- #1
nhalford
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Homework Statement
A bead of mass [itex]m[/itex] is threaded around a smooth spiral wire and slides downwards without friction due to gravity. The [itex]z[/itex]-axis points upwards vertically. Suppose the spiral wire is rotated about the [itex]z[/itex]-axis with a fixed angular velocity [itex]\Omega[/itex]. Determine the Lagrangian and the equation of motion.
Homework Equations
[tex]L = T - V[/tex]
[tex]\frac{\partial L}{\partial x} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)[/tex]
The Attempt at a Solution
This is related to a previous problem in which the wire is not rotating and its shape is given as [tex]z = k\psi, \hspace{3mm} x = a\cos\psi, \hspace{3mm} y = a\sin\psi[/tex] where [itex]a[/itex] and [itex]k[/itex] are both positive. For that problem, the resulting equation of motion is [itex]\ddot{\psi} = -\frac{gk}{a^2 + k^2}[/itex]
We still have [itex]z = k\psi[/itex], but now [itex]x = a\cos(\psi + \Omega t)[/itex] and [itex]y = a\sin(\psi + \Omega t)[/itex]. This gives [itex]\dot{z} = k\dot{\psi}[/itex], [itex]\dot{x} = -a(\dot{\psi} + \Omega)\sin(\psi + \Omega t)[/itex] and [itex]\dot{y} = a(\dot{\psi} + \Omega)\cos(\psi + \Omega t)[/itex]. Then the kinetic energy is
[tex]
T = \frac{1}{2}m\left(\dot{x}^2 + \dot{y}^2 + \dot{z}^2\right) = \frac{1}{2}m\left(a^2\left(\dot{\psi}^2 + 2\dot{\psi}\Omega + \Omega^2\right) + k^2\dot{\psi}^2\right)
[/tex]
and the potential energy is [itex]V = mgz = mgk\psi[/itex]. Then the Lagrangian becomes:
[tex]
L = T - V = \frac{1}{2}m\left(a^2\left(\dot{\psi}^2 + 2\dot{\psi}\Omega + \Omega^2\right) + k^2\dot{\psi}^2\right) - mgk\psi.
[/tex]
This gives
[tex]
\frac{\partial L}{\partial \psi} = -mgk, \hspace{3mm} \frac{\partial L}{\partial \dot{\psi}} = m\left(a^2\dot{\psi} + a^2\Omega + k^2\dot{\psi}\right), \hspace{3mm} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\psi}}\right) = m\left(a^2 + k^2\right)\ddot{\psi}
[/tex]
so plugging into Lagrange's Equation gives [itex]-mgk = m\left(a^2 + k^2\right)\ddot{\psi}[/itex] or [itex]\ddot{\psi} = -\frac{gk}{a^2 + k^2}[/itex], which is the exact same equation of motion as in the case with the coil not rotating. Obviously this isn't correct. Where am I going wrong here? My instinct is that there might be a problem with my choice of coordinates; in particular, [itex]\psi[/itex] is rotating, but I'm not sure if there is a better choice of coordinates.