Asymmetric Free Top: Euler's Equations & Stability

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In summary, the conversation discusses a homework problem involving the asymmetric free top with the solution of constant angular velocity for two of the axes and a small perturbation for the third axis. It also mentions the system of linear equations for the perturbation and asks for the general solution and its interpretation in terms of stability. One person expresses confusion and helplessness, but later states they have figured it out.
  • #1
Shafikae
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Consider the asymmetric free top with I1 [tex]\neq[/tex] I2 [tex]\neq[/tex] I3

1) Show that [tex]\omega[/tex]1 = [tex]\Omega[/tex] = const. and
[tex]\omega[/tex]2 = [tex]\omega[/tex]3 = 0 is a solution to Eulers equations.

2) Consider a small perturbation about the spin of the form
[tex]\omega[/tex]1 = [tex]\Omega[/tex] + v1
[tex]\omega[/tex]2 = v2
[tex]\omega[/tex]3 = v3
and assume that the vk are small. What is the system of linear equations for the vk?

3) Find the general solution to the system of equations and interprete the result in terms of stability of the motion.
 
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  • #2
You stated your homework problem clearly. So what is your question?
 
  • #3
I don't understand any of it! I don't know what to do, I don't know how to begin solving it. I'm a helpless case :(
 
  • #4
Shafikae said:
I don't understand any of it! I don't know what to do, I don't know how to begin solving it. I'm a helpless case :(
Use the template for homework posts:

Homework Statement



Homework Equations



The Attempt at a Solution


It would lead you to the first thing I'd suggest here...
Namely since the the problem refers to Euler's Equations, why don't you post those to show us you know what they are.
 
  • #5
Schaefer would be displeased!
 
  • #6
I got it thank you!
 

FAQ: Asymmetric Free Top: Euler's Equations & Stability

What is an Asymmetric Free Top?

An asymmetric free top is a rigid body that is free to rotate about a fixed point in space. It is asymmetric because its mass distribution is not symmetrical, meaning it has a different moment of inertia about different axes. This makes its rotation behavior more complex compared to a symmetric top.

What are Euler's Equations?

Euler's equations are a set of equations that describe the rotational motion of a rigid body. They are used to calculate the angular velocity and angular acceleration of the body about its principal axes, using its moment of inertia and external torques acting on it.

How do you solve Euler's Equations for an Asymmetric Free Top?

To solve Euler's equations for an asymmetric free top, you need to use the equations of motion for a rigid body and then apply them to the specific case of an asymmetric top. This involves finding the principal axes of the top and calculating its moment of inertia about each axis. Then, you can use the equations of motion to solve for the angular velocity and acceleration of the top.

What is the stability of an Asymmetric Free Top?

The stability of an asymmetric free top is determined by its moment of inertia and its angular velocity. If the top has a large moment of inertia and a high angular velocity, it will be more stable and less likely to tip over. However, if the top has a small moment of inertia and a low angular velocity, it will be less stable and more likely to tip over.

How does the stability of an Asymmetric Free Top change with different initial conditions?

The stability of an asymmetric free top can change with different initial conditions, such as the initial angular velocity and orientation of the top. For example, if the top is initially spinning at a high angular velocity, it may be more stable compared to when it is initially at rest. Similarly, if the top is initially oriented in a more stable position, it may be less likely to tip over compared to when it is oriented in a less stable position.

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