Is this arc length?

  1. Could someone please explain why PQ in the diagram below is rΔθ? Isn't rΔθ arc length?

    The best reason I can think of is that it's only an approximation for when the angle is very small, so PQ≈arclength=rΔθ. Not 100% sure though.

    [​IMG]

    The diagram is from the first volume of the Feynman lectures in 18-3, in the section where he talks about rotation of rigid bodies.
     
  2. jcsd
  3. mfb

    Staff: Mentor

    It is an approximation for small ##\Delta \theta##, right. I would expect that it is used in a differential somewhere, where the approximation gets exact.
     
  4. Thanks mfb
     
  5. rcgldr

    rcgldr 7,514
    Homework Helper

    Start with the actual distance:

    PQ = sqrt(Δr2 + (r sin(Δθ))2)

    If this is circular motion, then Δr = 0, and as Δθ -> 0, then sin(Δθ) -> Δθ, and you end up with lim Δθ -> 0 of sqrt((r sin(Δθ))2) -> sqrt((r Δθ)2) -> r Δθ.

    If r is some function of θ, then as long as Δr approaches zero more rapidly than r sin(Δθ), then lim Δθ -> 0 of f(Δr, r sin(Δθ)) -> f(0, r sin(Δθ)).
     
    Last edited: Mar 11, 2013
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