Pendulum experiment systematic errors

[haruspex]

is that for a cylindrical bob?

Yes.
It's unusual for me/us to supply such a complete solution, but I feel it is appropriate in this case. However, I'd like to believe you understand how these formulae come to be . Please ask me to explain any parts you don't get.

 

[Shyanne]

Yes.
It's unusual for me/us to supply such a complete solution, but I feel it is appropriate in this case. However, I'd like to believe you understand how these formulae come to be . Please ask me to explain any parts you don't get.

I think I understand all the components, however I do have one last question (sorry): I understand that the torque is τ=mghsin(θ) but I do not understand why we are multiplying the masses by both X and (L/2)-D instead of only X.

 

[haruspex]

I think I understand all the components, however I do have one last question (sorry): I understand that the torque is τ=mghsin(θ) but I do not understand why we are multiplying the masses by both X and (L/2)-D instead of only X.

The two mass centres are different distances from the pivot. The bob is at distance X, the rod's centre is at L/2 from the end above the pivot, so at distance L/2-D from the pivot.
(I have assumed the rod stays fixed and the bob slides on the rod.)

 

[mfb]

With the approach of post #29 and values from the thread, I get values between 9.86 and 10.33 m/s^2. That looks reasonable.
If the bob length is reduced to 3 cm, the values fit even better with a range from 9.82 to 9.99.

Who designed this experiment and who wrote instructions for it?
Looks like a horrible experiment design to me.

 

[Shyanne]

The two mass centres are different distances from the pivot. The bob is at distance X, the rod's centre is at L/2 from the end above the pivot, so at distance L/2-D from the pivot.
(I have assumed the rod stays fixed and the bob slides on the rod.)

I see, thank you! I've plotted the results and they seem MUCH better than before! Is there are any particular reason that the 0.05cm pendulum length yielded the most in

The two mass centres are different distances from the pivot. The bob is at distance X, the rod's centre is at L/2 from the end above the pivot, so at distance L/2-D from the pivot.
(I have assumed the rod stays fixed and the bob slides on the rod.)

Ah I see, thank you! Also, according to this site (http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html), shouldn't the mass of interia of the bob be I= M((R^2/4)+(H^2/4))?

 

[mfb]

The formula haruspex used is correct for a solid cylinder. You can make it more precisely by using the formula for a hollow cylinder: replace $R^2$ by $R_1^2 + R_2^2$ with the outer and inner radius. It won't change the result much.
Wikipedia has a large collection of formulas.

 

[haruspex]

according to this site (http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html), shouldn't the mass of interia of the bob be I= M((R^2/4)+(H^2/4))?

The second formula on that page is the one to use. It says M(R²/4+L²/12). That is the same as I quoted, but written a bit differently.

 

[haruspex]

You can make it more precisely by using the formula for a hollow cylinder:

Quite so. I assume that is to allow for the rod's shaft. I did mention I was ignoring the rod's radius, and that was one consequence. A refinement to the rod's moment would be another.

 

[Shyanne]

Quite so. I assume that is to allow for the rod's shaft. I did mention I was ignoring the rod's radius, and that was one consequence. A refinement to the rod's moment would be another.

Ah I see, I'll just include that as an error because I didn't measure that value as well.
I've plotted the results and they look MUCH better than the first time, however, the percentage errors do not follow a pattern like I had expected them to:
0.05: 2.145%
0.10: -0.483%
0.15: -1.749%
0.20: -2.236%
0.25: -2.569%
0.30: -2.371%

Does this merely indicate that any systematic error that occurred with measurements has affected the higher values to a greater degree? But then the 0.30m pendulum has a lower percentage error that 0.25.

 

[haruspex]

Ah I see, I'll just include that as an error because I didn't measure that value as well.
I've plotted the results and they look MUCH better than the first time, however, the percentage errors do not follow a pattern like I had expected them to:
0.05: 2.145%
0.10: -0.483%
0.15: -1.749%
0.20: -2.236%
0.25: -2.569%
0.30: -2.371%

Does this merely indicate that any systematic error that occurred with measurements has affected the higher values to a greater degree? But then the 0.30m pendulum has a lower percentage error that 0.25.

I assume you are quoting percentage errors in estimates of g.
The SHM approximation takes the restoring force as proportional to θ, whereas in reality it is proprtional to sin θ. So for the large angle you are using, the restoring force is overestimated. This will lead to an underestimate of g, probably by a constant factor. This could explain the errors at the longer pendulum lengths.
This suggests another source of error which tends to overestimate g more at the shorter lengths.
Of course, if you are quoting percentage errors in the expected periods, that all gets turned on its head.

 

[Shyanne]

I assume you are quoting percentage errors in estimates of g.
The SHM approximation takes the restoring force as proportional to θ, whereas in reality it is proprtional to sin θ. So for the large angle you are using, the restoring force is overestimated. This will lead to an underestimate of g, probably by a constant factor. This could explain the errors at the longer pendulum lengths.
This suggests another source of error which tends to overestimate g more at the shorter lengths.
Of course, if you are quoting percentage errors in the expected periods, that all gets turned on its head.

Unfortunately, I'm quoting percentage errors of the expected periods. :(

 

[mfb]

If the angle was not the same in every repetition, this can easily explain a 1-2% difference. As discussed on page 1, a 45 degree angle (without an appropriate correction) leads to an 8% error on g. If the angle varies a bit between experiments, you get a difference that is some fraction of those 8%.