Uncertainty in area of a circle

  1. 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
     
  2. jcsd
  3. rock.freak667

    rock.freak667 6,221
    Homework Helper

    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.
     
  4. 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.
     
  5. rock.freak667

    rock.freak667 6,221
    Homework Helper

    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.
     
  6. 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?
     
  7. rock.freak667

    rock.freak667 6,221
    Homework Helper

    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.

    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
     
  8. 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
     
  9. rock.freak667

    rock.freak667 6,221
    Homework Helper

    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
     
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