Actually trying to answer: “I set the string swinging like a pendulum” to me reads like the person pulls the ice cube back and then either lets go or gives it a little push. I expect it’s quite hard to do either of these while ensuring that the net momentum of the ice cube is exactly along a line that runs directly below the point at which the ice cube is attached to the branch. If it starts off with any momentum perpendicular to that line, you get an ellipse and not a line. As it loses energy and traverses a smaller ellipse it fills in the ellipse. If this happens quickly enough the final shape would be less of an ellipse than a splattering of drips in a vaguely elliptical pattern, with a strong concentration in the center. The cooler the day the more that happens, and possibly the day needs to be improbably hot before you get anything other than a few dots and a point?
Also, from the mechanical, historical perspective—a drop that landed at the dead center beneath the pendulum’s contact with the branch would have had to leave the cube in a brief moment of time before passing over the center, with exactly enough forward velocity at the moment it left the cube such that it would hit the center by the time it reached the ground (depends on how far up it’s hung)… which is a tiny portion of total drips, I assume?
Actually trying to answer: “I set the string swinging like a pendulum” to me reads like the person pulls the ice cube back and then either lets go or gives it a little push. I expect it’s quite hard to do either of these while ensuring that the net momentum of the ice cube is exactly along a line that runs directly below the point at which the ice cube is attached to the branch. If it starts off with any momentum perpendicular to that line, you get an ellipse and not a line. As it loses energy and traverses a smaller ellipse it fills in the ellipse. If this happens quickly enough the final shape would be less of an ellipse than a splattering of drips in a vaguely elliptical pattern, with a strong concentration in the center. The cooler the day the more that happens, and possibly the day needs to be improbably hot before you get anything other than a few dots and a point?
Slight adjustment to your scenario:
the ice-cube’s residence-times are maximized at the extrema, so your drips would concentrate toward the two extremes.
Also, from the mechanical, historical perspective—a drop that landed at the dead center beneath the pendulum’s contact with the branch would have had to leave the cube in a brief moment of time before passing over the center, with exactly enough forward velocity at the moment it left the cube such that it would hit the center by the time it reached the ground (depends on how far up it’s hung)… which is a tiny portion of total drips, I assume?