10.2. Purely mechanical machines[[Student version, January 17, 2003]] 359
a b c
θ θ
Rw (^1) w w 1 w 1
2
w 2
αβ
Figure 10.6:(Schematics.) Three simple macroscopic machines. In each case the weights are not considered part of
the machine proper. (a)Acoiled spring exerting torqueτlifts weightw 1 ,driving an increase in the angular position
θ.The spring is fastened to a fixed wall at one end, and to a rotating shaft at the other; the rope holding the weight
winds around the shaft. (b)Aweightw 2 falls, lifting a weightw 1 .(c)As(b), but the shafts to whichw 1 andw 2
are connected are joined by gears. The angular variablesαandβboth decrease asw 2 liftsw 1.
10.2.1 Macroscopic machines can be described by an energy landscape
Figure 10.6 shows three simple, macroscopic machines. In each panel, external forces acting on the
machine are symbolized by weights pulled by gravity. Panel (a) shows a simple one-shot machine:
Initially, cranking a shaft of radiusRin the direction opposite to the arrow stores potential energy
in the spiral spring. When we release the shaft, the spring unwinds, increasing the angular position
θ. The machine can do useful work on an external load, for example lifting a weightw 1 ,aslong
asRw 1 is less than the torqueτexerted by the spring. If the entire apparatus is immersed in a
viscous fluid, then the angular speed of rotation, dθ/dt,will be proportional toτ−Rw 1.
Your Turn 10b
Explain that last assertion. [Hint: Think back to Section 5.3.5 on page 161.]When the spring is fully unwound, the machine stops.
Figure 10.6b shows a cyclic analog of panel (a). Here the “machine” is simply the central shaft.
An external source of energy (weightw 2 )drives an external loadw 1 against its natural direction of
motion, as long asw 2 >w 1 .This time the machine is a broker transducing a potential energy drop
in its source to a potential energy gain in its load. Once again we can imagine introducing enough
viscous friction so that kinetic energy may be ignored.
Figure 10.6c introduces another level of complexity. Now we have two shafts, with angular
positionsαandβ. The shafts are coupled by gears. For simplicity, suppose the gears have a 1:1
ratio, so that a full revolution ofβbrings a full revolution ofαand vice versa. As in panel (b), we
may regard (c) as a cyclic machine.
Our three little machines may seem so simple that they need no further explanation. But for
future use, let us pause to extract from Figure 10.6 an abstract characterization of each one.
One-dimensional landscapes Figure 10.7a shows a potential energy graph, orenergy land-
scape,for our first machine. The lower dotted line represents the potential energy of the spring.
Adding the potential energy of the load (upper dashed line) gives a total (solid line) that decreases
(^4) Strictly speaking, living cells constantly recycle actin and tubulin monomers by depolymerizing filaments and mi-
crotubules and “recharging” them for future use, so perhaps we should not call this a one-shot process. Nevertheless,
Figure 10.4 does show polymerization force generated in a one-shot mode.