Question about torque as a cross product

[MarkWot.]

So yeah, I understand that you can calculate torque as F*d, and you get a "number".
But when you calculate a cross product of torque, r x F, what does that actually give you? It is a vector, perpendicular to F and r, but what "is" that? I mean, is it like an axis around which the object is rotating? And why does it have a magnitude, does it matter ever?

 

[SteamKing]

So yeah, I understand that you can calculate torque as F*d, and you get a "number".
But when you calculate a cross product of torque, r x F, what does that actually give you? It is a vector, perpendicular to F and r, but what "is" that? I mean, is it like an axis around which the object is rotating? And why does it have a magnitude, does it matter ever?

r × F gives you a moment, or torque, vector. Like all vectors, torque vectors have a magnitude and a direction. The axis around which the torque acts is located at the base of the r vector, perpendicular to the plane formed by r and F.

And yes, the magnitude and the direction do matter, especially when you are combining several torque vectors to find a resultant, just like combining several force vectors to find a resultant.

The cross product is a generalized way to keep track of all the Fd sin θ components of an arbitrary torque.

 

[BvU]

So yeah, MarkWot, welcome to PF !

Yes, torque is a vector (actually, it is a pseudo-vector or axial vector). Comes in handy when we draw a parallel with force, acceleration and such. (Unfortunately the link isn't explicit in the vector character...). A whole set of http://bama.ua.edu/~jharrell/PH105-S03/exercises/rot_mot_eqs.htm for translational motion has a one-to-one relation with a same set for rotational motion.

And yes, the magnitude is important: $$\vec \tau = {d\vec L\over dt} = I\,\vec\alpha$$ in the same way as the magnitude of a force is important ($\vec F = m\vec a$).

 

[BvU]

If you want some nitty-gritty, you can look at the wedge product or exterior product and at bivectors. (found that in this thread). Well, as you see, even first questions can shake loose a whole lot of in-depth answers...

"howdy" is Texan for how do you do ?

 

[MarkWot.]

Ahh it is still kinda difficult to understand... so, you are saying that the torque vector represents torque, but nothing is actually going on in that direction, and it is used when you have more torques and you need to calculate the resulting force?

p.s. I'm sorry for my bad english, I hope you understand what I am tying to say :)

@BvU - I don't know really, I just picked it up on tv hehe

 

[SteamKing]

Ahh it is still kinda difficult to understand... so, you are saying that the torque vector represents torque, but nothing is actually going on in that direction, and it is used when you have more torques and you need to calculate the resulting force?

p.s. I'm sorry for my bad english, I hope you understand what I am tying to say :)

@BvU - I don't know really, I just picked it up on tv hehe

A torque vector is no different from any other kind of vector. The magnitude of a torque vector has units of force × length. It can be decomposed into component torque vectors which act about the various coordinate axes. It can be combined vectorially with other torque vectors to produce a resultant torque vector.

One of the attractions of expressing a general torque calculation in vector form is that it reduces the amount of arithmetic calculation over using the definition of T = Fd sin θ

In effect, it simplifies the calculation of torque and reduces the chance of error creeping into the calculation.

 

[David Lewis]

I also had the same question. A vector is a geometric object (abstract idea) often used to represent a force (physical reality). Vectors correspond more intuitively to linear forces than they do to moments of forces. Vector properties got mapped onto moments so as to make mathematical manipulation easier, more concise and logically consistent. It wasn't designed to make it easier to visualize what's really going on.