consider a bead of mass m constrained to move on a fricitonless wire helix whose equations in cylindrical polar coords is(adsbygoogle = window.adsbygoogle || []).push({});

z = a phi where a is some constant

the bead is acted upon by a force which deends on the distance from the cneter only.

Formulate the problem using s the distance along the helix as your generalized coordinate.

for distance s along the helix [itex] s = a \phi_{0} [/itex]

but r would not be constnat

[tex] T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\phi_{0}}^2 + \dot{z}^2) [/tex]

z dot is zero and phi0 is a constnat soso

[tex] T = \frac{1}{2} m \dot{r}^2 [/tex]

the force is dependant on the distance from the center only

thus [tex] \vec{F} = -k\vec{r} = -\nabla V [/tex](say)

then [tex] V = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + a^2 \phi_{0}^2) [/tex]

[tex] L = T - V = \frac{1}{2} m\dot{r}^2- \frac{1}{2} k (r^2 + a^2 \phi_{0}^2) [/tex]

is this formulation correct?

Or am i totally off?

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# Lagrangian mechanics of a bead of mass

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