Uncertainty in area of a circle

[zero13428]

Homework Statement

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.

Homework Equations

A=(pi)(r^2)
(pi)(r^2)=642.4cm

 

[rock.freak667]

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]

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]

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]

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]

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.

"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]

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]

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