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(adsbygoogle = window.adsbygoogle || []).push({});

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:104

I 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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# Homework Help: Lagrangian dynamics problem.

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