What it shows:
Conservation of angular momentum and the exponential increase in friction are what save the coffee mug from smashing into the floor. Use this entertaining demonstration to introduce either of those physics concepts.
How it works:
You need a pencil, a pen, a china cup (we use a china cup to add suspense and a threat of disaster), and about 1 meter of string. Tie one end of the string to the cup and the other to the pen. Hold the pencil in one hand and drape the string over it so the cup hangs down a few centimeters. Hold the pen with your other hand (arm stretched out to the side), slightly below the level of the pencil. When you let go of the pen, will the cup fall and crash into the floor?
When you let go of the pen, it falls down in an arc about the pencil. At the same time, the cup accelerates downward, reducing the radius of the pen's arc dramatically. As the pen spirals inward around the pencil, its velocity increases rapidly (conservation of angular momentum) and it very quickly winds around the pencil several times. It takes only three or four windings around the pencil to stop the cup from falling any further. The friction between the string and pencil increases exponentially with the number of windings. (see "Rope Friction Around Pole" demo for details)
The demonstration has also become an instructional laboratory experiment and modeled as follows. Assuming an initially horizontal rope, the model gives the following relation between:
r0 (distance of small mass to rod axis, i.e., length of the horizontal rope) and
r1 (the distance the heavy mass falls before it rests):
r1 = (2/3) r0 sqrt(m/M) where (m/M) is the ratio of the small mass to the large mass. This is the same result obtained by Griffiths (eqn 22) in one of the AJP papers cited below.
Notes:
1) The above model does not include the small overhang distance of the large mass (i.e., initial distance between the rod and the large mass -- usually small).
2) The above model assumes point masses so doesn't include the extra length of an extended object.
3) The momentary upward tension (due to what Griffiths calls the "centrifugal jerk" as the acceleration changes sign) at the moment of arrest is much (??) larger than the weight of the heavy mass.
Three AJP papers on this experiment are referenced below.
Setting it up:
This is pretty simple, but you should try it out beforehand for dependable execution. The only times it has failed is when the pencil was not firmly held in a horizontal position. It's possible for the string to slip off the pencil if it's tilted down slightly and that invites disaster!
Comments:
We were first introduced to this demo by Martin Gardner in his Physics Trick of the Month column in The Physics Teacher 28(6), 1990, p.390. He presented it with a bunch of keys and an empty match folder for the objects. Gardner claims that it was invented by Stewart James, a Canadian magician, who explained it in a magiacian's magazine, Linking Rings, in 1926. It's still just as counterintuitive and entertaining!
A.R. Marlow, "A surprising mechanics demonstration," Am J Phys 59(10), 951-952 (1991).
David J. Griffiths and Tyler A. Abbott, Comment on "A surprising mechanics demonstration," by A.R. Marlow [Am J Phys 59, 951-952 (1991)]. Am J Phys 60(10), 951-953 (1992).
R.E. Sears, Comment on "A surprising mechanics demonstration," by A.R. Marlow [Am J Phys 59, 951-952 (1991)]. Am J Phys 63(9), 854-855 (1995).