Now that we have a mathematical frame-work for describing rotational motion we proceed to ask what can cause rotation. To use fancy words we go from a kinematical theory to a dynamical theory of rotation. I emphasize that there is nothing new here except that we are writing Newton's second law in a form which is convenient for dealing with rotation.
Consider a particle with mass m which moves on a horizontal frictionless plane. The particle is attached to a rigid but massless rod which pivots about the origin.
By design this system can only perform circular motion. For
convenience we introduce a coordinate system which rotates with the particle and is oriented with the 
axis parallel to the
rod and the 
axis pointing in the direction of motion.
Assume that we manage to induce rotation then as we do this there will be angular acceleration and concomitantly a tangential acceleration
In order for this to be possible Newton's second law says that there
must be a component of force acting in the tangential
direction with magnitude
We seek expressions with rotational variables only so we introduce Eq. 1 to get
At this stage I dropped the vector notation because I want to keep it simple the first time around on this. Again we only want rotational variables so we multiply by r to be able to introduce the moment of inertia of the particle about the point of rotation:
In general it may be that the force applied is not in the tangential direction in which case we resolve it as follows
The tangential force component is
Thus we can write
The quantity on the right hand side is denoted torque and given the
fine greek symbol 
and thus we have Newtons second law in rotational form:
In words torque is the tangential component of force multiplied by
the distance from the axis of rotation to the point where the
force is applied.
The equation
also holds for a composite body characterized by moment of
inertia, I.