# Help with determining errors on Kater's Pendulum experiment?

1. Feb 15, 2015

### J.Sterling47

1. The problem statement, all variables and given/known data
Hey guys, so I'm doing the an exercise on the Kater's pendulum, to calculate g. I've gotten down my g calculation to g = 9.80658m +/- 0.00054 using equation 1 below. The errors taken into account are just on the kater period T and the distance between the two pivot points (L) using a vernier scale. However I need to take into account a few more sources of error from the following list:

Finite amplitude/width of pendulum
Buoyancy of air
Damping due to friction
Imperfect knife edges
Temperature variations during measurement
Elastic variations of pendulum length
Flexibility of pendulum
Altitude of measurements

So I'm expected to give a qualitative and quantitative explanation on how these could affect my calculation. I could not due these experimentally due to the time constraint and lack of equipment (except altitude), but somehow I'm still supposed to estimate how these variables will affect my calculation quantitatively. Some of these are negligible, but for the ones that aren't, how can I go about estimating the errors?

2. Relevant equations
1.) g = (2π)2(L/T2)

Actually this was the only equation given.

3. The attempt at a solution

I tried looking up the actual experiment done by Kater. According to the wiki page Kater made corrections for temperature, finite width, atmospheric pressure and altitude, but I couldn't find the methods he used. Either way, since I can't do these myself, I don't know where to start.

2. Feb 16, 2015

### BvU

That looks like a very small error to me. 0.000055 relative error.

About 50 μm if the length of the pendulum is 1 m and T is exact.
If I remember well, this length error is the biggest contribution.
We used an invar rod and a vernier and let four different people measure. Forgot the result (it was 44 years ago...)

If L is exact, 0.000055 relative error in g is 0.06 ms when T is 2 s.
T follows from the intersection of two curves where a small weight is placed in different postions with turns of a screw.
You can do a polynomial fit and calculate, but looking at the plot is usually just as good.

The amplitude correction can be found here

Friction has a very small influence. See here for ω, estimate γ from amplitude decrease over a large number of periods.

The others you wave away with estimates (length change from elasticty, air, etc.)

I always found it an interesting topic. Nowadays you can google around and even find that this device was once used (1824) to define the yard. Never knew that.

3. Feb 18, 2015

### J.Sterling47

Ok thanks, although for the amplitude correction I do not have an angle as I did not measure it when I had the chance. That will still help me estimate it though!

As for altitude, the measurement was taken at about 115m above sea level, how could I take this into account? Would this and the temperature difference be waved away with estimates as well? Wiki specifically mentions those two (along with the amplitude) so I'm guessing they are more significant?

4. Feb 18, 2015

### BvU

Google ! But it's an obvious dependence.

5. Feb 19, 2015

### J.Sterling47

Hey thanks for your help so far, I got almost everything down on the list except the fluid mechanics. I was reading this which gave three effects (this is for the clock pendulum but it seems pretty generic and would be the same for this case): http://en.wikipedia.org/wiki/Pendulum#Atmospheric_pressure

It says air resistance can be neglected for the pendulum clock, and for the hyperphysics link, I don't have a way to quanitfy the amplitude decrease.
As for the first two:buoyancy and atm pressure, what would be a good way to estimate?

6. Feb 20, 2015

### BvU

Isn't it so that the first has to do with the mass of the bob, and that doesn't appear in the expression for g ? Just the fact that the two T are equal makes the length of the equivalent mathematical pendulum equal to the distance between the knives ?

Idem the second for mass of pendulum ? And the reversing principle makes that go away ? (I'm not sure).

And the third one is in the damping. Estimate that amplitude decreases by a factor of 2 (or e) in, say, 20 or 30 minutes and see what the effect on $\omega$ is.

Same thing with amplitude correction: estimate. 2 cm? 4 cm ?

Any correction is better than none. Even if the error on the correction is big, it still brings the final value closer tot he true value.
And don't worry too much about the litterature value either (is at the same latitude as your location ?). The aim of the experiment is not to reproduce that litterature value, but to carry out an independent determination.