# Why do we need torque separately from force?

I'm studying physics for a couple of month now and and I am currently finding it a bit unsatisfying how the basic physical concepts are presented, meaning often times we only get a formula ($\tau=r \times F$, for example) without much discussion or any derivation. So I was trying to build up a bit of background knowledge and intuition from the Feynman lectures.

From what I understand he derived torque (firstly without vectors) simply by inserting angular coordinates into the displacement in the work formula and rearranged it:

\begin{equation} \Delta W=F_x\,\Delta x+F_y\,\Delta y. \end{equation}

angular coordinates ("angular displacement"?): \begin{equation} \Delta x=-PQ\sin\theta=-r\,\Delta\theta\cdot(y/r)=-y\,\Delta\theta. \end{equation} \begin{equation} \Delta y=+x\,\Delta\theta. \end{equation}

inserted:

\begin{equation} \Delta W=(xF_y-yF_x)\Delta\theta. \end{equation}

He then calls the part without the angle "torque".

So isn't the torque just a special kind of force, one that acts on a circular displacement? Why do we treat force and torque so seperately when torque just seems to emerge when we work with angular coordinates? Isn't this just a special case, why can't we not use just force all the time (and not separate force/momentum/... and torque/angular momentum/... so strictly)?

Obviously I'm thinking about it the wrong way and have some major misunderstandings regarding the concept of torque and thus angular momentum etc. Are there any "better"/other derivations of this concept? After weeks of frustration I signed up here, maybe you have a better way of getting some intuition with torque. Thanks.

• You could use force all the time, but then you would have to tell everyone what radius that force acts on when you are talking about radial movements and you would have to lug $2\pi r$ trough every formula describing such movement, just to see it drop out at the end of many calculations. – CuriousOne Dec 31 '14 at 2:34