Simple Pendulum - Length Estimation

Classical Mechanics Level 3

In mechanics, a solver is usually presented with a description of a physical system and is expected to analyse it using first principles and assess how well the results correlate with experiments. A frequent assumption is made which is that the parameters of the system are known accurately. However, in reality, this may not be the case. It then becomes useful to estimate the parameters of a system using given measurements (obtained from experiments) following which further analysis can be carried out.

Example: To analyse using mechanics, with a fair degree of accuracy, how the movement of a car evolves with time, its mass is required to be estimated. With fuel being consumed at time-varying rates and passengers' masses and their distributions within the vehicle being unpredictable, this task is far from trivial.

In this problem, the solver is provided with a set of 'very accurate measurements' of a simple pendulum's angular position vs. time. The length of the pendulum is unknown. The task at hand is to estimate the length of this pendulum using this given information. The data of angle $\theta$ vs. time $t$ can be found in this Pastebin link.

Take $g = 10 \ m/s^2$ .

Initial conditions: $\theta(0) = \frac{\pi}{3} \ rad$ , $\dot{\theta}(0) = 0 \ rad/s$ . Ignore dissipative forces. Treat the bob as a point mass.

There is more than one way to solve this problem. One may use pure brute force or a more sophisticated technique. I look forward to seeing your solutions.

The answer is 3.1416.

1 solution

Steven Chase
Feb 8, 2020

These "raw data" problems are always fun. The dynamics of the pendulum are:

$\ddot{\theta} = \frac{-g \sin \theta}{L}$

Re-arranging into a more useful form gives:

$L = \frac{-g \sin \theta}{\ddot{\theta}}$

We are given the $(t, \theta)$ data in column form. What remains is the following:

1) Use difference quotients and the data from Columns 1 and 2 to derive $\dot{\theta}$ in Column 3
2) Use difference quotients and the data from Columns 1 and 3 to derive $\ddot{\theta}$ in Column 4
3) Calculate the sine of $\theta$ in Column 5
4) Calculate the estimated value for $L$ in Column 6

The estimate isn't very good, since the time step is quite large. Ordinarily, we would want to look at time steps two or three orders of magnitude smaller than this. But it is reasonable to guess from the estimated values that the true value is $\pi$ .

Thanks for posting. I will try to cook up a follow up problem involving more than one parameter.

Karan Chatrath - 1 year, 4 months ago

Sounds good. Perhaps a bit more resolution too?

Steven Chase - 1 year, 4 months ago

Yes, I will try. I initially used a much smaller time step to get these results. But simply pasting them on the link was troublesome due to the volume of data. There may be a workaround but this is my best guess right now.

Karan Chatrath - 1 year, 4 months ago

The beautiful thing though is that the calculation only requires two or three data points

Steven Chase - 1 year, 4 months ago

@Steven Chase – True. Your approach is quite simple. I solved it by carrying out a linear regression between the quantities $\sin{\theta}$ and $\ddot{\theta}$ . This way, I made use of the entire data-set.

Karan Chatrath - 1 year, 4 months ago

@Karan Chatrath – Interesting. Yeah, I later realized that I could have done curve-fitting of the sort you described, and that method wouldn't be sensitive to the low resolution. So there is a sensible trade-off there; a more sophisticated approach can tolerate low-res data.

Steven Chase - 1 year, 4 months ago

Simple Pendulum - Length Estimation

The answer is 3.1416.

1 solution

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