Velocity of Overshot Wheels

Now, suppose the machine so loaded, by making the rising buckets more capacious, that it makes only two turns in a minute, or one turn in 30 seconds; then each descending bucket must contain six cubic feet of water. If each bucket on the rising side contained three cubic feet, the motion of the machine would be the same as before. This is a point none will controvert. When two pounds are suspended to one end of a string which passes over a pulley, and one pound to the other end, the velocity of descent of the two pounds will be the same with that of a four pound weight, which is employed in the same manner to draw up two pounds.

Our machine would therefore continue to make four turns in a minute, and would deliver 90 cubic feet during each turn, and 360 in a minute. But, by supposition, it is making only two turns in a minute ; which must proceed from a greater load than three cubic feet of rising water in each rising bucket. The machine must, therefore, be raising mare than 90 feet of water during one turn of the wheel, and more than 180 in a minute.

Thus it appears that if the machine is turning twice as slow as before, there is more than twice the former quantity in the rising buckets ; and more will be raised in a minute for the same expenditure of power. In like manner, if the machine go three times as slow, there must be more than three times the former quantity in the rising buckets, and more work will be done.

But further we may assert, that the more we retard the machine to a certain practical extent, by loading it with more work of a similar kind, the greater will be its performance; and the truth of the assertion may be thus demonstrated:

Let us call the first quantity of water in the rising bucket, Q; the water raised by four turns in a minute will be 4 x 30 x Q = 120 Q.

The quantity in this bucket, when the machine goes twice as slow, has been shown to be greater than
2Q; call it 2 Q + x; the water raised by two turns in a minute will then be 2 x 30 x (2Q + x) = 120 + 60 x.

Suppose next, the machine to go four times as slow, making but one turn in a minute ; the rising bucket must now contain more than twice the quantity 2 Q + x, or more than 4 Q + 2 x, call it 4 Q + 2?- + ?.

The work done by one turn in a minute will now be 30 x (4Q + 2x + y) = 120 Q.+ 60 x + 30 y.

By such an induction of the work accomplished, with any rates of motion we choose, it is evident that the performance of the machine increases with every diminution of its velocity that is produced by the mere addition of a similar load of work, or that it does the more work the slower it goes. This, however, is abstracting from the effects of the friction upon the gudgeons of the wheel, a cause of resistance which increases with the load, though not in the same ratio.

We have also supposed the machine to be in its state of permanent uniform motion. If we consider it only in the beginning of its motion, the result is still more in favour of slow motion : for, at the first action of the moving power, the inertia of the machine itself consumes part of it, and it acquires its permanent velocity by degrees, during which the resistanees arising from the work, friction, &c., increase, till they exactly balance the pressure of the water ; and after this the machine no longer accelerates.

Now, the greater the power and the resistance arising from the work are, in proportion to the inertia of the machine, the sooner, it is obvious, will it arrive at its state of permanent velocity. The preceding discussion only demonstrates in general the advantage of slow motion ; but does not point out in any degree the relation between the rate of motion and the work performed, nor even the principles on which it depends. But this is not necessary for the improvement of practical mechanics.