Need help finding uncertainty for this equation

[User849483]

Homework Statement

Finding uncertainty of formula

Relevant Equations

Uncertainty

this is the formula
v is velocity
g= gravity
h= height
ro= outer radius of cylinder
ri = inner radius of cylinder
please help

 

[User849483]

I have tried using the y = ab/c uncertainty formula
for the (ro^2 + ri^2)/4(ro^2) but not sure where to go from there
Im unsure how to combine uncertainties

 

[User849483]

Forgot to mention, but I am attempting to find the velocity at which a hollow cylinder with a particular inner radius will roll down an incline plane

 

[PeroK]

I have tried using the y = ab/c uncertainty formula
for the (ro^2 + ri^2)/4(ro^2) but not sure where to go from there
Im unsure how to combine uncertainties

If $r_0$ is at its maximum possible value, compared to its minimum possible value, what does that do to the calculated velocity?

 

[User849483]

If $r_0$ is at its maximum possible value, compared to its minimum possible value, what does that do to the calculated velocity?

the velocity decreased as the inner radius increased
the outer radius stayed constant throughout the experiment

 

[PeroK]

the velocity decreased as the inner radius increased
the outer radius stayed constant throughout the experiment

Can you calculate the max and min values for $v$ based on the uncertainties in the other measurements?

 

[User849483]

Can you calculate the max and min values for $v$ based on the uncertainties in the other measurements?

i do have the v values yes

 

[kuruman]

The formula itself does not have an uncertainty. It is exact. If you measured $v$ for different values of the inner radius, then there is uncertainty in the measurements of the two radii and the velocity. So are you asking to find the uncertainty in the velocity given the uncertainties in the radii and the formula?

 

[User849483]

The formula itself does not have an uncertainty. It is exact. If you measured $v$ for different values of the inner radius, then there is uncertainty in the measurements of the two radii and the velocity. So are you asking to find the uncertainty in the velocity given the uncertainties in the radii and the formula?

yes i meant the uncertainty in the radii and formula

 

[kuruman]

yes i meant the uncertainty in the radii and formula

That doesn't help much. Can you tell us what task you were given as was given to you? It would also help if you briefly describe the experiment that you did and how you did it. Experimental uncertainties depend on what you did and how.

 

[User849483]

That doesn't help much. Can you tell us what task you were given as was given to you? It would also help if you briefly describe the experiment that you did and how you did it. Experimental uncertainties depend on what you did and how.

I had to find whether the inner radius of a cylinder affects its velocity when it rolls down an inclined plane. I timed each trial and I found there was a negative correlation. The formula I provided is a derived equation. I found the velocity for each inner radii value based on this formula, and now I have to find the uncertainty

 

[haruspex]

When addition and subtraction are involved, the formulae for combining fractional uncertainties don't work. E.g. if x has an uncertainty of 3%, what is the uncertainty in 1+x? If x is very large it is still 3%, if very small in magnitude then much less, but if negative it could be huge.

One way is to plug in the extreme value of the variables, but in principle you would need to try all the combinations. For n independent variables, $2^n$ combinations. Even then it is not guaranteed because of nonlinearity.

If you can, arrange the formula such that there is only one instance of each variable. Then, for each variable, try to figure out whether its maximum value corresponds to the maximum value of the result or the minimum.

 

[User849483]

When addition and subtraction are involved, the formulae for combining fractional uncertainties don't work. E.g. if x has an uncertainty of 3%, what is the uncertainty in 1+x? If x is very large it is still 3%, if very small in magnitude then much less, but if negative it could be huge.

One way is to plug in the extreme value of the variables, but in principle you would need to try all the combinations. For n independent variables, $2^n$ combinations. Even then it is not guaranteed because of nonlinearity.

If you can, arrange the formula such that there is only one instance of each variable. Then, for each variable, try to figure out whether its maximum value corresponds to the maximum value of the result or the minimum.

I dont believe that the formula allows for there to be only 1 instance of each variable
I have tried and it did not work

 

[haruspex]

I dont believe that the formula allows for there to be only 1 instance of each variable
I have tried and it did not work

Can you arrange $\frac{x+y}x$ so that there is one instance of each?

 

[User849483]

Can you arrange $\frac{x+y}x$ so that there is one instance of each?

i cant seem to figure it out, i've been trying but im not sure how to.
is there a way for me to arrange it with only 1 variable each?

 

[kuruman]

I had to find whether the inner radius of a cylinder affects its velocity when it rolls down an inclined plane. I timed each trial and I found there was a negative correlation. The formula I provided is a derived equation. I found the velocity for each inner radii value based on this formula, and now I have to find the uncertainty

So you have pairs of velocities and inner radii at fixed $h$ and $r_o$? If so, did you make a plot of velocity vs. inner radius? You must have otherwise you wouldn't know that there is "a negative correlation." For each of the measurements of $v$ you can use the formula to calculate a maximum and minimum value given your estimated uncertainty in $r_i$ and draw error bars on the graph. Then you can add a continuous theoretical solid line using the formula and see how close it comes to your measured values.

 

[haruspex]

is there a way for me to arrange it with only 1 variable each?

Not sure what you mean by "only 1 variable each". What you want is that each variable occurs only once in the expression. For $\frac{x+y}x$ that really is trivial.

What I should have mentioned before is that the extreme value method gives what I call the Engineer's answer. Engineers care about tolerances, so want to know the worst case. Physicists generally want the statistical answer. This treats the input uncertainties as standard deviations of random variables and computes the standard deviation of the result.
You can do that in a spreadsheet. I'll show you how if it is of interest, but it will be a few hours before I can get back to this.

 

[User849483]

So you have pairs of velocities and inner radii at fixed $h$ and $r_o$? If so, did you make a plot of velocity vs. inner radius? You must have otherwise you wouldn't know that there is "a negative correlation." For each of the measurements of $v$ you can use the formula to calculate a maximum and minimum value given your estimated uncertainty in $r_i$ and draw error bars on the graph. Then you can add a continuous theoretical solid line using the formula and see how close it comes to your measured values.

what do you mean by a "continuous theoretical solid line using the formula"? As in a line of best fit?

this is the graph I currently have, which displays my experimental values with an uncertainty of plus/minus 0.1 m/s.
Sorry, I do not understand what you mean.

 

[kuruman]

Sorry, I do not understand what you mean.

This is what I mean.

You have the equation $$v=\sqrt{\frac{gh}{\frac{1}{2}+\frac{ r_o^2+r_i^2}{r_o^2} }}$$
You also have values for $h$, $g$ and $r_o$. Put them in the equation.
Use the equation in a spreadsheet (or something like it) to calculate $v$ for 30-40 values of $\cancel{r_o}~r_i$ between 0.004 m and 0.012 m.
Connect the dots with a straight line.

The line so obtained is your theoretical prediction on which the velocity data points that you measured should lie within experimental error.

 

[WWGD]

Maybe this is what you're looking for
https://en.m.wikipedia.org/wiki/Propagation_of_uncertainty
?

 

[haruspex]

Use the equation in a spreadsheet (or something like it) to calculate $v$ for 30-40 values of $r_o$ between 0.004 m and 0.012 m.

Did you mean values of $r_i$?

 

[kuruman]

Yes, I did. Good catch.

 

[haruspex]

Forgot to mention, but I am attempting to find the velocity at which a hollow cylinder with a particular inner radius will roll down an incline plane

Ok, I misunderstood your need. In post #1 I thought you wanted to find how uncertainties in the radius values combine to form uncertainty in v.

There are several common situations.
In one, you have a single set of inputs with an estimate for the error in each, and you want an estimate for the error in the output.
In another, you have multiple sets of inputs which are in principle the same, no estimate for the error in each, and you want an estimate for the error in the mean output.

You now seem to be saying that your case is this: you have multiple sets of inputs and outputs in which the output and one input vary, you possibly have a priori estimates for the error in each, and you want an estimate for the error in the slope relating the variable input to the output.

That does seem strange because it means your equation is really $$v=k\sqrt{\frac{gh}{\frac{1}{2}+\frac{ r_o^2+r_i^2}{r_o^2} }}$$
and you want to find k and assess the error in that value. That is strange because you know k=1. So now I don’t know what you are trying to do, but if that is right, read on:

With no a priori estimates for the input or output errors, you can use https://saturncloud.io/blog/how-to-calculate-slope-and-intercept-error-of-linear-regression/.
If you have a priori estimates for the input and output errors you should in principle be able to do better, but I have never found a general method that handles that.
At a guess, your greatest source of experimental error is in measuring the velocity.

 

[BvU]

i do have the v values yes

Can you tell us how you obtained these $v$ values? Cylinders usually are not equipped with speedometers

the velocity decreased as the inner radius increased
the outer radius stayed constant throughout the experiment

From your relevant equation (was it given or did you deduce it? -- how?) I gather you mean the instantaneous velocty at the bottom end of the ramp.
I would be surprised if your experiment involved drilling ever bigger holes in one single cylinder, so 'the outer radius stayed constant' might be an euphemism for 'we assumed $r_o$ was constant and didn't bother to measure it more than once' (and yes, I remember being told something similar, a long time ago )

We need to know a little bit more about how you conducted your experiment. How you kept constants constant and what exactly you measured.

Then there are two schools of thought for the analysis phase: only plot direct observations, or manipulate until you can expect a linear relationship. Often you do both (in particular the first during measurements and the second afterwards)

Your plot in #18 might be in the first category if there really were speedometers. But then you don't want to fit to a straight line. What you can see from the graph is that 0.1 m/s is probably too pessimistic for the error in $v$.

A straight line would be expected in a plot of $1/v^2$ as a function of $r_i^2$ (can you see why?).

Re error propagation: the wiki lemma is a bit intimidating. Can you understand the simplfication paragraph idea ?

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