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Error estimation using differentials

  1. May 15, 2014 #1
    1. The problem statement, all variables and given/known data

    A force of 500N is measured with a possible error of 1N. Its component in a direction 60° away from its line of action is required, where the angle is subject to an error of 0.5°. What (approximately) is the largest possible error in the component?

    2. Relevant equations



    3. The attempt at a solution

    The component force is [tex] F_x = F cos \theta [/tex]

    so [tex] lnF_x~=~lnF~+~lncos\theta[/tex]

    applying differentials: [tex]\frac{dF_x}{F_x} = \frac{dF}{F} + \frac{d~cos\theta}{cos\theta} (-sin\theta) [/tex][tex] =\frac{dF}{F} + \frac{sin^{2} \theta}{cos\theta}d\theta[/tex]

    plugging in values [tex] \frac{dF_x}{F_x} = \frac{1}{500} + \frac{3}{4} \frac{2}{1} \frac{1}{2} \frac{\pi}{180} = 0.002 + 0.013 = 0.015[/tex]
    so the error is [itex] (0.015)(500)cos(60) = 3.75N[/itex]

    The solution says 4.28N, however, which I confirmed by checking each error combination. Where am I going wrong here?

    Thanks in advance.
     
  2. jcsd
  3. May 15, 2014 #2

    vela

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    You differentiated incorrectly. You should have
    $$\frac{dF_x}{F_x} = \frac{dF}{F} + \frac{d\theta}{\cos\theta}(-\sin\theta).$$
     
  4. May 15, 2014 #3

    Gotcha, I figured it would be something simple like that! Thanks a lot!
     
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