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Calculating uncertainty when subbing in to a formula?

The volume V of a cylinder of height h and radius r is given by the expression V = πr2h.
In a particular experiment, r is to be determined from measurements of V and h. The uncertainties in V and in h are :V ±7%, h ± 3%.
Why this is 5% not 10%?
Reply 1
Avatar for Pangol
Pangol
15
I think that the percentage uncertainty should be 10%. Who says that it is 5%? Is this a past paper we can look at?
Reply 2
Avatar for Louiskn
Louiskn
OP
Original post by Pangol
I think that the percentage uncertainty should be 10%. Who says that it is 5%? Is this a past paper we can look at?


Maybe so. It's from some online revision notes on the 7th page, it gives the answer below.

http://mrsmithsphysics.weebly.com/uploads/1/2/1/5/12150755/1.2_errors_and_uncertainties_notes.pdf
Original post by Louiskn
The volume V of a cylinder of height h and radius r is given by the expression V = πr2h.
In a particular experiment, r is to be determined from measurements of V and h. The uncertainties in V and in h are :V ±7%, h ± 3%.
Why this is 5% not 10%?


Hello there,

If we rearrange the equation, we find that . . .

[br][br]r=Vπh[br][br][br][br]r = \sqrt{\dfrac{V}{\pi h}}[br][br]

. . . or alternatively . . .

[br][br]r=(Vπh)12[br][br][br][br]r = \bigg(\dfrac{V}{\pi h}\bigg)^{\frac{1}{2}}[br][br]

This implies that the percentage uncertainty, p(Δr) p(\Delta r) , of the radius can be calculated by . . .

[br][br]p(Δr)=12[p(ΔV)+p(Δh)][br][br][br][br]p(\Delta r) = \frac{1}{2}[p(\Delta V) + p(\Delta h)][br][br]

. . . since the percentage uncertainty within the parentheses is multiplied by the exponent. Therefore . . .

[br][br]p(Δr)=12[(7%)+(3%)][br][br][br][br]p(\Delta r) = \frac{1}{2}[(7\%) + (3\%)][br][br]

[br][br]p(Δr)=5%[br][br][br][br]p(\Delta r) = 5\%[br][br]

. . . r r has a percentage uncertainty of 5% 5\% .

I hope that this has been helpful. :smile:
Reply 4
Avatar for Louiskn
Louiskn
OP
Original post by Smithenator5000
Hello there,

If we rearrange the equation, we find that . . .

[br][br]r=Vπh[br][br][br][br]r = \sqrt{\dfrac{V}{\pi h}}[br][br]

. . . or alternatively . . .

[br][br]r=(Vπh)12[br][br][br][br]r = \bigg(\dfrac{V}{\pi h}\bigg)^{\frac{1}{2}}[br][br]

This implies that the percentage uncertainty, p(Δr) p(\Delta r) , of the radius can be calculated by . . .

[br][br]p(Δr)=12[p(ΔV)+p(Δh)][br][br][br][br]p(\Delta r) = \frac{1}{2}[p(\Delta V) + p(\Delta h)][br][br]

. . . since the percentage uncertainty within the parentheses is multiplied by the exponent. Therefore . . .

[br][br]p(Δr)=12[(7%)+(3%)][br][br][br][br]p(\Delta r) = \frac{1}{2}[(7\%) + (3\%)][br][br]

[br][br]p(Δr)=5%[br][br][br][br]p(\Delta r) = 5\%[br][br]

. . . r r has a percentage uncertainty of 5% 5\% .

I hope that this has been helpful. :smile:


Perfect! Thanks!:smile:
Reply 5
Avatar for Pangol
Pangol
15
Ah - the lack of formatting in the question made me look right through the square (I thought it was just a 2). Nice explanation!
Original post by Louiskn
Perfect! Thanks!:smile:


You're very welcome. :smile:
Original post by Pangol
Ah - the lack of formatting in the question made me look right through the square (I thought it was just a 2). Nice explanation!


Thank you,

Yes, indeed- if the equation was as it was written in the question post, then you would have been correct. :smile:

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