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lylos
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Homework Statement
Consider a smooth hemisphere of radius a placed in the Earth's magnetic field. Place a small point mass on the top of the sphere and provide an initial small displacement as to allow the mass to slide down the sphere. Calculate the point where it falls off the sphere.
This is from chapter 2.4 of Goldstein.
Homework Equations
[tex]L=T-V[/tex]
[tex]\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}=\frac{\partial L}{\partial q}[/tex]
The Attempt at a Solution
First, I followed Goldstein in using a coordinate axis that is centered at the bast of the hemisphere with z pointing to top of sphere. The resulting motion can be contained in the xz plane if we consider the initial velocity in y to be zero.
[tex]L=\frac{1}{2}m(\dot{x}^2+\dot{z}^2)-mgz+\lambda(\sqrt{x^2+z^2}-a)[/tex]
When I transform this to spherical coordinates, keeping in mind that R is constant, I have:
[tex]L=\frac{1}{2}m(R^2\dot{\theta}^2)-mgRCos(\theta)+\lambda(R-a)[/tex]
Which yields the following equations:
[tex]mR\dot{\theta}^2-mgCos(\theta)+\lambda=0[/tex]
[tex]mR^2\ddot{\theta}=mgRSin(\theta)[/tex]
[tex]R-a=0[/tex]
Goldstein states that you would solve the 2nd then solve the 1st and you can then solve for lambda. I am wondering what trick you must use to solve for the 2nd equation. I feel that the small angle approximation won't work here. Please enlighten?