Is centripetal acceleration independent of linear acceleration in accelerated circular motion? Can we say that there is a relationship between them, or are they independent of each other? why?
Like does $a_c=v^2/r$ imply $a_c$ and $a_{tangential}$ are related?
I am very confused by this statement taken from a book.

 A: They are somewhat related. Remember that $a_t=\frac{dv}{dt}$, so that $\frac{da_c}{dt}=\frac{2v}{r}\frac{dv}{dt}$, that is, the change in centripetal acceleration is proportional to the tangential acceleration. You can also get eliminating $v$:
$a_c=(\frac{\dot{a_c}}{2a_t})^2r$
A: The acceleration vector is the vector sum of both components which are independent for sure, unless you impose a constraint (in this case the constraint is that the motion should happen in a circular trajectory with constant radius).
If you constraint motion to a circle of a fixed radius as the tangential velocity increases/decreases (due to a non-zero $a_{tangential}$) the value of $v$ will change in time, and that in turn will change the needed centripetal acceleration $a_c = v^2/r$ for mantaining the circular trajectory.
If, for example, we assume that $a_{tangential} = constant$ then you should expect a linear increase in tangential speed $v_{tangential} = \int a_{tangential} dt = a_{tangential}\cdot t$ as time, $t$, progresses. Since there is no radial speed ($v_{radial} = 0$) due to the fact that we want to mantain a circular trajectory with constant radius, $r = R = constant$ (or otherwise the object would radially get closer or farther from the center of the circunference), the total speed is
$v = \sqrt{v_{tangential}^2+v_{radial}^2} = |v_{tangential}| = |a_{tangential}\cdot t|$
Therefore we end up having 
$a_{c} = a_{tangential}^2\cdot t^2 / R$
Which means that given a constant tangential acceleration the speed will increase linearly and the centripetal acceleration needed to mantain the circular trajectory will increase geometrically.
A: Acceleration is a vector so we can resolve them into linear and circular acceleration. For example, think of swirling a bob on a thread whilst accelerating in a car or train, or plane for that matter.
