When a particle is performing uniform circular motion attached to a string about a fixed centre, at any instant of time its acceleration is directed towards the centre but the centre has no acceleration. But I was taught in school this is not possible because of the string constraint:
The accelerations of the ends of a string are the same if the string is not slack.
Where am I wrong?
Best Answer
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!