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Combining uncertainties - percentage and absolute. Brief summary.

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Reply 20
Original post by Stonebridge
To find the absolute uncertainty in ½at2
1. find the % uncertainty as I explained.
2. Then multiply the actual value you have calculated for ½at2 by (the % uncertainty / 100). This is how you find actual uncertainty from % uncertainty for any value.

Normally % uncertainties are expressed to the nearest %. Never 4.8% for example. Call it 5%


Thanks stonebridge!
Original post by Stonebridge
No use rule 2.
you are multiplying the value by a constant.


If I'm using the formula 43πr3 \frac {4}{3}\pi r^3 how would I multiply an uncertainty of r±0.0000025m?
Original post by Mutleybm1996
If I'm using the formula 43πr3 \frac {4}{3}\pi r^3 how would I multiply an uncertainty of r±0.0000025m?


Rule 4 in my first post.
3 x %uncertainty in r
(edited 9 years ago)
If you have a function of many variables you have to derive the expression and find the partial derivatives but that's something you normally do at University, not in edexcel, As or A levels.
Combining uncertainties seems to be quite a common source of problems in the new spec A Level. I've made a video explaining the rules and going through an example here: https://www.youtube.com/watch?v=JCdukQADyc8
image.pngI don't know how to calculate this answer , can someone please help

1.

Hey guys. I need a help for something thats keeping me sleepless since days. What are the accuracy and precision values of these instruments/apparatus?
(a)micrometer screw gauge(b)vernier calliper(c)meter rule(d)ammeter(e)voltmeter(f)ohm meter(g)thermometer(h)digital mass balance/weighing machine

In MSC they mention that they expect us to write about these apparatus's accuracy and precision values. But i dont know exactly how and what is that for each of these. Please somebody help exams are on 11th of may :l

so if i had a question like - (4Pi^2)/(0.034+-0.004 cm) would I divide 4Pi^2 by the uncertainty +-0.004 to get the answer?
This the first time ever that I say forum thread that answers itself, but you are right to do so because judging by the number of questions you received people are still confused about this.

I may also provide some help
First of all
In general, the uncertainty in a single measurement from a single instrument is half the least count of the instrument.

percentage uncertainty = uncertainty in weight / value for weight * 100%

I believe that more information you could find at this article and also try out this blog
(edited 6 years ago)
Thank you so much! This was extremely useful and cleared up things my teacher didn't really explain. It helped a lot. Thanks!
How to find percentage uncertainity for y=w1÷(w1-w2)? If percentage uncertainity for w1 is 1.5% and w2 is 1.5%.
Reply 31
Note that you don't always need to consider the uncertainty for a constant:

From the ideal gas law:

R a constant known with much precision, so we do not even consider itscontribution to uncertainty.


Also, Stonebridge, I think you yourself gave the best answer here (if this is you!):


You would normally ignore it, as the value of the constant found in tables would have a very small uncertainty. Any uncertainty in the constant would need to be of the same order of magnitude as your experimental uncertainty for it to be necessary to include it. In your example, if it was water you were using, the value of c can be found from tables to be 4186 J/kg. This is presumably accurate to over one part in 4000. Much more than your readings. If your experiment used data that was more accurate than this, you would need to get the value of c to even greater accuracy.



https://www.physicsforums.com/threads/when-working-out-the-uncertainties-what-to-do-with-the-constants.379160/
(edited 4 years ago)
Reply 32
Y

Original post by Stonebridge
In view of the fact that this question is being asked again and again on this forum, and to save me time posting the same answer again and again, this is a summary of how you combine uncertainties at A Level.*

You have two values, each with an absolute ± uncertainty.

1. If you add or subtract the two (or more) values to get a final value
The absolute uncertainty in the final value is the sum of the uncertainties.
eg.
5.0 ± 0.1 mm + 2.0 ± 0.1 mm = 7.0 ± 0.2 mm
5.0 ± 0.1 mm - 2.0 ± 0.1 mm = 3.0 ± 0.2 mm

2. If you multiply one value with absolute uncertainty by a constant
The absolute uncertainty is also multiplied by the same constant.
eg.
2 x (5.0 ± 0.1 mm ) = 10.0 ± 0.2 mm
The constant can be any number. eg Pi

3. If you multiply or divide two (or more) values, each with an uncertainty
You add the % uncertainties in the two values to get the % uncertainty in the final value.
eg
5.0 ± 0.1 mm x 2.0 ± 0.1 mm

This is
5.0 ± 2% x 2.0 ± 5%

Result
10.0 ± 7%

This is 10.0 ± 0.7 mm2
(0.7 is 7% of 10.0)

4. If you square a value
You multiply the % uncertainty by 2
If you cube a value you multiply the % uncertainty by 3
etc
If you need the square root of a value, you divide the % uncertainty by 2.
This is because square root in index form is to the power ½
√x = x½

The general rule is
Multiply the % uncertainty by the index.


What happens to % uncertainty when I multiply by a constant?

The % uncertainty doesn't change. The absolute uncertainty is multiplied by the constant. (see 2 above)
In the example given above we multiplied 5.0 ± 0.1 by a constant, 2
2 x (5.0 ± 0.1 mm ) = 10.0 ± 0.2 mm
The absolute uncertainty is multiplied by 2.
The original % uncertainty was 5.0 ± 2%
In the final value of 10.0 ± 0.2 mm
the % uncertainty is still 2%

Note: This is consistent with 3. above.
When you multiply a value by a constant, it is assumed the constant has no uncertainty. We do not associate an uncertainty with the value of Pi or the number 2, for example. So they have a % uncertainty of zero.
So when you multiply the value by the constant and add the % uncertainties, there is only the % uncertainty in the value itself and zero in the constant. Result: no change in % uncertainty.

What if the formula I use to calculate my final value has both adding and multiplication/division?

Let's take an example. Assume you have all the uncertainties in the values in the formula and we want the uncertainty in s

s = ut + ½at²

Step 1.
Add the % uncertainties in u and t to find the % uncertainty in ut
Step 2.
Multiply the % uncertainty in t by 2 (Rule 4 above) and add it to the % uncertainty in a to find the % uncertainty in ½at² (The constant ½ has no uncertainty)
Step 3.
Convert those % uncertainties to absolute uncertainties in ut and in ½at²
Step 4.
Add the absolute ± uncertainties in ut and ½at² found in 3. above to get the absolute uncertainty in the final value of s


*If you would like a more advanced treatment of this topic I recommend the following.
http://www.rit.edu/cos/uphysics/uncertainties/Uncertaintiespart1.html#systematic.

Wtf is this.I have had only up to GCSE maths education so this sounds like some next level stuff😲(mind throughly confused )
Reply 33
Original post by Laotsu
Y


Wtf is this.I have had only up to GCSE maths education so this sounds like some next level stuff😲(mind throughly confused )

Say for example you did an experiment and measured several variables, a,b,c,d. Lets look at two of those. You measured the circumference if a circle as a and the radius of the circle as b. You then used this information to estimate pi (lets say you didn't know if the formula for it is true). You know the formula is circumference = 2pi * radius. So you measured the circumference and the radius but both these values have uncertainty - its an experiment so you can't ever measure things absolutely precisely. So if they both have uncertainty, how much uncertainty is there in your measured value of pi? Well you can use these rules to combine those uncertainties to give you an answer.
Thank you
Reply 35
Original post by 0le
Say for example you did an experiment and measured several variables, a,b,c,d. Lets look at two of those. You measured the circumference if a circle as a and the radius of the circle as b. You then used this information to estimate pi (lets say you didn't know if the formula for it is true). You know the formula is circumference = 2pi * radius. So you measured the circumference and the radius but both these values have uncertainty - its an experiment so you can't ever measure things absolutely precisely. So if they both have uncertainty, how much uncertainty is there in your measured value of pi? Well you can use these rules to combine those uncertainties to give you an answer.

Makes sense
The really perverse form that uncertainties take in A Level physics should be proof enough that calculus and physics are not separable.
As a check, if your errors are reasonably small, then just finding the upper bound of the result and subtracting the mean should give you close enough to the answer.
eg if you have to multiply (10+/- 1) and (8 +/- 2)
(10 + 1)(8 + 2) - 10 x 8 = 30.
using the above rules gives you an error of 28. (interestingly, if you ignore the 1 x 2 term when you expand the product out you get the right answer. this is not a coincidence.) If you really don't trust yourself to find the error correctly then it might be a good idea to have a quantitative way to check if your answer at least makes sense.
It has been 7 years, and this is still helping A-Level Physics students like myself. Thank you sooooo soooo much StoneBridge!!!


Original post by Stonebridge
In view of the fact that this question is being asked again and again on this forum, and to save me time posting the same answer again and again, this is a summary of how you combine uncertainties at A Level.*

You have two values, each with an absolute ± uncertainty.

1. If you add or subtract the two (or more) values to get a final value
The absolute uncertainty in the final value is the sum of the uncertainties.
eg.
5.0 ± 0.1 mm + 2.0 ± 0.1 mm = 7.0 ± 0.2 mm
5.0 ± 0.1 mm - 2.0 ± 0.1 mm = 3.0 ± 0.2 mm

2. If you multiply one value with absolute uncertainty by a constant
The absolute uncertainty is also multiplied by the same constant.
eg.
2 x (5.0 ± 0.1 mm ) = 10.0 ± 0.2 mm
The constant can be any number. eg Pi

3. If you multiply or divide two (or more) values, each with an uncertainty
You add the % uncertainties in the two values to get the % uncertainty in the final value.
eg
5.0 ± 0.1 mm x 2.0 ± 0.1 mm

This is
5.0 ± 2% x 2.0 ± 5%

Result
10.0 ± 7%

This is 10.0 ± 0.7 mm2
(0.7 is 7% of 10.0)

4. If you square a value
You multiply the % uncertainty by 2
If you cube a value you multiply the % uncertainty by 3
etc
If you need the square root of a value, you divide the % uncertainty by 2.
This is because square root in index form is to the power ½
√x = x½

The general rule is
Multiply the % uncertainty by the index.


What happens to % uncertainty when I multiply by a constant?

The % uncertainty doesn't change. The absolute uncertainty is multiplied by the constant. (see 2 above)
In the example given above we multiplied 5.0 ± 0.1 by a constant, 2
2 x (5.0 ± 0.1 mm ) = 10.0 ± 0.2 mm
The absolute uncertainty is multiplied by 2.
The original % uncertainty was 5.0 ± 2%
In the final value of 10.0 ± 0.2 mm
the % uncertainty is still 2%

Note: This is consistent with 3. above.
When you multiply a value by a constant, it is assumed the constant has no uncertainty. We do not associate an uncertainty with the value of Pi or the number 2, for example. So they have a % uncertainty of zero.
So when you multiply the value by the constant and add the % uncertainties, there is only the % uncertainty in the value itself and zero in the constant. Result: no change in % uncertainty.

What if the formula I use to calculate my final value has both adding and multiplication/division?

Let's take an example. Assume you have all the uncertainties in the values in the formula and we want the uncertainty in s

s = ut + ½at²

Step 1.
Add the % uncertainties in u and t to find the % uncertainty in ut
Step 2.
Multiply the % uncertainty in t by 2 (Rule 4 above) and add it to the % uncertainty in a to find the % uncertainty in ½at² (The constant ½ has no uncertainty)
Step 3.
Convert those % uncertainties to absolute uncertainties in ut and in ½at²
Step 4.
Add the absolute ± uncertainties in ut and ½at² found in 3. above to get the absolute uncertainty in the final value of s


*If you would like a more advanced treatment of this topic I recommend the following.
http://www.rit.edu/cos/uphysics/uncertainties/Uncertaintiespart1.html#systematic.
Original post by TheSecretSayer
It has been 7 years, and this is still helping A-Level Physics students like myself. Thank you sooooo soooo much StoneBridge!!!

You are most welcome. :smile:
It's been 2 years since I been here. I do a level physics and i get this now.
Thank you

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