Thread: Can you reduce the uncertainty when measuring different lengths

Hi everyone,

I don't know if this is basic enough to be in the pre-university section, but I guess it can be moved if not.

Imagine you are watching a pendulum and taking a note of the time each time the pendulum reaches the right hand side. There will be an uncertainty in the measurement of this time: Delta[t]

Following the rules of uncertainty propagation, this means that the uncertainty in measuring the period of a swing (P = t₂ - t₁) will be sqrt[2] x Delta[t].

Now if I imagine that the period P is constant, I can lower the uncertainty in P (Delta[P]) by measuring the time over say 100 periods. Delta[P] will then = sqrt[2] x Delta[t] / 100 right?

My question is this:

In the real word, the period P will not be constant, and will increase on each swing as the pendulum loses energy. Is it still okay to use the method above? Isn't it only okay to divide by N when you are measuring the same thing each time? I can't justify in my own mind, that it is still okay to divide by N, even when P may have different values each time you measure it.

My immediate thought, is that if you were measuring things of very different lengths, then you cannot divide by the number of measurements N to reduce your uncertainty on any one object. But if we go back to the pendulum, the change in period will be very small and so perhaps it is still reasonable to divide by N in this case?

I hope this makes sense!

Any help on this would be great!
Thank you.

I think you need to consider what it is that you're trying to measure. If you record the total time to make multiple cycles you will determine the average cycle time more accurately than if you time just one cycle. Bit if your goal is to get the most accurate representation of the time for teh first cycle only this won't help. You're better off measuring the first cycle many times over, then determining the uncertainty due random variation. You can also get an estimate for systemic variation by using your timing mechanism on a well-known, calibrated system - this will tell you whether your measurements are consistently high or low.

In the system I am thinking about (I am using the pendulum as a less complicated analogy), the period will increase very slightly each time. I am actually more concerned with what the uncertainty of the measurement of P will be, rather than the value of P itself.

I will actually measure the P separately each time on any one swing, because it will change slightly - i.e. I will have a value for P[1], P[2], P[3] etc., I won't be taking the mean after 100 swings. Does this mean that I am not permitted to divide the uncertainty of any one P measurement by 100 to decrease the uncertainty? Am I only permitted to do that if I am first dividing my P values by 100 to get the mean P?