Probability density function of a pendulum displacement

In summary, the conversation revolves around a question about why most photos of pendulum clocks show the pendulums at the turning points rather than crossing the mid point. The solution provided involves modeling the pendulum motion as a sinusoidal function and using a mathematical formula to explain the frequency of the pendulum at the turning points. The conversation also touches on the difference between a physicist's explanation and a mathematical model for the pendulum motion.
  • #1
venkatmn
3
0
Hi, I need a verification for this question. Can some one help me?

Question: A man enters the pendulum clock shop with large number of clocks and takes a photograph. He finds that most of the pendulums were at the turning points and only a few were captured crossing the mid point. Why is it so?

Soln:

If we project the pendulum motion on a horizontal line it can be modeled as

x=a.sin(wt+phi)

where 'a' is the amplitude and 'w' is the angular frequency of the motion. Now consider the case we have lots of pendulum clocks hung on the wall of a shop. Then the inital phase will be a random variable whose value falls between (-pi to pi). Now if we apply the formula for function of single random variable we would get for x as

f(x)=1/(pi.sqrt(a^2-x^2)) -a<x<a

From the previous equation we observe that the density function diverges for the turning points and hence the pendulum should be spending most of the time there. Thats why naturally most of the photos show the pendulum at the extremas.

Thank you
 
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  • #2
The logic is OK. In the original question, no mathematics was asked for. A physicist would simply say that the speed in the region near the extrema is slower than the speed in the middle, and so the bob spends more time near the extrema per cycle, and hence the probability is more of finding it near the end points than near the middle.
 
  • #3
Thanks for the verification but I am looking for a mathematical model describing the pendulum motion...
 

1. What is a probability density function (PDF) of a pendulum displacement?

A probability density function is a mathematical function that describes the probability of a pendulum being at a certain displacement at any given time. It shows the relative likelihood of the pendulum being at a particular displacement, with higher probabilities assigned to more likely displacements.

2. How is the PDF of a pendulum displacement calculated?

The PDF of a pendulum displacement is calculated using the formula:
f(x) = (1/√(2πσ2)) * e-(x-μ)2/(2σ2),
where f(x) is the PDF, σ is the standard deviation, and μ is the mean displacement.

3. What does the shape of the PDF of a pendulum displacement indicate?

The shape of the PDF of a pendulum displacement is a bell curve, also known as a Gaussian distribution. This indicates that the most likely displacement of the pendulum is at the mean, and the probability decreases as the displacement moves away from the mean in either direction.

4. How does the length of the pendulum affect the PDF of its displacement?

The length of the pendulum affects the PDF of its displacement by changing the values of the mean and standard deviation. A longer pendulum will have a higher mean displacement and a smaller standard deviation, resulting in a narrower and taller bell curve. A shorter pendulum will have a lower mean displacement and a larger standard deviation, resulting in a wider and shorter bell curve.

5. Can the PDF of a pendulum displacement be used to predict future displacements?

Yes, the PDF of a pendulum displacement can be used to predict future displacements. By knowing the mean and standard deviation, we can use the PDF formula to calculate the probability of the pendulum being at a certain displacement at any given time. However, it should be noted that the PDF assumes ideal conditions and may not accurately predict future displacements in real-world situations due to external factors such as air resistance and friction.

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