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Friday, January 29, 2016

Lagrangian function of a sphere on a cylindrical surface

Question:
There is an open cylinder of radius R., which is stationary. A hollow sphere of radius \rho and mass m rolls on the surface without slipping. Find the Lagrangian function.

Answer:
The Lagrangian function is defined as
L = KE - PE
where KE and PE represent kinetic energy and potential energy, respectively.
The mechanical potential energy is the gravitational force \times height, which is taken from the center. The height varies with the angle \theta. Thus,
PE = -mg(R-\rho)\cos\theta
The negative sign indicates the height is directed to the center.
Now the kinetic energy has two parts, translational and rotational kinetic energies.
KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 
The translational velocity, v, is along with the trace of circle whose radius is R-\rho. Therefore,
v = (R-\rho)\omega
The angular velocity can be expressed by
\omega = \theta' = \frac{d\theta}{dt}
The moment of inertia of hollow sphere is
I = \frac{2}{3}m\rho^2
Plug everything into above.
L = \frac{1}{2}m(R-\rho)^2 \theta '^2 + \frac{1}{2}\frac{2}{3}m\rho^2 \theta '^2 + mg(R-\rho)\cos\theta 
We can rewrite it as
L = \frac{m\theta'^2}{6}(3R^2 -6R\rho + 5\rho^2)+ mg(R-\rho)\cos\theta

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