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My problem is with linear momentum of a particle in circular motion. If we imagine a particle moving around a circle, if there are no torques acting, then we can say its angular momentum is conserved, i.e. $mvr=k$ (where $m$ is the mass, $v$ the velocity, $r$ the radius and $k$ a constant value). It means that if the radius decreases, then the velocity will increase and viceversa.
Since during all the process the force (which is centripetal) is only responsible for changing the direction of the velocity the particle (it makes it move around a circle) and is always perpendicular to it, the magnitude of linear momentum should not change. Then why does the velocity (or momentum) change with a change in the radius (by conservation of angular momentum)? Why is linear momentum not conserved in circular motion? Correct me if something of what I said is wrong. If you are able to explain this, please try to give an intuitive explanation.