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Deriving Torque=(Force)(Distance)

  1. Jan 24, 2016 #1
    So there was a really old thread about this, but I don't think the matter was ever really resolved, which is why I'm making this thread now.

    I'm trying to derive the equation τ = Fd but I've run into a bit of trouble. I started with x=θr where x is the arclength on a circle (since any point on a rotating rigid body is going to follow a circular path) and then, from that, v=ωr and differentiating again to a=r(dθ/dt) . Now multiple by m on both sides and you have
    ma=rm(dθ/dt)
    ∴F=r τ [Perhaps this is the source of my error, but I'm taking f=ma as general and applying it to mass times angular acceleration to give angular force (Torque)]
    Which is not what we wanted at all!

    (PS I'm new to the forum, I made an account just for this question, so I realise that my formatting needs work, I couldn't easily see how to write the derivative nicely as a fraction like I've seen in other threads, I couldn't decide whether to write my working line by line or not etc etc, please be nice XD I also didn't know which prefix to use, but where I live there's no calculus in our high school physics course and most people in my classes aren't really questioning to this extent, so I thought I'd put Intermediate)
     
  2. jcsd
  3. Jan 24, 2016 #2
    I tend to think about the cross product of the moment arm and force vectors as the definition of torque rather than something that can be derived.
     
  4. Jan 24, 2016 #3
    Right, I see that and its the answer that almost every source I've seen gives, but it seems like there should be some justification? Also, if you could point out where I went wrong that would put my mind at rest as well.
     
  5. Jan 24, 2016 #4
    The analogy to Newton's 2nd for rotational motion is

    Torque = I (rotational acceleration)

    NOT

    Torque = m (rotational velocity)

    which is what your "derivation" seems to suggest.
     
  6. Jan 24, 2016 #5
    a=r(dθ/dt) is not the time derivative of v=ωr. Both of the right hand sides are equal to each other, but the left hand sides are not.

    Remember: ω simply is (dθ/dt).
     
  7. Jan 25, 2016 #6
    a=r(dθ/dt) These equation is not correct one .If it was written as a second derivative it becomes true it seems for me it is a typing error.
     
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