical forces are balanced: gravity down from the centre of mass and the normal force up through the stand. This, however, creates a "torque", which you would normally expect to rotate the object about the point of contact on the stand. Torque creates a change in angular momentum. As the gyroscope is spinning you have to take the change in that spin angular momentum into account, when you see that the torque is in the wrong direction to rotate a spinning gyroscope down.

The gyroscope is an example of the full vector nature of angular momentum. And that fact that you cannot ignore spin angular momentum in a rigid body.

nnunn and alan123hk
alan123hk said:
I'm looking for an as simple and easy to understand as possible way for deriving the equations of gyroscope to help me better understanding the basic principle of it.

If linear momentum is more intuitive than angular momentum to you, it might help to decompose the gyroscope into point masses, and consider their linear momentum:

The angular momentum explanation mentioned by @PeroK is a bit more abstract:
http://hyperphysics.phy-astr.gsu.edu/hbase/rotv2.html

PeroK
A.T. said:
If linear momentum is more intuitive than angular momentum to you, it might help to decompose the gyroscope into point masses, and consider their linear momentum:

The angular momentum explanation mentioned by @PeroK is a bit more abstract:
http://hyperphysics.phy-astr.gsu.edu/hbase/rotv2.html

There's a critical point at 2:25 where he equates the torque generated by the vertical forces to a torque generated by horizontal forces. That equivalence would appear to require that the wheel is supported by a rigid joint with the rope. In which case that equivalence would apply to a non-spinning wheel as well. Why can he equate the different torques for a spinning wheel but not for a non-spinning wheel? That would be my question.

Edit: to answer my own question, it doesn't require the rigid joint. In the case of the static wheel, the wheel rotates and falls; and, in the case of the spinning wheel, the wheek precesses. And the faster the wheel is spinning the more pronounced the precession and the less pronounced the falling.

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Hi Perok, AT

Thanks for your valuable and informative replies.

I just found some very cool videos which explained the strange behavior of gyroscopes, and would like to share them to those who want to understand the basic physics of gyroscope. It should be useful for layman like me but certainly not for expert.
I think these videos are very cool because they clearly explained the principle of gyroscope in detail by means of simple and beautiful maths.

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nnunn

## 1. What is a gyroscope?

A gyroscope is a device that is used to measure or maintain orientation and angular velocity. It consists of a spinning wheel or disc, mounted on an axis, that maintains its orientation regardless of any movement or rotation of the mounting.

## 2. What are the equations of a gyroscope?

The equations of a gyroscope are the Euler's equations of motion, which describe the rotation of a rigid body in three-dimensional space. They are:
- ωx = (Izz - Iyy) * ωy * ωz / Ixx
- ωy = (Ixx - Izz) * ωx * ωz / Iyy
- ωz = (Iyy - Ixx) * ωx * ωy / Izz

## 3. How do you derive the equations of a gyroscope?

The equations of a gyroscope can be derived using the Lagrangian method, which involves calculating the kinetic and potential energies of the spinning wheel or disc. The Euler-Lagrange equations are then used to obtain the final equations of motion.

## 4. What are the assumptions made in the derivation of the equations of a gyroscope?

The main assumptions made in the derivation of the equations of a gyroscope are:
- The gyroscope is a rigid body
- The rotation of the gyroscope is about a fixed axis
- There is no external torque acting on the gyroscope
- The gyroscope is spinning at a constant angular velocity

## 5. What are the applications of the equations of a gyroscope?

The equations of a gyroscope have various applications in navigation, robotics, and aerospace engineering. They are used in devices such as gyrocompasses, gyroscopic stabilization systems, and gyroscopic sensors for measuring angular velocity and orientation. They are also used in spacecraft and satellites for attitude control and stabilization.

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