# Using Lagrange to solve rotating parabolic motion and equilibrium

• relativespeak
In summary, we considered a bead sliding on a wire bent in a parabolic shape and being spun with a constant angular velocity. Using Lagrangian and cylindrical polar coordinates, we found the equation of motion for the bead. To find equilibrium positions, we set the derivatives of the polar radius to zero and determined that equilibrium occurs when w^2 = 2kg or when p = 0. Further analysis is needed to determine the stability of these equilibrium positions.
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## Homework Statement

Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity w about its vertical axis. Use cylindrical polar coordinates and let the equation of the parabola be z = kp2. Write down the Lagrangian in terms of p as the generalized coordinate. Find the equation of motion of the bead and determine whether there are positions of equilibrium, that is, values of p at which the bead can remain fixed, without sliding up or down the spinning wire. Discuss the stability of any equilibrium positions you find.

L=E-U
v1=pw
v2=dp/dt
v3=2kp(dp/dt)
U=mgkp^2
E=mv^2/2

## The Attempt at a Solution

I found the equation of motion using the lagrangian, which matched the answer in the back of the book. I just don't know how to use Lagrance to find equilibrium, and I've tried several things such as setting d2p/dt2=0 and d2U/dp2=0.The equation of motion is: (g=accel. due to gravity)

pw2-4k2p(dp/dt)2-2kgp=(d2p/dt2)(1+4k2p2

Last edited:

dp/dt=0, d2p/dt2=0

so equil. occurs when w2=2kg, or when p=0. Is that right? And then how can I find which is stable/unstable?

Have you studied any equilibrium conditions/criteria?

)-2k2p(dp/dt)2

To find equilibrium, we can set the right side of the equation to zero, which gives us:

dp/dt=0 and p=0 or dp/dt=2pw/3k and p=1/2pw2/3k2

These values of p represent positions where the bead can remain fixed without sliding up or down the spinning wire. We can determine the stability of these equilibrium positions by looking at the second derivative of the potential energy, which is related to the curvature of the parabola. If the second derivative is positive, the equilibrium position is stable, and if it is negative, the equilibrium position is unstable.

In this case, the second derivative of the potential energy is positive for both equilibrium positions, indicating that they are both stable. This is because the curvature of the parabola increases as p increases, making it more difficult for the bead to slide up or down the wire.

Overall, using Lagrange to solve for the equation of motion and determine equilibrium positions in rotating parabolic motion is a useful tool for analyzing and understanding the dynamics of this system.

## What is Lagrange's method?

Lagrange's method is a mathematical approach used to solve problems involving motion and equilibrium. It is named after the famous Italian mathematician Joseph-Louis Lagrange and is based on the principle of least action.

## How is Lagrange's method used to solve rotating parabolic motion?

Lagrange's method is used to find the equations of motion for a rotating parabolic system by considering the kinetic and potential energies of the system. It involves setting up a Lagrangian function and applying the Euler-Lagrange equations to find the equations of motion.

## What is the significance of using Lagrange's method for rotating parabolic motion?

Lagrange's method is significant because it provides a more elegant and efficient way to solve problems involving rotating parabolic motion. It also allows for the inclusion of constraints and can be applied to systems with multiple degrees of freedom.

## Can Lagrange's method be used to solve equilibrium problems?

Yes, Lagrange's method can also be used to solve equilibrium problems by setting the equations of motion to zero. This allows for the determination of the equilibrium points of a system.

## What are the limitations of using Lagrange's method for solving rotating parabolic motion?

Lagrange's method may be limited in its application to more complex systems with non-conservative forces, such as friction. It also requires a good understanding of calculus and mathematical principles to apply effectively.

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