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I Physical significance of integral of F cross dr

  1. May 1, 2016 #1
    In the vector calculus course, I calculated integrals like,
    ##\int \vec F \times \vec{dr} ##
    Does this kind of integrals have physical significance or practical application other than Biot-Savart's Law?
     
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  3. May 1, 2016 #2

    Andrew Mason

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    In Newtonian mechanics the integral represents torque or rate of change of angular momentum.

    The cross-product as a mathematical tool was invented probably because of its practical application. Although its development as a mathematical tool was post-Newton, it is very useful in mechancs.

    AM
     
  4. May 1, 2016 #3
    Isn't torque defined as ##\vec r \times \vec F## ?
     
  5. May 1, 2016 #4

    PeroK

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    ##\vec r \times \vec F = - \vec F \times \vec r##
     
  6. May 1, 2016 #5

    Andrew Mason

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    Yes. That is just a convention. The difference is the sign or direction of the torque vector.

    AM
     
  7. May 1, 2016 #6
    That was not my point. ##d \vec r## represents infinitesimal change in position vector, while ##\vec r## represents position vector. Could you please give me a practical example where the net torque is calculated by ##\int \vec F \times d \vec r## ?
     
  8. May 1, 2016 #7

    PeroK

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    The line integral doesn't represent the torque on a body. The total torque on a body would be a volume integral:

    ##\vec{\tau} = -\int \vec F \times \vec r dV##

    I'm not sure when you would use the line integral.
     
  9. May 2, 2016 #8

    Andrew Mason

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    I see your point. PeroK is quite right that ##\int \vec F \times d \vec r## does not represent torque. I am not sure what it would represent. I also don't see how it applies even to the Biot-Savart Law.

    AM
     
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