How Does Measuring Position Impact Speed Measurement Accuracy in a Pendulum?

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SUMMARY

The discussion centers on the impact of measuring both the speed and position of a pendulum on the accuracy of speed measurements. Participants explore the relationship between measurement uncertainties, specifically questioning whether incorporating position measurements improves speed accuracy. The consensus is that measuring position alongside speed can reduce overall measurement error, contrary to the initial assumption that it might increase uncertainty. The conversation references the Heisenberg Uncertainty Principle, although it is deemed not directly applicable in this classical mechanics context.

PREREQUISITES
  • Understanding of classical mechanics, specifically pendulum motion
  • Familiarity with measurement uncertainty and error analysis
  • Knowledge of the Heisenberg Uncertainty Principle and its implications
  • Basic mathematical skills for combining measurements and uncertainties
NEXT STEPS
  • Research the mathematical relationship between position and velocity in pendulum motion
  • Study measurement uncertainty principles and their applications in physics
  • Explore the use of photogates for measuring instantaneous speed in pendulums
  • Investigate advanced error analysis techniques for combined measurements
USEFUL FOR

Physics students, educators, and researchers interested in measurement systems, error analysis, and classical mechanics, particularly in the context of pendulum dynamics.

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Homework Statement


We have a pendulum in thermal balance with the surroundings - no damping. We measure the speed of the pendulum. The accuracy of each speed measurement is known. How does the accuracy of the speed change, if we decide also measure the position of the pendulum with accuracy 10% of the amplitude?

Homework Equations


Just anything you can think of.

The Attempt at a Solution


Believe me, I would be more than happy to show anything. It has been quite some time since I first came across this problem but I still have no idea how to continue.
 
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Is this a Heisenberg Uncertainty Principle oriented problem?
 
Not necessary but it could also be included.
The topic I am dealing with is "Measurement and measuring systems in Physics". The goal is more to simply use all of the knowledge he have, if solving this problem includes using Heisenberg Principle, than so be it.
 
You could try writing an expression for the pendulum's velocity in terms of its position along its trajectory. Then find the uncertainty in velocity due to the uncertainty of position by taking a differential. I'm assuming it's not a Heisenberg uncertainty problem since quantum effects would be negligible (unless your pendulum is very very small).
 
What do you measure ? The speed of a pendulum ? That's a function of time. Or the period ? That depends on the amplitude.
See e.g. http://web.mit.edu/8.01t/www/materials/modules/chapter24.pdf, appendix.
 
I assumed the instantaneous speed could be measured with a photogate of some kind (with some uncertainty in itself) but if there is also uncertainty in the pendulum's position when the speed is measured, it would add additional error to the speed measurement.

Yes the speed is a function of time but so is the angle (thus position), so you can eliminate the time variable. It is possible I'm misinterpreting something.
 
Miles Whitmore said:
I assumed the instantaneous speed could be measured with a photogate of some kind (with some uncertainty in itself) but if there is also uncertainty in the pendulum's position when the speed is measured, it would add additional error to the speed measurement.

I am almost 100% sure this is not the case. Of course each measurement of the velocity has it's well defined uncertainty BUT if I can (at the same time) measure also it's position, that this should improve my result not make the error even bigger.

The first case, measuring the velocity only and having no idea of pendulum's position, gives me a certain value of it's velocity.

However measuring the position of the pendulum and using some basic math, I improve my knowledge of the system and what exactly is happening, therefore if I combine both measurements (as the problem states) the error should reduce. Or am I wrong?
 

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