Deriving Torque=(Force)(Distance)

1. Jan 24, 2016

Cyrus Hafezparast

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. Jan 24, 2016

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.

3. Jan 24, 2016

Cyrus Hafezparast

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.

4. Jan 24, 2016

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.

5. Jan 24, 2016

mfig

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).

6. Jan 25, 2016

Alemayehu worku

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.