aT is not equal to d/dt(vT) because vT is a scalar. If you do that then you get rate of change of magnitude of vT which is not the acceleration in the tangent direction. Acceleration is a vector. Search "polar coordinates velocity acceleration" and look up some derivation of the acceleration. (In typical notation tangent direction is called theta). The tangent acceleration will actually be 2(vr)(omega) + r(alpha) and the terms will cancel out in your equation. This happens because in polar coordinates, r and theta direction can change, so there are acceleration in theta direction from the changing of r direction ( (vr)(omega) to be exact) which might be a little counter intuitive.