2007 AAPT Summer Meeting
Greensburo, NC
GalileoÕs Paradox
Thomas
B. Greenslade, Jr.
Department of Physics
Kenyon College
Gambier, OH 43022
740 4272989
GREENSLADE@KENYON.EDU
Abstract:
In a 1602 letter
Galileo noted that a body sliding freely down a chord from the edge of the
circle reaches the lowest point on the circle at the same time as a body
released simultaneously from the top. This is a modern, lowcost version of the
demonstration in which the circle is composed of a bicycle rim mounted on edge.
Construction of Apparatus:
The
lower edge of the rim of a bicycle wheel is screwed to a wooden block to hold
the rim vertically. A wire is stretched between the hole at the bottom of the
rim to the top of the rim. A second wire is stretched from a hole at about the
3 oÕclock position to the one of the bottom. Metal beads, large enough to be
visible to the class, slide freely on these wires. Construction time is well
under an hour once the parts are on hand.
The
only cost was $0.50 for the metal beads from a bead shop. The wire, screws,
wood block and bicycle rim came from my scrap pile.
Use of Apparatus:
The
beads are released simultaneously from the top points of their wires. They meet
at the bottom with a click (which can be heard by the entire class) showing
that, despite the difference in distances, the travel times are the same. This
result seems to be paradox to most observers.


To show the equality of these times, assume that the diameter of the circle is D, the length of the chord is L, and that the angle between the chord and the horizontal is θ. The acceleration of the body sliding down the chord is a = g sin θ , and the time, t, for the trip down the chord is found using
L = ½at^{2}. Putting in the expression for the acceleration, and solving for t^{2} gives t^{2} = 2L/(g sin θ). We know that the triangle bounded by the chord and the diameter of the circle is a right triangle, so
sin θ = L/D,
and t^{2} = 2D/g. But this is just what we would expect for a body
falling freely through a vertical distance D, and the paradox is resolved.
In
another form, this demonstration appears on pg 42 of SuttonÕs 1937 book,
ÒDemonstration Experiments in Physics.Ó