How can a particle in circular motion about a fixed point accelerate, if the point doesn't too?
I'm guessing you mean the string constraint that Tension must be equal in both directions at all points in the string except the endpoints, where the tension at the endpoints must be equal and opposite?
So for an object moving in circular motion around a fixed point attached to a string, you're right that the object is moving in a circle because of the tension from the rope giving centripetal force. I think your confusion is coming from, shouldn't the center point also feel a tension and thus accelerate?
So the answer comes from the definition of a "fixed point"! In real life this means nailing something to the ground, or gluing it down, or placing it between a rock and a hard place, etc. This means that the center will indeed feel tension, but it will also feel some resistive force (usually normal or frictional forces) that will keep it from accelerating.
If the center point was not "fixed", then the circular motion would immediately stop, the string would go immediately slack, and the problem would become much more complex.
Hope that answered your question!
To distill (previous answers are correct, but perhaps unnecessarily long if I understand your point of confusion):
For a rigid system moving without rotation, all points in the system move with the same acceleration.
In any other case (i.e. if there is any change in orientation of the system) this is no longer true.
accelerations of both the ends along the string is same if the string is not slacked
Now, I understand your problem.
If you have a string placed in the shape of s in vacuum and if you start pulling it from one end it finally becomes l i.e straight.Here you can say that string isn't slackened because acceleration of both ends in same.
In case of circular motion i.e particle rotating about a fixed centre the string provides the necessary force to keep the particle moving around the centre and this force is called as tension.
Now,since string isn't slackened, is the acceleration of both ends same?
I want to explain what's happening in terms of forces rather than acceleration:
Newton's third law of action and reaction states that if the string exerts an inward centripetal force on the particle, the particle will exert an equal but outward reaction upon the string,the reactive centrifugal force.
The string transmits the reactive centrifugal force from the particle to the centre, pulling upon the centre. Again according to Newton's third law, the centre exerts a reaction upon the string, pulling upon the string. The two forces upon the string are equal and opposite, exerting no net force upon the string (assuming that the string is massless), but placing the string under tension i.e no slackening of the string.
The reason the centre appears to be "immovable"(not accelerating) is because it is fixed. If the rotating ball was tethered to the mast of a boat, for example, the boat mast and ball would both experience rotation about a central point.