Register to reply

Uncertainty in area of a circle

by zero13428
Tags: circle, uncertainty
Share this thread:
zero13428
#1
Oct1-10, 07:53 PM
P: 7
1. The problem statement, all variables and given/known data
The radius of a circle is measured to be 14.3+-0.3cm. Find the circle's area and the uncertainty in the area.

I don't understand how to correctly apply uncertainty equations with sigma and partial derivatives to these types of problems.

2. Relevant equations

A=(pi)(r^2)
(pi)(r^2)=642.4cm
Phys.Org News Partner Science news on Phys.org
World's largest solar boat on Greek prehistoric mission
Google searches hold key to future market crashes
Mineral magic? Common mineral capable of making and breaking bonds
rock.freak667
#2
Oct1-10, 08:34 PM
HW Helper
P: 6,206
Well then, we have A=πr2. If we take ln of both sides we will get

lnA=ln(πr2)=lnπ+2lnr

Now just take differentials

dA/A = 2*dr/r

dA is nothing but the error in A. Same with dr. Just substitute the numbers.

I really could not explain it properly without showing you the differentials.
zero13428
#3
Oct1-10, 08:47 PM
P: 7
You said to take the ln of both sides. As in the natural log? I didn't know these had anything to logs or am I reading something wrong.

rock.freak667
#4
Oct1-10, 10:20 PM
HW Helper
P: 6,206
Uncertainty in area of a circle

Quote Quote by zero13428 View Post
You said to take the ln of both sides. As in the natural log? I didn't know these had anything to logs or am I reading something wrong.
Well normally, to get the error, you would just add the relative errors. I showed you how to do it.

So if you had A=r3 then dA/A = 3*dr/r

It comes out the same if you just differentiate it normally.
zero13428
#5
Oct2-10, 10:47 AM
P: 7
I know at the beginning I asked how to use sigma and partial derivatives to solve this type of problem but I don't really know much about them yet. We haven't gotten to them in my math class. This problem is coming from an intro to physics lab course that focuses on propagation of error and uncertainty in measurements made and then using Excel functions like STDEV and (chi^2) to figure out stuff related to uncertainties.

Is there a standard formula to use if given a measurement or multiple measurements and the uncertainity in them?

"dA/A", is that supposed to be a partial derivative?
rock.freak667
#6
Oct2-10, 02:34 PM
HW Helper
P: 6,206
Quote Quote by zero13428 View Post
I know at the beginning I asked how to use sigma and partial derivatives to solve this type of problem but I don't really know much about them yet. We haven't gotten to them in my math class. This problem is coming from an intro to physics lab course that focuses on propagation of error and uncertainty in measurements made and then using Excel functions like STDEV and (chi^2) to figure out stuff related to uncertainties.

Is there a standard formula to use if given a measurement or multiple measurements and the uncertainity in them?
In that case, you can just find the areas with the radii given and then find the standard deviation, which would be how much the measurement deviates from the mean. Measuring its error.

Quote Quote by zero13428 View Post
"dA/A", is that supposed to be a partial derivative?
If you had like one value alone and you wanted to get the error,

dA would be the error in A.
A would be the actual measurement.

The relative error in A would then be dA/A
zero13428
#7
Oct2-10, 03:08 PM
P: 7
Actually I think I got it worked out. Let me know if this looks right.

A=(∏)(r)^2
∂(A)/∂(r) = 2(∏)(r)

sigma_A=√(((∂A/∂r)^2)(sigma_r)^2))

sigma_A=√(((2∏(14.3))^2)(0.3)^2))= 26.9cm

Area = 642.4cm
Uncertainty = 26.9cm
rock.freak667
#8
Oct2-10, 03:57 PM
HW Helper
P: 6,206
Quote Quote by zero13428 View Post
Actually I think I got it worked out. Let me know if this looks right.

A=(∏)(r)^2
∂(A)/∂(r) = 2(∏)(r)

sigma_A=√(((∂A/∂r)^2)(sigma_r)^2))

sigma_A=√(((2∏(14.3))^2)(0.3)^2))= 26.9cm

Area = 642.4cm
Uncertainty = 26.9cm
If you wanted to use the partial derivative ∂A, as the error in A, then it should read like this

∂A/∂r= 2πr or ∂A=2πr ∂r

Now if we divide both sides by A (which is πr2 as well)

∂A/A = 2πr/πr2 ∂r

or ∂A/A = 2∂r/r


Register to reply

Related Discussions
How do you find the approximate uncertainty of a circle of radius 3.8x10^4? Introductory Physics Homework 3
Derivative of the Area of a Circle Calculus & Beyond Homework 2
Area of the Circle and Probability General Math 4
Circle area Computing & Technology 24
Area of a circle General Math 2