What is the Effect of a Lorentz Force on a Pendulum's Gyroscopic Stabilization?

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In summary: Yes, something like that is going on. But a pretty thing is: the Lorentz force does not do the work but it turns unstable equilibrium into the stable one :)This is not a very famous effect but it is really amazing. Consider a pendulum which consists of a massless rod of length ##r## and a point of mass ##m##; the system is in the standard gravitational field ##\boldsymbol g##. So the point ##m## moves on the sphere of radius ##r##.It is clear, the equilibrium when the point rests in the North Pole of the sphere is unstable. However, if you add a damping term,
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wrobel
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This is not a very famous effect but it is really amazing. Consider a pendulum which consists of a massless rod of length ##r## and a point of mass ##m##; the system is in the standard gravitational field ##\boldsymbol g##. So the point ##m## moves on the sphere of radius ##r##.
It is clear, the equilibrium when the point rests in the North Pole of the sphere is unstable.

Introduce a Cartesian inertial frame ##OXYZ## with origin in the point of suspension and the axis ##OZ## is vertical such that ##\boldsymbol g=-g\boldsymbol e_z##.
Now let us switch on a Lorentz force ##\boldsymbol F=\boldsymbol B\times\boldsymbol v## which acts on ##m##. The vector ##\boldsymbol B=B\boldsymbol e_z## is constant.

Theorem. Assume that ##B## is sufficiently big:
##\frac{B^2}{8m}>\frac{mg}{2r}##
then the North Pole equilibrium is stable.
 
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No I just thought that it would be interesting for PF. I planned to write the proof.
 
  • #4
wrobel said:
This is not a very famous effect but it is really amazing.
Intuitively, when B is very strong it will force m on tiny circles anytime it tries to fall from the the North Pole.
 
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Oh, didn't notice the username.

Yes, magnetic fields tend to stabilize charged things, that should not be surprising.
 
  • #6
A.T. said:
Intuitively, when B is very strong it will force m on tiny circles anytime it tries to fall from the the North Pole.
Yes, something like that is going on. But a pretty thing is: the Lorentz force does not do the work but it turns unstable equilibrium into the stable one :)

mfb said:
es, magnetic fields tend to stabilize charged things, that should not be surprising.
so it makes wonder only me
 
  • #7
It is not a fully stable point in the classical sense: if you displace the pendulum a bit, it won't come back to the center, it will rotate around it in a sequence of small "u"-patterns.

If you add a damping term, no matter how small, the stable point should become unstable.
 
  • #8
That surprises me with the damping. Gravity and the magnetic field do not change the total energy, while damping can only reduce it. And the north pole is the state of maximal energy. I would expect some downwards spiral if the initial position is slightly off.

m=1g, r=1m, B=1T leads to 8.8 mC charge (B should be Bq I guess). Hmm, not practical on a large scale.
 
  • #9
O, I am sorry, I deleted my last post containing error.
mfb said:
If you add a damping term, no matter how small, the stable point should become unstable.
this is true

mfb said:
It is not a fully stable point in the classical sense: if you displace the pendulum a bit, it won't come back to the center, it will
in accordance with definition it is not obliged to come back https://en.wikipedia.org/wiki/Stability_theory
 
  • #10
mfb said:
It is not a fully stable point in the classical sense
Seems similar to the stability of Lagrange points, which aren't minima of the graviational+centrifugal potential, but stuff stays there due to the Coriolis force, which is analogous to Lorentz force here.
 
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the proof
 

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FAQ: What is the Effect of a Lorentz Force on a Pendulum's Gyroscopic Stabilization?

1. What is gyroscopic stabilization?

Gyroscopic stabilization is a method of maintaining balance and stability in an object through the use of gyroscopic forces. This is achieved by spinning a gyroscope, which is a spinning wheel or disk mounted on an axis, at high speeds.

2. How does gyroscopic stabilization work?

Gyroscopic stabilization works by utilizing the principles of angular momentum and precession. When a gyroscope is spinning, it resists any change in its orientation, and this resistance helps to stabilize the object it is attached to. As the object moves or tilts, the gyroscope's axis of rotation will also shift, causing a force that counteracts the movement and maintains stability.

3. What are the applications of gyroscopic stabilization?

Gyroscopic stabilization has a wide range of applications, including aircraft and spacecraft stabilization, navigation systems, gyroscopic compasses, and stabilizers for cameras and other equipment. It is also commonly used in vehicles such as motorcycles and bicycles to improve balance and stability.

4. What are the advantages of using gyroscopic stabilization?

One of the main advantages of gyroscopic stabilization is that it does not require any external power source. The spinning gyroscope provides its own stability, making it a reliable method in situations where power may not be readily available. Additionally, gyroscopic stabilization is highly effective at maintaining stability, even in harsh and unpredictable environments.

5. Are there any limitations or drawbacks to gyroscopic stabilization?

While gyroscopic stabilization is a useful technique, it does have some limitations. One of the main drawbacks is that it is only effective in maintaining stability along a single axis. This means that multiple gyroscopes may be needed for objects that require stability in multiple directions. Additionally, gyroscopic stabilization can be affected by external factors such as vibrations or changes in temperature, which may impact its effectiveness.

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