If a stone is rotated with the help of a rope, then both centripetal force and centrifugal force will be liberated. The magnitude of that forces are same, but direction of them are opposite to each other.
Again, We know that, centripetal force works toward centre of circle. So why stone does not come back/reach to the centre of circle?
Suppose, for any time $t$ during rotating, stone is situated at point $A$ on the circumference of circle. By drawing a tangent on point $A$ we can show the direction of velocity of stone for that point. Again suppose that after very short interval of time stone reaches at point $B$, and by sketching another tangent on point $B$ we can show the direction of velocity of stone for that point. Will anybody tell me from which two velocity vector this resultant velocity vector for point $B$ will be obtained? If it is possible, please show that with image, or picture.
Best Answer
You have two questions.
Here is the answer to your first question, "So why stone does not come back/reach to the centre of circle?"
The answer is that the stone is accelerating toward the exact center of the circle. Velocity is composed of speed AND direction. As you noted, the velocity is always changing, however the speed is not. This is related to your second question.
"Will anybody tell me from which two velocity vector this resultant velocity vector for point B will be obtained?"
Assume point B is an extremely small distance along the circle away from point A. You will note, after you draw this, that the Vector B is ever so slightly angled toward the center of the circular path. The tip of Vector A touches the tail of the extremely tiny vector that joins Vector A to Vector B. In other words, Vector A plus the tiny Vector result in Vector B. Keep in mind we're talking about an infinitesimally small increment.
If you draw this on paper, you'll see that, theoretically, the very tiny vector points directly to the center of the circular path. That tiny vector represents the change in velocity or acceleration. The stone is accelerating toward the center of the circle. It's constantly moving toward the center of the circle, but it never reaches it. The acceleration $\frac{v^2}{r}$, can be derived geometrically from the radius vectors and the velocity vectors.
The stone never reaches the center of the circle because as it makes a move toward the center of the circle due to the slight vector rotation, it is also tending to move in a straight line tangentially, taking it away from the center of the circle. This constant move inward is opposed by a constant move outward resulting in no progress to the center of the circle. The stone constantly wants to move in a straight line away from the center, but the force in the string is pulling it inward an equal amount for each increment of time.