454 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
θ∆ha
b
Figure 12.5:(Schematic.) Mechanical analogy to the action potential. A heavy chain lies in a tilted channel, with
twotroughs at heights differing by ∆h.(a)Anisolated kink will move steadily to the left at a constant speedθ:
successive chain elements are lifted from the upper trough, slide over the crest, and fall into the lower trough. (b)A
disturbance can create a pair of kinks if it is above threshold. The two kinks then travelawayfrom each other.
Figure 12.5 shows a molding such as you might find in a hardware store. The cross-section of
the molding is shaped like a rounded letter “w.” We hold the molding with its long axis parallel to
the floor, but with its cross-section tilted, so that one of the two grooves is higher than the other.
Call the height difference between the bottoms of the two troughs ∆h.
Suppose now that we lay a long, flexible chain in the higher groove, and immerse everything in
aviscous fluid. We pull on the ends of the chain, putting it under a slight tension. In principle, the
chain could lower its gravitational potential energy by hopping to the lower groove. The difference
in height between the two grooves amounts to a certain stored potential energy density per length
of chain. To release this energy, however, the chain would first have to move upward, whichcosts
energy. What’s more, the chain can’t hop over the barrier all at once; it must first form a kink.
The applied tension discourages the formation of a kink. Hence the chain remains stably in the
upper groove. Even if we jiggle the apparatus gently, so that the chain wiggles a bit, it still stays
up.
Next suppose we begin laying the chain in the upper groove, starting from the far left end,
but halfway along we bring it over the hump and continue thereafter laying it in the lower groove
(Figure 12.5a). We hold everything in place, then let go at time zero. We will then see the
crossover region moving uniformly to the left at some velocityθ. Each second, a fixed length of
chainθ×(1s)rises over the hump, pulled upward by the weight of the falling segment to its right.
That is, the system displays traveling-wave behavior.
Each second the chain releases a fixed amount of its stored gravitational potential energy. The
energy thus released gets spent overcoming frictional loss (dissipation).
Your Turn 12b
a. Suppose that the chain’s linear mass density isρ(1d)m,chain.Find the rate at which gravitational
potential energy gets released.
b. The speed at which the chain moves is proportional toθ,and hence so is the retarding frictional