Apparatus Competition

2007 AAPT Summer Meeting

Greensburo, NC

 

Galileo’s Paradox

 

Thomas B. Greenslade, Jr.

Department of Physics

Kenyon College

Gambier, OH 43022

 

740 427-2989

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, low-cost 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 o’clock 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 = ½at2. Putting in the expression for the acceleration, and solving for t2 gives t2 = 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 t2 = 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 Sutton’s 1937 book, “Demonstration Experiments in Physics.”