How many cycles of time should be measured for the most accurate T?

[merrypark3]

Homework Statement

Measuring the period of a sine wave using oscilloscope, for the best accuracy, is it better to measure only one cycle?

Homework Equations

The Attempt at a Solution

Let [itex]t=T_{1}+T_{2}+\cdots + T_{n}[/itex] (Measuring n cycle)

From error propagation formula,
[itex]\delta t = \sqrt{(T_{1})^2+(T_{2})^2+\cdots + (T_{n})^2}[/itex] (or [itex]\delta t= n \delta T_{1} [/itex] ?)


but as [itex] T_{1}, T_{2}, \cdots , T_{n}[/itex] are independent so the value of each [itex]\delta T_{n} [/itex] is same. So, let [itex] \delta T= \delta T_{1} = \delta T_{2}= \cdots = \delta T_{n} [/itex]

This [itex] \delta T [/itex] is a error for a period.

Therefore,

[itex] \delta t= \sqrt{n(\delta T)^{2}}=\sqrt{n} \delta T[/itex]

But, [itex] \delta t \propto n[/itex] (as n increases, the scale for one div the screen become smaller,
which increase your error)

So [itex] \delta t = \sqrt{n} \delta T \propto n [/itex]

So, [itex] \delta T \propto \sqrt{n} [/itex]


Therefore, measuring the least number of cycle is best

Is it correct?

 

[Quantum Braket]

I think you inverted something there.

Read through this a bit: http://courses.washington.edu/phys431/propagation_errors_UCh.pdf

Afterwards, ask yourself how repeated trials affects the error.

 

[Borek]

Common logic tells me measuring once length of several cycles should yield a better result (as the measurement error is effectively divided by the number of cycles, which in an exact number).

Caveat: if the cycles are not identical, this way you will lose the information about variability (variance).

 

[voko]

Measuring multiple cycles gives better accuracy if your clock is accurate. If the clock itself is drifting, then it is a nasty question. I think we need more context here.

 

[paranormal]

I would say that it depends on the specific circumstances and goals of the measurement. In general, measuring more cycles of a sine wave can improve the accuracy of the measurement, but it also increases the potential for error. If the goal is to get a rough estimate of the period, then measuring only one cycle may be sufficient. However, if a more precise measurement is needed, it may be beneficial to measure multiple cycles and take an average. Ultimately, the number of cycles to measure should be determined based on the desired level of accuracy and the potential sources of error in the measurement.