Lamina Pendulum Experiment help I've been tasked with researching ways to find the gravitational acceleration g and its uncertainty together with the moment of inertia and and the moment of inertia's uncertainty. I have figured out everything with the equations provided except how to calculate the moment of inertia for the lamina pendulum. I will be using period and a length from the center of mass for my readings. The equation for period will be linearised. I just want to know how to calculate the moment of inertia about the center of mass.
 A: Well, this will certainly depend of the shape and mass distribution of your pendulum. For instance, for a pendulum made of a rod and a thin cylindrical disk, the moment of inertia of the rod (about its own symmetry axis) would be $\frac{1}{12}M_rL^2$, and the moment of inertia of the disk $\frac{1}{2}M_sR^2$. If you have a solid sphere instead of the disk, then it'll be $\frac{2}{5}M_dR^2$, etc. You can easily find tables of different moments of inertia just by googling.
In any case, don't forget to apply Steiner's theorem when necessary!
A: There is no need to calculate the moment of inertia I, which could be quite difficult for an irregular lamina.  You can find both I and g from the same experiment.  
The period T of small amplitude oscillations of a compound pendulum is give by (see Source below) :
$(\frac{T}{2\pi})^2 = \frac{k^2+h^2}{gh}$
where k is the radius of gyration (related to moment of inertia through the centre of gravity) and h is the distance of the pivot from the CG.
This equation can be re-arranged as :
$h^2 = gh(\frac{T}{2\pi})^2 - k^2$.
So if you plot values of $h^2$ against $h(\frac{T}{2\pi})^2$ you will get a straight line $y=ax+b$ with slope $a=g$ and y intercept $b=-k^2$, from which you can find the moment of inertia about the CG :
$I = mk^2$
where m is the mass of the lamina.
So the procedure is :
1.  Suspend the lamina freely from 3 corners in turn (each as far from the estimated CG as possible), marking a plumb line for each corner used, and identify where they cross (CG).  Only 2 lines are needed, but the 3rd is a check on the accuracy of the other two.  The lines should cross at about 60-90 degrees.
2.  Time small amplitude oscillations when the lamina is suspended from each of the 3 corners; also when it is suspended from other points at various distances h from the CG.  Make sure the lamina swings freely each time.
3.  Carefully measure the distances h of each pivot from the CG and record alongside the corresponding values for T.
4.  Plot corresponding values of $h^2$ against $h(\frac{T}{2\pi})^2$.  Find the slope and intercept.
If you plot the data in a curve-fitting package and select a straight line fit, the program will give you not only the slope and intercept but also the error estimates for these, which you can use to get your errors in g and I.
If you don't have access to a curve-fitting package, search online for details about linear regression (statistics).  You can do your own calculations in a spreadsheet - more accurate than making measurements from a graph.
http://www.schoolphysics.co.uk/age16-19/Mechanics/Rotation%20of%20rigid%20bodies/text/Compound_pendulum/index.html
