Determining Best Value of k for Compound Pendulum

[discombobulated]

Homework Statement

I have calculated the value of the radius of gyration, k for a bar (and moment of inertia) and got three different values from three different methods. Now i need to determine which is the best value. I'm confused about how to do this, and how do i get the %uncertainty for k.

Value of k/m [Moment of Inertia / kgm²]
0.282 [8.93 x 10-3] (method 1)
0.289 [9.37 x 10-3] (method 2)
0.291 [9.49 x 10-3] (method 3)

(sorry i tried to separate this but for some reason it didn't work, so i used brackets to try and separate them)

Method 1 is from the intercept of the period squared x distance from centre of mass graph
Method 2 is from the dimensions of the bar
Method 3 from using the minimum time period

Homework Equations

T²D = 4/gπ² D² + 4/gπ² k² used in method 1

D = distance from centre of mass
T = radius of gyration
k = radius of gyrat
g= acceleration due to gravity

k = (1/12)(l² + w²)^1/2 used in method 2

k = (T²g)/ 8π² used in method 3

The Attempt at a Solution

I don't think that the experimental value in method 1 is very accurate because of the problems in measuring the period, so would the best value of k be from method 2?
Is there a set value that k is meant to be for a compound pendulum consisting of a wooden bar pivoted at different holes?
I'm quite confused about this and would be really grateful for any help in understanding.

 

[kuruman]

Each of the methods has its own measurement uncertainties that lead to uncertainties in the final answer which you can find by error propagation. Ideally, you should figure out each one separately and see if they overlap.

For example, it looks like the value from method 1 is an outlier. You need to examine it carefully and see whether there something that you did (or didn't do) to minimize the uncertainty. The intercept of the graph depends greatly on the distance from the point of support to the CoM. How accurately did you determine where on the object the CoM is? How well did you measure its distance from the point of support? You should come up with a number, say ± 3 mm. Then you can do two additional calculations using method 1 with the higher and lower values. See if either one comes closer to the other values. You mention that you think that there were "problems" measuring the period. What kind of problems? Usually, one measures the time for a good number of oscillations, say 10-20, and divides the total time by that number. One also starts the clock as the pendulum goes through the vertical at maximum speed because at maximum angle there is greater uncertainty as to when exactly the pendulum is instantaneously at rest.

This is just an example of what I mean when I say that you have to examine critically what you did with respect to introduction of uncertainties. Only you know what you did and didn't do so only you can figure that out.