A thin uniform rod of length L is laid horizontally on top o

A thin uniform rod of length L is laid horizontally on top of a fixed horizontal cylinder of radius R, so that the middle of the rod is in contact with the top op the cylinder, and so that the rod is perpindicular to the cylinder axis. The rod is allowed to rock without slipping on the cylinder.

a.) Find the frequency of small rocking oscilations

b.) Now suppose the rod is placed as before, but at a 45 degree angle to the axis of the cylinder. What is now the frequency of small oscilations?

hint: http://physics.stackexchange.com/questions/57828/a-small-oscillations-of-a-rod-on-the-cylinder

Solution

Your energy conservation is of the form: E=A2+B2=constE=A2+B2=const

Take the time-derivative, then: 2A+2B¨=02A+2B¨=0

Follows SHO equation: ¨=(A/B)¨=(A/B) ,

and the angular frequency is =A/B=A/B

If you want to use the Lagrangian then L=TV=B2A2L=TV=B2A2

The Euler-Lagrange equation is ddt(L/q)=L/qddt(L/q)=L/q

so we arrive at the same result: ddt(2B)=2Addt(2B)=2A