What happens inside a body when it rotates? I'm studying rigid body dynamics lately. I came across the definition of torque, and though I've found a lot of explanations as to why there is an r there (the moment), all of them are mathematical (equating work and so on). None of them explained physically and I still couldn't figure out why the distance from axis of rotation increases the net effect, or torque. 
So I thought about this and came to this line of thought -
Rotation can be thought of as a rigid body have all the infinitesimal masses performing circular motion about a fixed axis. There is pure rotation, hence angular velocity is constant. Thus velocity which is omega times r increase with the distance from the rotation axis. So if force is applied at more distance, this implies more velocity of the point of the application, and since the body is rigid, all the other mass connected to the point of application goes along through inter-atomic interactions and hence more rotational effect. Is this line of thought correct?
So what happens inside a body when it rotates? Do the rest of the atoms go along due to electromagnetic attraction and if so, can someone explain exactly what happens inside the body when it rotates and where does that r come from from an inter-atomic point of view?
 A: Why would there be no radius in torque? A real torque is a real force that acts on a real rotating body (rigid or not) at a real radius. If you look at e.g. rotating machine parts, they all have a finite diameter. That diameter is of enormous importance for the design of a part, because together with the material constants it determines just how much torque can be translated trough that part. A thin axle will break much easier than a thick one, for instance. A thin axle will deflect far more than a thick one for the same amount of torque. The radius matters, big time!
Now, if a force moves an object a certain distance, what do you get? Work. That definition holds, whether that movement is linear, or not. So if a force moves an object around a perimeter (equal to $2\pi r$), then you still get a work $W=2\pi r F$ performed, right? Well, in practice we then proceed to hide that $2\pi$ in the definition of the angle and the angular velocity. 
Rigidity of bodies and point masses/mass distributions are just approximations to make life easier. None of that really exists. Don't waste too much time on getting used to them, because these are useless concepts outside of Newtonian mechanics and almost all of physics lives outside of Newtonian mechanics. 
A: The physical explanation for why torque increases with r is that the longer the lever arm is the greater angular acceleration you can cause for a given force F.  If a screw is stuck because it was screwed in too hard (ie with too much torque), you need to get a longer wrench.  With the longer wrench (ie, larger r_w) you can generate greater force at the edge of the screw r_s by the ratio r_w/r_s and thus start to accelerate the screw out of the hole.  This is also the concept of the fulcrum or a seasaw, (ie, if you sit farther back on a seasaw than the other person you can overcome their weight even if they are heavier than you.)  
"Assuming the lever does not dissipate or store energy, the power into the lever must equal the power out of the lever. As the lever rotates around the fulcrum, points farther from this pivot move faster than points closer to the pivot. Therefore a force applied to a point farther from the pivot must be less than the force located at a point closer in, because power is the product of force and velocity" [Uicker, John; Pennock, Gordon; Shigley, Joseph (2010). Theory of Machines and Mechanisms, 4th ed.. Oxford University Press, USA].
A: It's part and parcel of the definitions of angular momentum and torque.
Consider a system of particles, not necessarily a rigid body. Without loss of generality, we can use an inertial frame that is instantaneously co-located and co-moving with the system center of mass. The angular momentum of the system with respect to the origin of this frame is defined as $$ \mathbf L = \sum_\alpha \mathbf r_\alpha \times m_\alpha \dot {\mathbf r}_\alpha $$
Differentiating with respect to time yields
$$\frac{d\mathbf L}{dt} =
\sum_\alpha 
  \dot {\mathbf r}_\alpha \times m_\alpha \dot {\mathbf r}_\alpha +
  \mathbf r_\alpha \times m_\alpha \ddot {\mathbf r}_\alpha $$
The first term inside the sum, $\dot {\mathbf r}_\alpha \times m_\alpha \dot {\mathbf r}_\alpha$, is identically zero, leaving
$$\frac{d\mathbf L}{dt} =
\sum_\alpha 
  \mathbf r_\alpha \times m_\alpha \ddot {\mathbf r}_\alpha =
\sum_\alpha 
  \mathbf r_\alpha \times {\mathbf F}_{net,\alpha} $$
where $ m_\alpha \ddot {\mathbf r}_\alpha \equiv {\mathbf F}_{net,\alpha}$ by Newton's second law. This net force acting on the $\alpha^{th}$ includes internal forces from other particles in the system and external forces that come from outside the system. Splitting this net force into internal and external forces, we have
$$\frac{d\mathbf L}{dt} =
\sum_\alpha 
  \mathbf r_\alpha \times
  \bigl({\mathbf F}_{ext,\alpha} + \sum_{\beta\ne\alpha} \mathbf F_{\alpha,\beta}\bigr) $$
where $\mathbf F_{\alpha,\beta}$ denotes the force exerted by the $\beta^{th}$ particle and the $\alpha^{th}$ particle. If all internal forces follow the strong form of Newton's third law (forces between pairs of particles are equal but opposite, and act along the line connecting the particles) then $\sum_\alpha \sum_{\beta\ne\alpha} r_\alpha \times \mathbf F_{\alpha,\beta}$ vanishes. In other words, the strong form of Newton's third law says internal torques do not contribute to change in angular momentum. The end result is
$$\frac{d\mathbf L}{dt} =
\sum_\alpha 
  \mathbf r_\alpha \times \mathbf F_{ext,\alpha} $$
This is true for any system of particles that obeys Newton's laws of motion, not just a rigid body.
So why are angular momentum and torque defined this way? The answer is simple: These concepts keep appearing over and over again. Physicists tend to give those key concepts that keep appearing over and over again names and standard definitions. 
