How can centripetal acceleration change the speed? In the circular motion of a particle, it has two acceleration components, first, the tangential one which is responsible for increment (positive or negative) in speed and is $$ \frac{d \omega}{dt} \times r$$ and the other is centripetal acceleration and is responsible for the change in direction and is $$\omega \times \frac{dr}{dt}$$. Now imagine that you are rotating a stone with a string tied to it and then you slowly start to increase the centripetal acceleration which means that you start pulling the string with your other hand then according to the equation of circular motion kinematics, the magnitude of the velocity shouldn't be affected but now according to a new law that I studied which is the conservation of angular momentum, the object must increase its magnitude of velocity in order to conserve angular momentum and I don't get that, aren't the laws contradicting themselves and how does increase the central force even results in an increment of speed ?
 A: When you say

according to a new law that I studied which is the conservation of angular momentum, the object must increase its magnitude of velocity in order to conserve angular momentum and I don't get that

Remember: What is angular momentom? Well its really just like linear momentum, right?
So if you have your rock out in outer space rotating around a pole via a string: What happens to the speed of the rock if you start to lengthen the string some?
Well, its going to slow down a bit, right? But how can that be? There is no tangential force pushing it backwards to slow it down, right? Actually wrong. There is a tangential force. Not because you added a rocket booster to the rock, but because while the string is extending, the rock is no longer going in a circle, and the string is pulling on it with a tangential component
How is that?
Consider this corner case to help make it more clear: Suppose  instead of gradually lengthening the string, you suddenly add 1 foot in length to the string instantaneously (string has no weight and does not stretch). Well, the rock is gonna go straight for a while until it hits the new string length again and start going in a circle
But when the string length is hit, there is a triangle between the straight path line segment of the rock (side a), the string when it first let go of the rock (side b) and the longer string when it catches the rock (side c, or the hypotaneuse of the right triangle)
So side c is the direction of the force that catches the rock again. Now, since the angle between side c and side a is not 90 degrees, there is a component of force pull back against the movement of the rock, which slows it down
Now try instead of lengthening by 1 foot, lengthen by 0.1 foot ten times in one second. Or instead of  that, lengthen by 0.01 foot a hundred times again in one second, and so forth. Keep making it smaller lengths but more number of times until when you get to lengthen by 1/inf foot...infinity times you get continuous lengthening motion and that shows you how it happens in the continuous manner
When you shorten the string, the opposite happens: The string pulls in the direction of the motion of the rock and speeds it up
So turns out that when you double the length of the rope, the speed gets cut in have. And if you cut the length of the rope in half the speed doubles
You can work out the math yourself and verify this is true. But in the simple case, suppose you double the length of the string instantaneously. Well then the triangle side b divided by side c is 0.5 right? Well, thats the sine of 30 deg
Yeah if physics is something you want to really have fun with it is best to really try to understand why all these formulas and rules are true.
