How Do You Formulate Lagrange Equations for a Mass on a Rotating Parabolic Path?

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Homework Help Overview

The discussion revolves around formulating Lagrange equations for a mass constrained to move along a rotating parabolic path defined by the equation z = a.r^2, where the mass is influenced by gravity. Participants explore two scenarios: one where the parabola is fixed and another where it rotates about the OZ axis.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the equations of position and velocity for both scenarios, questioning the implications of fixing the angle θ and its effect on the components of motion. There is also a debate regarding the correct expression for the derivative of z.

Discussion Status

Some participants have provided feedback on the proposed equations, indicating that part (b) appears correct while seeking clarification on part (a). There is an ongoing exploration of the implications of setting θ to zero and how it affects the equations.

Contextual Notes

Participants are considering the constraints of the problem, such as the fixed nature of θ in part (a) and the rotational aspect in part (b). There is uncertainty regarding the correct formulation of the derivative for z, which remains under discussion.

Fabio010
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A point of mass m, affected by gravity, is obliged to be in a vertical plan on a parabola with equation z = a.r^2

a is a constant and r is the distance between the point of mass m and the OZ vertical axis. Write the Lagrange equations in the cases that the plan of the parabola is :

a) is fixed

b) it rotates with angular speed ω about the OZ axis.
http://www.google.pt/imgres?q=parabola+lagrangian+mechanics&um=1&hl=pt-PT&sa=N&biw=1097&bih=521&tbm=isch&tbnid=AbK_S7_Po3-jSM:&imgrefurl=http://stochastix.wordpress.com/2007/12/11/a-bead-sliding-on-a-rotating-parabola/&docid=vdbKHvoDOKa0bM&imgurl=http://stochastix.files.wordpress.com/2007/12/parabola.jpg%253Fw%253D450&w=300&h=300&ei=LjCNUJuvPMHDhAfL34CoCw&zoom=1&iact=hc&vpx=200&vpy=117&dur=1276&hovh=225&hovw=225&tx=132&ty=131&sig=102710367222874968480&page=1&tbnh=129&tbnw=129&start=0&ndsp=16&ved=1t:429,r:12,s:0,i:104I just need to know the equations of the position and speed.

in a) i considered:

x = r cosθ ||||| x' = r'cosθ - rθ'sinθ
y = r sinθ ||||| y' = r'sinθ + rθ'cosθ
z = a.r^2 ||||| z' = a.r^2' in b)

x = r cos(wt) ||||| x' = r'cos(wt) - rwsin(wt)
y = r sin(wt) ||||| y' = r'sin(wt)+ rwcos(wt)
z = a.r^2 |||||| z' = a.r^2' Is that right?
 
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For part (a) θ is fixed, hence θ' = ? I would think that without loss of generality you could let θ be zero.

Part (b) looks good to me.
 
TSny said:
For part (a) θ is fixed, hence θ' = ? I would think that without loss of generality you could let θ be zero.

Part (b) looks good to me.


considering that θ = 0, then part a) is not going to have the y component :)


And other question:

z' = a.r^2' or z' = 2.a.r'.r ? is that the same?
 
Fabio010 said:
considering that θ = 0, then part a) is not going to have the y component :)


And other question:

z' = a.r^2' or z' = 2.a.r'.r ? is that the same?

Correct!
 
Thanks for all your the help!