Lets say that a particle connected to a string is rotating about an axis O with a uniform tangential velocity of u. Then, the string is pulled towards the axis to decrease its radius (say from r1 to r2). Now according to the Law of Conservation of Angular Momentum, the tangential velocity of the particle must increase since its mass remains constant.
But, as I understand it, the force responsible for changing the radius of motion of the particle is the tension of the string, which always acts perpendicular to the velocity of the particle, i.e, in the radial direction. So, how is it possible that this force is responsible for increasing the tangential velocity of the particle?
I thought that the motion (or the trajectory) of the particle, while its radius is changing, can be thought to be circular at any instant with radius r (where r1<r<r2) which changes by infinitesimal amounts at each instant starting from r1 and ending at r2. But if this were the case, then the tension cannot be responsible for changing the tangential velocity of the particle. So does that mean that any instant the trajectory of the particle is not perfectly circular?
