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Suppose we have a small mass attached to a string that has been fed through a hole in the friction-less table on which the mass is rotating. Pulling the string downwards thus decreases the radius of the mass's circular motion.

In this scenario, several things seem to be true:

  1. The system's angular momentum is conserved.
  2. The kinetic energy of the mass increases as the radius of its circular motion decreases.
  3. Pulling on the string does work, which is presumably translated into the kinetic energy gained by the mass.

It would appear that the object is in circular motion at any given instant, which implies that the tension force provided by the string is always perpendicular to the velocity of the mass. But what force or force component is then responsible for increasing the tangential velocity of the mass as the radius decreases?

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marked as duplicate by sammy gerbil, John Rennie, Kyle Kanos, stafusa, Jon Custer Sep 19 '18 at 12:43

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It would appear that the object is in circular motion at any given instant, which implies that the tension force provided by the string is always perpendicular to the velocity of the mass.

Mostly true, but not quite. If the string is held steady, then the object is in circular motion. But by pulling the string harder than that, it moves off of circular by a tiny amount.

Since it starts at one distance and reaches a closer point, it cannot be moving in circles but in a spiral. So the path has a (small) radial component. That means the radial string and the not-exactly-tangential velocity vector are no longer exactly perpendicular.

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