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RMS error in volume of a cylinder

  1. Feb 8, 2013 #1
    1. The problem statement, all variables and given/known data
    The length and radius of a perfect cylinder are measured with an RMS error of 1%. The RMS error of the inferred volume of the cylinder is.. ?


    2. Relevant equations
    V=∏r2h
    Hence dV/V=2dr/r+dh/h

    3. The attempt at a solution
    I took dr/r and dh/h as 1%. So got the final answer as 2x1%+1%=3%
    But the question came with 4 options(one of which is correct):
    1.7%, 3.3%, 0.5% and 1%

    So my answer is wrong ! What am I overlooking ? Is there any special precautions to be taken about RMS error ?
     
  2. jcsd
  3. Feb 8, 2013 #2

    mfb

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    The uncertainties of the two measurements should be independent, so I would add those contributions in quadrature and not linear. But then the result is 2.2% :D.
    1.7% would be the answer if all three dimensions had been measured independently.
     
  4. Feb 8, 2013 #3
    Could you please explain that, or give some link where it has been explained ? I didn't quiet understand what you meant by that.. X-)
     
  5. Feb 8, 2013 #4

    haruspex

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    The two errors might add up, but also they might partly cancel. So the RMS error in the volume will be less than simply 3 x one dimension of error. Instead, you use a root-sum-squares way of adding them up. But mfb is right that the answer should not be 1.7 either. That would be the answer for volume of a rectangular block with independent 1% errors in each of the three dimensions: √(12+12+12) = √3.
    In the present case, there are only two measurements. An error of x% in the radius will produce an error of 2x% in the cross-sectional area, so we have √(12+22) = √5 ≈ 2.2, which is not in the list.
     
  6. Feb 9, 2013 #5
    Ok. So √3 which matches with one of the given options doesn't really make much sense.

    Thanks a lot for the help guys. What was more important to me was how to find the RMS error(of any measurement, not just the volume of a cylinder) rather than just matching answers with options. I guess I learnt that. :-)
     
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