Determining How To Measure Position Langrange's Equation

In summary, Figure 7.16 depicts a smooth horizontal wire hoop rotating at a fixed angular velocity ω around a vertical axis through point A. A bead with mass m is threaded on the hoop and can move around it, with its position described by the angle θ relative to the diameter AB. The Lagrangian for this system can be obtained using θ as the generalized coordinate. Using the Lagrange equation of motion, it can be shown that the bead oscillates about point B like a simple pendulum. The frequency of these oscillations, with small amplitude, is ω. The system has no potential energy as the hoop rotates in a horizontal plane with no gravity. The kinetic energy of the bead is dependent on both its
  • #1
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Figure 7.16 is a bird's-eye view of a smooth horizontal wire hoop that is forced to rotate at a
fixed angular velocity co about a vertical axis through the point A. A bead of mass m is threaded on the hoop and is free to move around it, with its position specified by the angle [itex]\phi[/itex] that it makes at the center with the diameter AB. Find the Lagrangian for this system using as your generalized coordinate. (Read the hint in Problem 7.29.) Use the Lagrange equation of motion to show that the bead oscillates about the point B exactly like a simple pendulum. What is the frequency of these oscillations if their amplitude is small?

I am sort of confused as to exactly what the motion is. Does the hoop rotate around the point A, and does this hoop then exert some force on the bead, causing it to rotate through angles around the loop, the angles being measured relative to the fixed point B? If this is true, is there someway of relating the angular velocity of the loop to the angular velocity of the bead?
 
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  • #2
It would be easier if you attached figure 7.16.
 
  • #3
Hmm, I distinctly remember attaching the figure. Let me try this again.
 

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  • #4
Yes, the hoop is "hinged" at point A, and the axis AB swings around point A with a fixed frequency, ω.

The hoop has a diameter D = length of AB. The bead has mass m and can be located by the angle it makes with the axis AB, with the angle θ being used to track it ... say it runs CCW with zero the line AB, with B at 12 o'clock.

We need a second angle to describe the hoop's position ... let vertical be 12 o'clock, and let ψ run CCW.

Then the bead is located in space by the angles θ,ψ.
 
  • #5
Now, is there a way we can write θ in terms of ψ, or ψ in terms of θ?
 
  • #6
The geometry of the system does not provide any relationship - they are independent.

There will be dynamical relationships (depending upon starting values and time); for these you must find the equations of motion.
 
  • #7
Okay. So, do I need to consider the kinetic energy of the bead and hoop, or just the bead? And would this system have potential energy, wouldn't it be zero?
 
  • #8
The mass of the hoop is not given, and it's motion is uniform (forced by some engine) - so only the bead's energy is required.

So pick some inertial reference frame (like when the hoop is vertical, ψ=0, and establish the kinetic energy of the bead in terms of the angle θ: this will include the tangential motion along the hoop, as well as the rotational energy from the motion of the hoop; this latter will depend on the angle θ due to the changing effective radius of this rotation. The angular velocity in this direction is fixed: ω.

The potential energy will be due to gravity, and for this you will need both angles.
 
  • #9
Is this hoop vertical? From the context of the question, and the pictures, I figured the hoop and bead lay in the horizontal plane. If this is the case, how does gravity effect either the bead or hoop?
 
  • #10
You are correct ... the hoop rotates about a vertical axis; thus there is no gravity.
 
  • #11
Okay, let me summarize what I think is going on, even though I might be repeating myself. As the hoop rotates around the point A, the bead will rotate around the hoop, having an angular position of [itex]\phi (t) [/itex] relative to the line AB; [itex]\dot{\phi}(t)[/itex] will be it's angular velocity relative to this line. So, it's angular velocity relative to just the point A, that is, its angular velocity relative to the vertical axis going through the point A, will be [itex]\psi = \dot{\phi} + \omega[/itex]. As the hoop rotates around the point A, the bead will rotates around the hoop, causing the distance between the bead and the point A to vary, which I imagine would result in its rotational inertia and rotational KE to vary.

So, the KE would be [itex]KE = \frac{1}{2}mv^2 + \frac{1}{2}I \psi^2[/itex], where v is the tangential velocity of the bead? Does this seem correct?
 
  • #12
Seems OK!
 

1. What is Langrange's equation?

Langrange's equation is a mathematical formula used in classical mechanics to describe the motion of a system of particles. It is based on the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action, a measure of the system's energy.

2. How is position measured in Langrange's equation?

In Langrange's equation, position is measured using generalized coordinates, which are variables that describe the position of a particle or system in terms of its degrees of freedom. These coordinates can be chosen based on the specific constraints and geometry of the system being studied.

3. What are the steps involved in determining how to measure position in Langrange's equation?

The steps involved in determining how to measure position in Langrange's equation include: identifying the degrees of freedom of the system, choosing appropriate generalized coordinates, writing out the kinetic and potential energies of the system in terms of these coordinates, and then applying the Euler-Lagrange equations to determine the equations of motion for the system.

4. Can Langrange's equation be used to describe the motion of any system?

Yes, Langrange's equation can be used to describe the motion of any system that can be defined in terms of generalized coordinates. This includes systems such as pendulums, planetary orbits, and rigid bodies.

5. What are some potential applications of Langrange's equation?

Langrange's equation has many applications in physics and engineering, including in the fields of mechanics, electromagnetism, and quantum mechanics. It is used to model and predict the behavior of complex systems and has been instrumental in the development of theories such as general relativity and quantum field theory.

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