Question

Can someone explain problem 1.6 regarding a physical pendulum for vibrations and waves?

I read the solutions and I don't understand how you can model the arc as two point masses. And how does one know the mass of each point mass is M/2

Additionally, how is cos(alpha/2) = l/D
D being the diameter. I recalled that cosine is adjacent / hypotenuse. Can someone help me see how this represents the equilibrium positions of the masses?

Thanks!

Answer 1

What is the question?

Answer 2

Problem 1.6 − A physical pendulum
A uniform rod of mass m is bent in a circular arc with radius R. It is suspended in the middle and 2 can freely swing about point P. The length of the arc is πR.

*insert picture

a) What is the period of small angle oscillations about P?
b) Compare your result with the period derived (and demonstrated) in lectures for a hoop with mass m and radius R.

http://ocw.mit.edu/courses/physics/8-03sc-physics-iii-vibrations-and-waves-fall-2012/unit-i-vibrations/pset-1/MIT8_03SCF12_OCW_PS01.pdf
Here is the link to view picture (ctrl+f problem 1.6)

Answer 3

mechanical oscillation can be start from the point that the force is derived from the relation F=-kx and the equation of motion is the second order differential equation with a solution like this x=asinwt the mechanical energy and potential energy oscillate