Physical Pendulum Formula Derivation

[Parto0o]

Homework Statement

A physical pendulum, consisting of a uniform rod (of mass M and length L) with an attached blob, can oscillate about an axis that goes through one end of the rod. The mass of the blob is also M. The distance of the blob to the rotation axis is x.
The aim is to derive a formula for the period T of small-amplitude oscillation versus the distance x.

Homework Equations

Small-amplitude oscillation: T = 2∏√(I/Mgd)

The Attempt at a Solution

At first I didn't even know what formula to begin with. So I asked my lecturer and he told me that I have to use the formula for small-amplitude oscillation; which would be T = 2pi srt(I/Mgd).
I guess, finding the center of mass of the physical pendulum is the hardest part for me. Now I really have no idea how to relate everything and derive the formula.

 

[rude man]

You can use your formula. Your main task is to determine the center of gravity of the compound pendulum consisting of the uniform rod and the blob. The c.g. will obviously be a function of x. d is then the distance between the pivot point and the c.g.

To find the c.g., think of taking the compound pendulum off the pivot and making a see-saw out of it, i.e using a knife-edge to balance the two halves. The location of the knife edge is the location of the c.g. It will obviously be a function of the blob position x.

http://en.wikipedia.org/wiki/Pendulum[/url]

If you haven't had differential calculus including simple ordinary differential equations, ignore the following.

More fundamentally, you can solve the ordinary differential equation (ODE) τ = I θ'' where
τ = restoring torque as a function of θ
I = rotational inertia of the compound pendulum
θ = the angle the pendulum makes with the vertical.

τ will comprise the restoring torque exerted on the bar by the bar itself as well as the restoring torque exerted on the bar by the blob. The latter will of course be a function of x.

By 'restoring torque' we mean the torque exerted by gravity to the bar and to the blob. The ODE is solved by assuming an initial θ = θ₀. You will need to assume sinθ ~ θ to solve the ODE.