Let is II represent one side of a vessel, A n the position of the oar when its blade enters the water, and E the fulcrum or aide of the row lock against which it presses; then, since the vessel will more forward during the time that a stroke of the oar is being made, let r be the position of the fulcrum and c D the position of the oar at the end of the stroke : if the vessel had remained at rest, the oar, at the end of the stroke, would have had the position cd, which may be considered as parallel to c D.
Now, n being the centre of percussion on the blade of the oar, the actual motion of n (supposed to be parallel to the keel of the vessel) may be represented by n d while the movement of the vessel is EP d); and therefore B n represents the relative movemeut of The lines BD and la d being proportional to the velocities of the oar and 'vessel, which velocities wo may represent by r and v' ; will express the relative velocity of the oar, and the effective power of the latter will vary with (v—e)r. Let a, in square feet, be the area of the blade of an oar, and let the pressure of water against a square foot of surface be lb. when the velocity is I foot per second; then 3 —a (v— dr will denote the force of the oar.
2 If, for simplicity, the prow of the vessel bo supposed to have the form of a wedge with plane faces mooting in a vertical lino, or cuts water, on putting a' for the area of the whole prow and 0 for the inclination of each face to a vertical plane passing through the keel, we shall have 3 a' 0. 2 for the resistance of the water against the prow. Therefore, n being tho number of oars all of which are supposed to act with equal forces ; wo have, when the versed has acquired a terminal velocity, a (v— a' 0, from which v' may be found. The velocity of a vessel moved by oars is, however, found to increase in a less ratio than the number of oars.
The power of the oar in rowing appears to be diminished by the reaction of the feet of the rowers in pressing against the foot-boards; this has a tendency to force tho vessel backwards, but it is compen anted by the greater velocity which tho centre of percussion in the ear thereby acquires. Some force, however, is lost in overcoming the inertia of the oar, and in bringing it forward against the air ; this last force is considerable when the vessel is rowed against a high wind, though it is to a certain degree diminished by the practice of feathering the oars.
The above equation might be used to determine the velocity of a vessel impelled by steam, in which paddle-wheels are employed, if it were passible to determine, nearly, the value of na, or the number of square feet of paddle which, on both irides of the vessel, arc at every moment acting efficiently against the water : the value of v would be, of course, determined by the number of revolutions which the wheel makes in a given time ; and it should be expressed by the number of feet per second which the centre of percussion in the paddle moves through in turning about the axle.
The momentum of water flowing horizontally against a plane, such as a float-board of an undershot wheel, depends on its velocity, on the area of the surface with which it comes in contact, and on the obliquity of that surface to the direction of its motion ; and in estimating the effect of water on such a wheel, it is necessary to determine from the dimensions of the channel and the velocity of the water in it the weight of water which descends vertically through a certain height, as one foot, in a given time. The product of this weight multiplied by the height actually descended, being compared with the product of the number of pounds which the wheel can raise to a certain height in the same time multiplied by that height, expresses the ratio of the power to the effect. From many experiments, Mr. Smeaton concluded that, when the quantity of water expended is the same, the useful result varies nearly with the head of water, or with the square of its velocity.
The power of an overshot wheel is estimated by the product of the weight of water expended in a given time multiplied by the whole weight of its descent—that is, by the sum of the diameter of the wheel and the height of water in the reservoir above the top of the wheel. The useful effect of an overshot wheel is said to be nearly double that of one which is undershot.
In determining the power of water on breast-wheels, it is customary to consider such a wheel as one compounded of an undershot and of an overshot wheel ; and its effect is conceived to be equivalent to that of an undershot wheel whose head of water is equal to the difference in height between the surface of the water in the reservoir and the point at which the water impinges on the wheel, together with the effect of an overshot wheel whose height is equal to the difference between the point of impact and the level of the tail-water.