Is linear monemtum conserved with in angular momentum?

I am trying to understand conservation linear momentum with in angular momentum

Herewith i just explain what i am looking for. Imagine an object of mass(M), velocity(V), is undergoing circular motion
with radius(R) Angular momentum L = RMV (R-radius M-mass- V- tangential velocity)

Mass is constant here

Since L is conserved so when R is decreased tangential velocity(V) has to be increased to keep angular momentum constant.

Does this mean linear momentum not conserved when radius of rotation changes?

When linear momentum is not conserved then what is the force that increases the velocity of object when R decreases.

Could anyone clarify this. or could any one say is my basic understanding about momentum is wrong. for easy understanding i am only considered magnitude of momentum's not cross product of vectors

Answer

Two ways of answering your question.

Consider the mass as the system which is acted on by a gravitational force - a central force.

The central force does not apply a torque on the mass and this means that the angular momentum of the mass is constant.
However even if the radius stays the same the linear momentum of the mass changes (in direction) because there is a net force on the mass.
However the speed and the kinetic energy of the mass does not change because the displacement of the mass is at right angle to the line of action of the force and so the force does no work on the mass.

If the radius of the orbit decreases then to conserve angular momentum the mass starts to move faster and so there is an increase in the kinetic energy and the magnitude of the linear momentum of the mass.
This comes about because as the radius of the orbit is decreased work is done on the mass by the central force because there is a displacement of the mass along the line of action of that force.
That work done increases the kinetic energy of the mass and the magnitude of the linear momentum of the mass.

Now consider as the system your mass and another mass which is providing the central force on your mass.
N3L tells you that there must also be a force on the other mass due to your mass which is equal and opposite.

Again angular momentum is conserved because there are no external torques acting on the two masses but this time the angular momentum is carried by both masses because they rotate about their common centre of mass.
The linear momentum of the two masses also does not change because there are no external force acting on the two masses.

Any change in linear momentum of one mass is compensated for by an equal and opposite change in the linear momentum of the other mass.
The equal and opposite forces on the two masses act for exactly the same tine on the two masses.
If the separation of the two masses decreases then both masses have more kinetic energy that energy coming from the decrease in the gravitational potential energy of the two masses.

It is often the case that the mass of one object is much greater than the other and the assumption is made that the orbit of the less massive object is centre at the centre of mass of the more massive object. This can simplify calculations without much loss os accuracy.