10.2. Purely mechanical machines[[Student version, January 17, 2003]] 361
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Figure 10.8:(Mathematical functions.) (a)Imagined potential energy landscape for the “bumpy rubber gears”
machine in Figure 10.6c, with no load nor driving. For clarity each gear is imagined as having only three teeth. The
twohorizontal axes are the anglesα, βin radians. The vertical axis is potential energy, with arbitrary scale. (b)The
same, viewed as a contour map. The dark diagonal stripes are the valleys seen in panel (a). The valley corresponding
to the main diagonal has a bump, seen as the light spot atβ=α=2(arrows).
preferred motions are along any of the “valleys” of this landscape, that is, the linesα=β+2πn/ 3
for any integern. Imperfections in the gears have again been modeled as bumps in the energy
landscape; thus the gears don’t turn freely even if we stay in one of the valleys. Slipping involves
hopping from one valley to the next, and is opposed by the energy ridges separating the valleys.
Slipping is especially likely to occur at a bump in a valley, for example the point (β=2,α=2)
(see the arrows in Figure 10.9b).
Now consider the effects of a load torquew 1 Rand a driving torquew 2 Ron the machine. Define
the sign ofαandβso thatαincreases when the gear on the left turns clockwise, whileβincreases
when the other gear turns counterclockwise (see Figure 10.6). Thus the effect of the driving torque
is to tilt the landscape downward in the direction of decreasingα,just as in the lower dashed lines
of Figure 10.7a,b. The effect of the load, however, is to tilt the landscapeupwardin the direction of
decreasingβ(see Figure 10.9). The machineslides down the landscape,following one of the valleys.
The figure shows the case wherew 1 >w 2 ;hereαandβdrive toward negative values.
Just as in the one-dimensional machine, our gears will get stuck if they attempt to cross the
bump at (β=2,α=2), under the load and driving conditions shown. Decreasing the load could
get the gears unstuck. But if we instead increased the driving force, we’d find that our machine
slipsanotch at this point, sliding from the middle valley of Figure 10.9 to the next one closer to
the viewer. That is,αcan decrease without decreasingβ.
Slipping is an important new phenomenon not seen in the 1-dimensional idealization. Clearly
it’s bad for the machine’s efficiency: A unit of driving energy gets spent (αdecreases), but no
corresponding unit of useful work is done (βdoes not increase). Instead the energy all goes into