Deriving Torque=(Force)(Distance)

[Cyrus Hafezparast]

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)

 

[Dr. Courtney]

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.

 

[Cyrus Hafezparast]

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.

 

[Dr. Courtney]

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.

 

[mfig]

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).

 

[Alemayehu worku]

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.