Uncertainties in fluid viscosity experimental trials

In summary: I think the most straightforward way to think about it is that there is some kind of correlation between the two, but it's not perfect. In your case, the squares of the two errors would sum to something larger than the sum of the squares of the individual errors. So the average would be influenced by the error in h, but also by the error in t.
  • #1
physicsdude123
2
0

Homework Statement


You are going to examine the viscosity of a fluid by dropping a ball bearing into a
cylinder of fluid and measuring the fall time. You will use a ruler to measure the
height, h, of the fluid column, and a stop watch to measure the fall time, t. The ruler
and stop watch have reading uncertainties of 0.5 cm and 0.05 s respectively. There
are four members in your group and you each do one measurement of each
quantity. Here are your results:
h1 = 36.0 cm t1 = 15.55 s
h2 = 36.0 cm t2 = 15.25 s
h3 = 36.0 cm t3 = 15.75 s
h4 = 36.0 cm t4 = 15.95 s
You would report your average value of h (in cm) and t (in seconds) as


Homework Equations



Don't know of any?

The Attempt at a Solution



I know to average the quantities, so for h=36.0 and t=15.63, but what I am unsure about
is how to do the uncertainties. Does the uncertainties stay the same as the measurement tool? So would they both be +/-0.5 or am I missing something?

Thanks!
 
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  • #2
You have 4 mesurements of h and four of t so the average of each of those two measurands would have their respective measurement errors reduced by a factor of 1/√4.

Since you will be determining avg. velocity = avg. h/avg. t you now need to combine the error on avg. h and the error on avg. t. What is the rule for error propagation for multiplication/division?
 
  • #3
You should treat the two kinds of measurement differently.
The height is fixed; the only uncertainty of it is in the measurement. For each measurer, the error range is ±0.5 cm. Even if they all work precisely, they will all generate the same error. The error in the ruler is not some random perturbation, it's just the limit of precision. Therefore the error in 36cm is still ±0.5 cm.
The time really does vary. You don't know the distribution for that, but you could take it to be normal. The error induced by the stopwatch, however, is uniform over the range ±0.05 s. You could approximate that with a normal distribution of the same variance. So the resulting time data come from the sum of two normal distributions, one with known mean and variance and the other with unknown parameters.
 
  • #4
haruspex said:
You should treat the two kinds of measurement differently.
The height is fixed; the only uncertainty of it is in the measurement. For each measurer, the error range is ±0.5 cm. Even if they all work precisely, they will all generate the same error. The error in the ruler is not some random perturbation, it's just the limit of precision. Therefore the error in 36cm is still ±0.5 cm.
.

Not so. h will vary randomly from reader to reader. I am neglecting the systematic error you're referring to, which implies an error in the ruler itself. I think the intent of this problem is to see how random errors propagate.
 
  • #5
rude man said:
Not so. h will vary randomly from reader to reader. I am neglecting the systematic error you're referring to, which implies an error in the ruler itself. I think the intent of this problem is to see how random errors propagate.
I admit the question could be more clearly worded, but I maintain that my interpretation is the more reasonable. I'm not saying there's an error in the ruler, only that the precision of reading is to the nearest 1cm. It's inevitable that that's how instrument readings work.
Under your interpretation, all the height readings being 36.0 would be somewhat unbelievable.
My guess is that recognising that the two sets of data should be treated differently is the whole point of the question.

The timings are more problematic. There's a bit of a mismatch between, on the one hand, the error being quoted as up to .05 s and, on the other, readings like 15.55s. With the obvious interpretation of the error range being "to the nearest .1 s", the readings should only have one decimal place. I think it possible the question is supposed to say the timing error is up to .025s.

The general statistical problem this raises is a very interesting one. We have a continuous r.v. X, and Y = [X], where the square brackets denote nearest integer. X and Y-X are clearly not independent. On the assumption that X is normal, I've been trying to derive the expectation and variance of Y, but it's messy.
 

1. What is fluid viscosity?

Fluid viscosity is a measure of a fluid's resistance to flow. It is a property of a fluid that describes its internal friction and is affected by factors such as temperature, pressure, and composition.

2. Why is it important to understand uncertainties in fluid viscosity experimental trials?

Understanding uncertainties in fluid viscosity experimental trials is important because it allows scientists to accurately measure and compare the viscosity of different fluids. It also helps to identify any sources of error in the experimental setup and ensure the reliability of the results.

3. How are uncertainties in fluid viscosity determined?

Uncertainties in fluid viscosity are determined through a process called error analysis. This involves identifying and quantifying all potential sources of error in the experimental setup, such as equipment limitations and human error, and then calculating the overall uncertainty using statistical methods.

4. Can uncertainties in fluid viscosity be reduced?

While it is not possible to completely eliminate uncertainties in fluid viscosity, they can be reduced by using more precise equipment, conducting multiple trials, and carefully controlling experimental variables. It is also important to properly document and analyze any sources of error to improve future experiments.

5. How can uncertainties in fluid viscosity affect the interpretation of experimental results?

Uncertainties in fluid viscosity can affect the interpretation of experimental results by introducing a margin of error. This means that the measured viscosity may differ from the true value by a certain amount, which can impact the accuracy and reliability of the results. Therefore, it is important to consider uncertainties when drawing conclusions from experimental data.

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