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Why is the constant excluded?

  1. Feb 20, 2010 #1
    For a question like In an experiment, the external diameter D and internal diameter d of a metal tube were found to be (64 +/- 2) mm and (47 +/- 1) mm respectively. What is the maximum percentage error for the cross-sectional area of the metal tube?

    I will need to find the external area.
    So, area = pi(1/2d)^2
    But I'm confused over whether should the uncertainty change?
    What I mean was 1/2(64 +/- 2) = (32 +/- 1) ??
    Or will the uncertainty remain at 2? If it remains at 2, the percentage uncertainty will definitely change.

    And why does the formula R=kAB, k is a constant and A and B are physical terms, has the uncertainty formula tR / R = tA / A + tB / B
    where tR / R, tA / A, tB / B are fractional uncertainty.
    Why is the constant excluded?
    R = kAB = AB + AB + AB ... + AB
    So won't the uncertainty add up?

  2. jcsd
  3. Feb 20, 2010 #2
    Re: uncertainty

    Yes. X = 64 +/- 2 means we are sure that X is between 62 and 66, which is equivalent to saying X/2 is between 31 and 33, hence X = 32 +/- 1. Multiplication by a constant always preserves relative error.

    The absolute uncertainty adds up. The relative uncertainty is unchanged since
    [tex]\frac{\Delta R}{R} = \frac{k \Delta(AB)}{k AB} = \frac{\Delta(AB)}{AB}[/tex]
  4. Feb 20, 2010 #3
    Re: uncertainty

    Ok. Thanks for clearing up things.

    "Multiplication by a constant always preserves relative error"

    This is also due to the constant being canceled right?
  5. Feb 20, 2010 #4
    Re: uncertainty

    Yes, that's right.