Now apply Σm = 0 to the worm wheel and we have
Kr´´ = Wr´´ or K = wr´´ (8).
Substituting the value of K in (7) in equation (8) we have
P2πν = Wr´´ or P2πν = Wr´´p (9).
Now it is evident that the distance p´ moved by W while K moves through the distance p is to p as r´´ is to r´ or p´ : p :: r´´ : r´ or
p´ = pr´´/r´ (10).
Substituting this value of pr´´/r´ in equation (9) we have P2πν = Wp´ or the condition Σwork = 0, since 2πν is the distance moved by (P) while W moves the distance p´.
No provision for friction has been made in any of the examples given, so that allowance must be made for this propensity whenever any of the foregoing rules are applied to practice. The amount of allowance required will vary and must be made to suit conditions.
An endless screw is sometimes used as a component part of graduating machines, counting machines, etc. It is also employed in conjunction with a wheel and axle to raise heavy weights. The distance between the threads of the screw is called the pitch or step. These threads are sometimes square, sometimes acutely pointed or edged, sometimes rounded off on the edges. Power is often applied by means of a lever or other contrivance attached to the end of the screw, or by a long handled wrench (a monkey wrench for instance), which, when turned, moves forward in the direction of its axis, overcoming resistance. In the case of the screw-jack, it may be used to raise a heavy weight. The relation between the force applied and the resistance to be overcome is important to note, for every time the screw performs one revolution it moves forward through a distance equal to the space between one thread and the next.
Fig. 35. Archimedian screw
The Archimedian screw we have read and heard so much about is simply a hollow pipe wound around a cylinder. It was often used in olden times for raising water, but is now only occasionally applied. The lower end of the spiral pipe is, of course, left open and immersed in water, as shown in the illustration ([Fig. 35]), a device for raising water, the supply stream being the motive power. The oblique shaft of the wheel has extending through it a spiral passage, the lower end of which is immersed in water; and the stream, acting upon the wheel at its lower end, produces its revolution, by which the water is conveyed upward continuously through the spiral passage and discharged at the top. An arrangement like this could easily be constructed at the edge of most rivers to raise water to irrigate the grounds, if so desired, and the little flutter wheel at the bottom of the inclined shaft would be powerful enough to lift all the water required. Fred thought that would be a great scheme, and determined to try his hand at it one of these days, but he was told that a wheel of that kind could only work at intervals, as the river's flow was often running in opposite directions owing to the inflow of the tidal water.
Fig. 36. Spiral conveyor