About increase of angular velocity when radius of circumnavigating motion is decreased.
The diagram below represents the trajectory of a point mass that is circumnavigating a central point, in the process of being pulled closer to the central point.
The dark gray arrow points represents the centripetal force that the point mass is subject to.
If the centripetal force would be precisely the required centripetal force to sustain circumnavigating motion then the radius of circumnavigation would remain constant.
Here there is a surplus of centripetal force. The dotted line represent the trajectory through one quadrant, and this motion is not circular; the mass is being pulled inward, so the motion is more like an inward spiral.
If the motion would be circular the centripetal force would be perpendicular to the instantaneous velocity. But the motion isn't circular.
In the diagram the total centripetal force is decomposed in two force components: one component perpendicular to the instantaneous velocity, one component tangent to the instantaneous velocity. The tangent force component is in the process of increasing the instantaneous velocity. In other words: during the contraction the centripetal force is doing work
The amount of work that the centripetal force is doing is such that when the radial distance is halved the angular velocity is quadrupled.
Here is a comparison to underline that in order to understand change of velocity you have to identify the force that is doing work.
For illustration I will first discuss change of linear velocity, with a deliberately wrong "explanation".
When a cannon is fired the projectile flies out of the cannont and the cannon recoils. This happens because momentum must be conserved. The projectile flies out of the cannon because of the recoil of the cannon, and the cannon recoils because the projectile flies out of it.
Clearly the above is nonsensical.
As we all know: to explain the change of velocity we must identify the force that is involved. The explosion of the charge creates a hot, high pressure gas, the expansion of that gas is doing work, causing the projectile to shoot out of the barrel and it causes the recoil.
The same principle applies in the case of angular acceleration. When there is a surplus of centripetal force the system will contract, during that contraction the centripetal force is doing work, causing angular acceleration.
Conversely, if the centripetal force that is provided is not enough to sustain circular motion then the radial distance will increase. Then the centripetal force is doing negative work, decreasing the angular velocity.
The expression for angular momentum: $m r^2 \omega$
As we know, with $\omega$ for angular velocity the expression for the conserved angular momentum is $m r^2 \omega$
While linear momentum is a property of motion along a line, angular momentum is property of motion in a plane. It can be circular motion or spiralling motion, the point is that the minimum space you need in order to have an angular momentum at all is two spatial dimensions: a plane.
As we know, when we are dealing with two spatial dimensions we are dealing with area. There is a correlation between angular momentum and area.
The first derivation in the Newton's Principia is a derivation of Kepler's law of areas from first principles.
(I reproduce the text from wikipedia, because I am the author of that image, and I wrote that exposition)
During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times.
At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve.
The above geometric reasoning shows there is a conserved quantity. This conserved quantity is proportional to the area of the triangle that are swept out. This explains why the expression for angular momentum, $m r^2 \omega$, is a quadratic expression.