Create a plastic track, shaped like an inverted cycloid.

Place in this track a steel ball bearing.

Add a c-shaped electromagnet around the center of the track, so that they can magnetically attract or repel the ball when energized.

Add a sensor in the middle of the track; also, add two more, one on either side of the first on, and both equidistant from it.

Add some electronics and a voltage source, so that the electromagnet get energized when the ball triggers the left or right sensor twice in a row. When the center sensor is triggered, the the electromagnet is turned off.

Each time the rolling magnet passes through the center, the electromagnet briefly pulls the ball, adding to it's speed.

If the ball were sliding frictionlessly, then it's period would depend solely on the radius of the cycloid; I honestly don't know how much effect there is due to the moment of inertia.

Since there's friction involved, the ball shouldn't go flying off the top edge of the track if you use a reasonably small voltage.

On the other hand, if the voltage *is* high enough to lift the ball all the way past the top, then this could be used as a lift for a rolling ball kinetic sculpture. In this case, the track doesn't need to be cycloid shaped; a parabola, or a U shape, or some odd 3-D shape, etc, will work... so long as the ball doesn't lose too much kinetic energy going back and forth.
-- goldbb, Jun 20 2015

//If the ball were sliding frictionlessly, then it's period would depend solely on the radius of the cycloid//

Ah, you're facing the classic escapement problem. If the action were frictionless, you wouldn't need the electromagnet/spring/weight to keep things going.

As soon as you add a driving force (in this case, the electromagnet), you change the period of the oscillator (in this case, the ball). If the voltage and timing of the pulse (relative to the ball's movement) were constant, then the clock would run at a steady rate, but this rate would differ from the natural frequency of the ball.

For precise timekeeping, you would want to trigger the electromagnet from a regulated source (eg, a quartz clock). But then the whole mechanical side of the device becomes a mere decoration.
-- MaxwellBuchanan, Jun 21 2015


On a related note, it should be possible to do something very interesting using a tautochrone curve. (A tautochrone curve is the same as a cycloid, but the emphasis is on the tautochronicity. It's a curve shaped such that a ball dropped onto it at any point will roll to the end in the same time, wherever it starts.)
-- MaxwellBuchanan, Jun 21 2015



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