Deriving Torque=(Force)(Distance)

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Discussion Overview

The discussion centers around the derivation of the equation for torque, τ = Fd, and the underlying principles of rotational motion. Participants explore various approaches to understanding torque, including its relationship to angular acceleration and the cross product of vectors.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant attempts to derive τ = Fd starting from the relationship between arclength and angular motion, but expresses confusion over the application of linear force equations to rotational dynamics.
  • Another participant suggests that torque is better understood as a definition involving the cross product of moment arm and force vectors rather than a derivation.
  • There is a clarification that the analogy to Newton's second law for rotational motion should involve torque equating to moment of inertia times rotational acceleration, not mass times rotational velocity.
  • Some participants challenge the correctness of the equations presented, particularly the relationship between linear and angular acceleration, indicating a potential misunderstanding in the derivation process.

Areas of Agreement / Disagreement

Participants express differing views on whether torque can be derived or should be defined through vector relationships. There is no consensus on the derivation process, and some participants highlight errors in the initial approach without agreeing on a definitive resolution.

Contextual Notes

Some limitations in the discussion include potential misunderstandings of the relationships between linear and angular quantities, as well as the need for clearer definitions of terms used in the derivation.

Cyrus Hafezparast
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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 realize 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)
 
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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.
 
Dr. Courtney said:
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.
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.
 
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.
 
Cyrus Hafezparast said:
v=ωr and differentiating again to a=r(dθ/dt)

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).
 
spamanon said:
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).
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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