Wednesday, June 22, 2011

Rubber Bands and Plumb Lines

The mechanics problem of what happens when the rubber band is loaded with the plumb line surprisingly is easiest to solve using complex numbers. Suppose the plumb line with a weight of mass m hangs on a rubber band which is attached at two fixed points deparated by a horizontal distance D. To first approximation the force exerted by the rubber band is linear and proportional to the relative amount it is stretched or |Δz|/|z|. The proportionality constant is k.

The sum of the two complex numbers, z_1 and z_2, representing the two sides of the rubber band is D which is fixed and not dependent on the mass of the plumb line. Writing a second equation with primed z's for the weighted position and subtracting tells us that the sum of the changes is zero.

So the change of one side is the negative of the change in the other. The balance of forces requires that the sum of the forces exerted by the rubber bands equals minus the force of gravity which in complex notation is, i mg. Substituting -Δz_1 for Δz_2 and rearranging tells us that the plumb line will only drop vertically after it is loaded.

Watching what happens as the rubber band is loaded with the plumb line at various positions shows that its motion actually is vertical.

Supplemental: One can solve the problem in a similar manner using vectors but what seems to be odd about the problem is that the force is dependent on which end of the vector is attached to the common point. One would use the unit vectors i and j in place of 1 and the "imaginary" unit i respectively. The tension in the rubber band is not a vector quantity but the internal forces are balanced. The problem is difficult to work in terms of vector components.

No comments: