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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.

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    $\begingroup$ 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. $\endgroup$ – CuriousOne Dec 31 '14 at 2:34
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That derivation of Feynman's is one of the best around. As you have worked out yourself, in principle you can think of everything in terms of force as long as you also know the position vector where the force acts. This is actually more information - often a great deal more - than you need to compute the dynamics and statics of a rigid body: you can slide a force vector along the straight line with its same direction and passing through its tail, and the effect of that force on a rigid body's dynamics is the same (although where the force acts is important for working out internal stresses on a body). The sum of forces acting at the centre of mass and the nett torque about the centre of mass is all the information you need to compute rigid body statics and dynamics. This is in general a great deal less information (three vectors: force, torque and position of centre of mass) than a specification of all the individual forces and their positions of action. So, if you like, this is an instance of data compression to make a description of dynamics more wieldy.

Another way of thinking of the split between force and torque is that they correspond to the natural, intuitive split between the Euclidean isometries of translation and rotation. Feynman's work calculation is splitting the work done by a system of forces into the resulting translational kinetic energy that results and the kinetic energy associated with rotation about the centre of mass.

Ultimately you will meet the notion of Lagrangian dynamics and Noether's Theorem. Feynman from memory touches on these notions in his famous lectures in a "Symmetry In Physics" chapter. From Noether's Theorem we understand that the conservation of various quantities arises because our description of physics does not change if we impart continuous transformations on our co-ordinate systems: because most physics does not change if we shift our time co-ordinate origin, we conclude through Noether's theorem that there is a conserved quantity which we call energy. Conservation of momentum arises because our physics is invariant with respect to translations of our co-ordinate systems (one component of momentum conservation for each component of co-ordinate translation) and conservation of angular momentum arises because our physics is invariant with respect to co-ordinate translations. So again we see a split of conserved quantities between those to do with translation (momentum) and those to do with rotation (angular momentum). The split thus arises very neatly and naturally from the notions of Eucledean isometries.

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Momentum (linear motion) and angular momentum (rotational motion) are conserved quantities of an isolated system. To change either you need an agent acting from the outside: to change the momentum you need force, to change angular momentum you need torque. These are independent actors. Even if the total force is zero, as in the case of a couple, you may have effective torque that can change the rotational state (angular momentum) while the linear momentum does not change.

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