Rope swinging with periodic radius changes

[David Carroll]

If one simply swings a rope with an object tied to the end of it, the object describes a circle. But if one were to create a contraption that caused the radius of the rope to periodically decrease 4 times every revolution, one could cause the path of the object to describe a square. My question is, why is the object's acceleration not infinite once it makes the 90 degree angle?

 

[mfb]

For a perfect square and with a finite velocity of the object, acceleration "is" infinite at the corners. And you would need an infinite force.

 

[David Carroll]

So the above experiment would be impossible?

 

[jbriggs444]

For a centrally-directed force such as a rope, yes it would be impossible to make the corners of the path perfectly square. The loophole that mfb left open ("finite velocity") could allow for a finite force if the velocity at the corners were zero. But with a centrally-directed force, the resulting trajectory cannot be a square of non-zero size. It would have to be a line headed straight toward the center.

The remaining loophole, a square of zero size, is probably not what you had in mind.

 

[David Carroll]

If the period of radius contraction were perfectly continuous, then the velocity at the corners couldn't be zero, could it? In other words, the centrally-directed force is a reel which reels in the rope slightly 4 times per revolution in such a way to create a trajectory of a square for a ball tied to the other end of the rope. And if the reel were connected to a perfectly timed motor, then the reeling would be smooth and continuous. But if that's the case, how could the velocity of the ball be zero at the corners? Wouldn't a motorized reel using a constant force result in constant velocity for the ball at the end of the rope?

Oh, wait a minute...when the motorized reel has extended the rope to its upper limit, the reel itself has reached zero velocity because it cannot go from reeling out to reeling in zero time. Otherwise the reel itself would have infinite acceleration. I just answered my own question!

 

[mfb]

If your force is central, then angular momentum is conserved and not zero, so the velocity can never get zero.

 

[David Carroll]

The velocity of the reel would be zero. The reel has an axis that is perpendicular to another contraption that is spinning that reel. That contraption would have a constant angular momentum. But the axis of the reel itself, qua reel, would change momentum once it ceased to be reeling out and started reeling in.