Angular momentum is defined as $L = r \times p$ where r is the center of rotation and p is the momentum. Since angular momentum is conserved, if r decreases then p must increase. And since p is m*v the velocity must increase. But that would change the momentum while keeping the angular momentum constant. Aren't both angular and linear momentum supposed to be conserved unless acted upon by an external force?
Angular momentum is conserved. If you have ever been in an office rolly-chair, you might (admit it) have spun around in it. If you stick out your legs, you will actually slow down. If you tuck them in, you will speed up.
This is just a consequence of the equation $$L = r \times p$$
Since $L$ is conserved, as $r$ -> $0$, $p$ -> $\infty$, and as $r$ -> $0$, $p$ -> $\infty$
Therefore, these forms of momentum are still conserved. This is only why an ice skater spins faster when they tuck their feet and arms together, and they slow down when they stick a leg out.
Linear momentum is, as the name suggests, LINEAR hence in case of rotation you need to take care of the tangential motion and not the curved motion which is nicely explained by angular momentum. When the radius increases an external force is used to push the body away and that is the reason why p is not conserved. This is because the p depends on a single linear dimension that of tangent. But when you increase radius in the perpendicular distance the vector is laterally displaced hence the value of p is unchanged. But since the entire frame is non-inertial and you want to conserve angular momentum as the first preference the tangential motion has to reduce. Hence the vector is laterally displaced so that the direction is constant but the magnitude changes. So you are changing the frame of reference while thinking about the p and L.