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## Homework Statement

Consider a bead of mass m moving on a spoke of a rotating bicycle wheel. If there are no forces other than the constraint forces, then find the Lagrangian and the equation of motion in generalised coordinates. What is the possible solution of this motion?

## Homework Equations

Lagrange's equation of motion

## The Attempt at a Solution

Use cylindrical coordinates for the problem.

Let the origin be at the centre of the wheel and let the wheel lie on the rθ-plane such that θ=0 at the top of the wheel and θ>0 in the clockwise direction.

So, [itex]T = 0.5m(\dot{r}^{^2} + r^{2}\dot{\theta}^{2})[/itex] and [itex]U = mgrcosθ[/itex].

So, [itex]\frac{∂L}{∂r} - \frac{d}{dt}(\frac{∂L}{∂\dot{r}}) = 0[/itex]

[itex]\Rightarrow (mr\dot{\theta}^{2} - mgcosθ) - m\ddot{r} = 0[/itex]

[itex]\Rightarrow \ddot{r} = r\dot{\theta}^{2} - gcosθ[/itex]

and [itex]\frac{∂L}{∂θ} - \frac{d}{dt}(\frac{∂L}{∂\dot{θ}}) = 0[/itex]

[itex]\Rightarrow (-mgrsinθ) - mr^{2}\ddot{θ} = 0[/itex]

[itex]\Rightarrow \ddot{θ} = - \frac{g}{r}sinθ [/itex]

Do you think I've got the equations of motion right?

I have no idea how to work out the possible solution of the motion. Thoughts?