Uncertainty of pendulum period and pendulum length

[dobbygenius]

Homework Statement

what is the uncertainty of time taken?

Relevant Equations

-

I'm doing a lab report where I manually measure the time taken for a bifilar pendulum to do 10 oscillations. Is there a rule or a method that I should follow to calculate its uncertainty? Or is the uncertainty just an estimation of human reaction time and judgment?

I also need to know the uncertainty of 1/length. First I measured the bifilar pendulum with a ruler of 0.001 m increment so the length uncertainty is 0.001/2=0.0005 m. However, in my investigation, I need to know the uncertainty of 1/L where L is length. Is there also a rule to calculate the uncertainty of 1/L? Or does it follow the uncertainty of length itself (0.0005)?

Thank you so much for any help

 

[mjc123]

If the uncertainty in length is << L, you can assume that the relative uncertainty in 1/L is the same as the relative uncertainty in L, i.e.
u(L)/L = u(1/L)/(1/L)
Thus u(1/L) = (1/L)*u(L)/L = u(L)/L².

 

[dobbygenius]

If the uncertainty in length is << L, you can assume that the relative uncertainty in 1/L is the same as the relative uncertainty in L, i.e.
u(L)/L = u(1/L)/(1/L)
Thus u(1/L) = (1/L)*u(L)/L = u(L)/L².

thank you so much!

 

[BvU]

I manually measure the time taken for a bifilar pendulum to do 10 oscillations. Is there a rule or a method that I should follow to calculate its uncertainty?

You could repeat the measurement a number of times and calculate an estimate of the standard deviation from the sample. See e.g. 'Sample standard deviation' here .
This only works if you don't get the same result for all measurements

Note that the oscillation period for a pendulum depends on the amplitude, so you want to do each measurement starting from the same -- small! -- amplitude.

First I measured the bifilar pendulum with a ruler of 0.001 m increment so the length uncertainty is 0.001/2=0.0005 m.

Sure ...
But you want to be clear on what exactly you are measuring. What you want is probably the $L$ in the formula. The distance from the suspension point to the center of mass of the pendulum (or some other point perhaps ?). That is two readings on your ruler (which itself must be reliably calibrated at the same temperature as the temperature at the time of your measurement).
Draw an end view of your setup and think about this $L$

This is not to scare you off but to encourage you to consider details and assign some (or no) weight to them.

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[haruspex]

Homework Statement:: what is the uncertainty of time taken?
Relevant Equations:: -

I manually measure the time taken for a bifilar pendulum to do 10 oscillations. Is there a rule or a method that I should follow to calculate its uncertainty?

There are two methods I see used.

With only one measurement, you would estimate the sources of error; in this case, accuracy of timing, accuracy of count (but I'm sure you can ignore the second).

For many samples, you would use the "standard error of the mean". This is different from the "Sample standard deviation". In particular, it diminishes as the sample size increases, as should be expected.
See https://www.investopedia.com/ask/an...andard-error-means-and-standard-deviation.asp.

In principle, there should be a method which melds the two so that it works for all sample sizes, but I have never found one.

 

[dobbygenius]

You could repeat the measurement a number of times and calculate an estimate of the standard deviation from the sample. See e.g. 'Sample standard deviation' here .
This only works if you don't get the same result for all measurements

Note that the oscillation period for a pendulum depends on the amplitude, so you want to do each measurement starting from the same -- small! -- amplitude.
Sure ...
But you want to be clear on what exactly you are measuring. What you want is probably the $L$ in the formula. The distance from the suspension point to the center of mass of the pendulum (or some other point perhaps ?). That is two readings on your ruler (which itself must be reliably calibrated at the same temperature as the temperature at the time of your measurement).
Draw an end view of your setup and think about this $L$

This is not to scare you off but to encourage you to consider details and assign some (or no) weight to them.

$\$

I see, thankyou so much again!

I got the uncertainty of the average time taken for 10 oscillations to be 0.9%. Now I need to know the uncertainty of the time taken for 1 oscillation, is there another method to calculate the uncertainty of this?

 

[dobbygenius]

There are two methods I see used.

With only one measurement, you would estimate the sources of error; in this case, accuracy of timing, accuracy of count (but I'm sure you can ignore the second).

For many samples, you would use the "standard error of the mean". This is different from the "Sample standard deviation". In particular, it diminishes as the sample size increases, as should be expected.
See https://www.investopedia.com/ask/an...andard-error-means-and-standard-deviation.asp.

In principle, there should be a method which melds the two so that it works for all sample sizes, but I have never found one.

I see, thankyou!

 

[BvU]

I see, thankyou so much again!

I got the uncertainty of the average time taken for 10 oscillations to be 0.9%. Now I need to know the uncertainty of the time taken for 1 oscillation, is there another method to calculate the uncertainty of this?

If the number ten is without error, the relative error for one oscillation is the same as the relative error for ten oscillations.

If you are familiar with differentiation:$$\Delta f(x) \approx {\partial f\over\partial x}\Delta x \Rightarrow \Delta 10x \approx 10\Delta x\Rightarrow {\Delta 10x \over 10x} \approx {\Delta x\over x}$$ (actually, squares or absolute values; see here and examples)

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