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Why is this formula incorrect?

  1. Aug 15, 2016 #1
    1. The problem statement, all variables and given/known data
    The formula for calculating the area of a curve in a polar graph is $$\large \rm \frac{1}{2}\int r^2~ d\theta $$ and is adapted from
    $$\large \rm Area = \frac{1}{2}r^2\times \theta $$
    But the formula to calculate the arc length is very different from $$\large \rm Length =\int r ~d\theta $$ which should've been adapted from $$\large \rm Length=r\times \theta $$ Why is the formula $$\large \rm Length =\int r ~d\theta $$ incorrect to calculate the arc length of a sector in a polar graph?
     
  2. jcsd
  3. Aug 15, 2016 #2

    jbriggs444

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    The incremental tangential change in position is given by ##r\ d\theta## but there is also an incremental radial change in position given by ##dr = \frac{dr(\theta)}{d\theta}\ d\theta##

    The radial change in position has no impact on the incremental area element (it would be a second order effect). But it does have an impact on the incremental perimeter element.
     
  4. Aug 15, 2016 #3
    Can you please lay down your explanation in simpler words? (I'm a pre-university student)
     
  5. Aug 15, 2016 #4

    jbriggs444

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    Draw a graph in polar coordinates from point A to point B. Make sure that point A and point B are not the same distance from the center. Does the distance between A and B increase if you move point A closer or farther away from the origin? Does the product of r (the distance of B from the origin) and theta (the angle between A and B as measured from the origin) change as you do so?
     
  6. Aug 15, 2016 #5
    Farther
    Yes the angle changes. But why are we taking into account angle AOB? Should it be BOInitial?
     
  7. Aug 15, 2016 #6

    jbriggs444

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    You wanted ##d\ \theta##. That's the change in polar angle between starting point and ending point of the incremental segment whose length we are concerned with. That segment starts at A and ends at B (or vice versa).
     
  8. Aug 15, 2016 #7
    Oh so right now you're trying to figure out the distance from A to B. Well then, please continue.
     
  9. Aug 15, 2016 #8

    jbriggs444

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    Distance from A to B can be computed using the pythagorean theorem: The square root of the sum of the square of the radial separation plus the square of the tangential separation.
     
  10. Aug 15, 2016 #9
  11. Aug 15, 2016 #10

    jbriggs444

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    You have a lot of good responses on that site. There is little that I can add to what has been said there. The incremental segments in a path are not always at right angles to a line drawn from the origin. The length of such a segment will not, in general, be given by r times the angle it subtends.
     
  12. Aug 15, 2016 #11

    Ray Vickson

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    Look at a simple example. What is the distance from ##A: r = 2, \theta = 30^o## to ##C: r = 2.1, \theta = 31^0##? Your formula ##r d \theta## gives the distance from ##A## to ##B: r = 2, \theta = 31^o##, if you replalce a very slightly curved arc by a straight line segment. It is the base ##AB## of the right-triangle ##ABC##, but you need the hypotenuse ##AC##.
     
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