826 



APPLIED MECHANICS. 



[WATEU-roWEB. 



bring double the number of particles in contact with the 

 float, and we also double the force of each particle in 

 triking it ; so that, upon the whole, we quadruple the 

 pressure. So, also, by taking 3 times the velocity, we 

 hare 9 time* the pressure ; and, generally, if we know 

 the pressure duo to one Telocity, such as 1 foot per 

 second, we can compute that due to another velocity, 

 such as 10 feet per second, by multiplying the velocity 



:self, and taking the pressures in those proportions. 

 By an investigation of the mechanical laws which 

 govern the motion of fluids, we ascertain that the velocity 

 with which a fluid flows from an orifice in any vessel, is 

 -;une as that which a heavy body would acquire by 

 falling through a height equal to that of the fluid column 

 above the orifice. This is found by the following rule : 

 Multiply the square root of the height (in feet) by 8 ; 

 tlio product is tin- velocity (in feet per second). Thus, 

 if a stone were dropped from a precipice 100 feet high 

 (neglecting the resistance of the air to its fall), its velocity 

 when it strikes the bottom would be about 80 feet per 

 second ; for 10 is the square root of 100, aud 8 x 10 = 

 80 feet per second. If, on the other hand, we knew the 

 velocity, we should be enabled to calculate the height of 

 fall by reversing the rule ; that is to say, divide the 

 velocity (in feet per second) by 8, and square the result 

 for the height (in feet). Thus, if we ascertained that a 

 stone had in its fall acquired a velocity of 80 feet per 

 second, we should reckon that it had fallen 100 feet, for 



N* 



- Q = 10, and 10 squared or 10 X 10 = 100. 



o 



These rules do not take into account the resistance 

 offered by the air to the motion of the body falling 

 through it. In falling through great heights, this re- 

 sistance has very great effect on the velocity ; but for 

 small falls, it may be neglected without material error. 

 If now, we apply these computations to the flow of cur- 

 rents of water, we can calculate the height due to a cer- 

 tain velocity of current ; that is, the head of water whose 

 pressure on its lower particles gives them the velocity in 

 question. Thus, if on measuring the velocity of a stream 

 at a rapid, we found it to be 16 feet per second, since 



1 (\ 



^- = 2, and 2 X 2 = 4, we should reckon that, in order 



o 



to have this velocity, it mast be pressed on by a column 

 or head of water 4 feet high. It must, therefore, press 

 upon any body immersed in it with the same force as it 

 is itself subjected to ; for it is the peculiar property of 

 fluids to convey pressures equally in every direction 

 through thein.t It does not necessarily follow that, at 

 the point where the velocity is measured, there is an ac- 

 tual fall of 4 feet : it is sufficient that the water has 

 somewhere fallen enough to acquire the velocity mea- 

 sured, or that any forces whatever have combined to give 

 it that velocity, or to reduce its velocity to that degree. 

 The velocity is, in fact, the expression of the result of all 

 the forces and resistances, that have acted on the water 

 up to the point where it is measured. Having computed 

 the head of water pressing, it is easy to compute the 

 amount of pressure exerted on a square foot. A cubic 

 foot of water weighs 62^ Ibs. ; therefore the bottom of a 

 cubical box, measuring 1 foot every way, is pressed on 

 by a force of 62J Ibs. when the box is filled with water ; 

 in other words, the pressure of a column of water 1 foot 

 liiL;b,iaG2i Ibs. on every square foot of bottom surface. 

 Were the ^height of the column increased, the pressure 

 would be increased in like proportion ; every additional 

 foot of height would throw an additional pressure of 

 C2i Ibs. on every square foot. The intensity of pressure 

 fin Ibe. per square foot), then, is 62J times the height in 

 feet Thus, the pressure due to a head of 4 feet, is 

 62J X 4 = 250 11)8. per square foot. We may now com- 

 Unu the two computations that for head due to given 

 velocity, and that for pressure due to given head into 

 one rule, for determining the pressure per square foot 

 due to a given velocity, as follows : As the square of 8 

 is f>4, which doe* not much differ from 62 J, we may, 

 without material inaccuracy, avoid dividing the velocity 

 It* o(, //yrfrodyiuuiuv, p. 7M. t Hoc ante, p. 7S1. 



by 8 before squaring the quotient, and again multiplying 

 by 62J, and thus have the very simple rule. 



Square the velocity (in feet per second), and the result 

 is nearly the pressure (in Ibs. per square foot). Thus, 

 when the stream is moving with a velocity of 16 feet per 

 second, its pressure per square foot is 1(5 X 16 = 256 Ibs. 

 nearly. By the former computation, we found 250 Ibs., 

 lesa by 6 ; that is, less than J, st part of the whole. So 

 far we have found the means of computing the pressure 

 on a float-board at rest, in a stream flowing with a given 

 speed ; but in the case of an undershot water-wheel, the 

 float is in motion in the same direction with the current, 

 and therefore the relative velocity of the latter, in acting 

 upon it, is so much diminished. The relative velo, 

 of the float and of the current may, of course, be varied 

 by applying more or less resistance to the motion of the 

 wheel. It is necessary to know the relation of the two 

 velocities, so as to derive the best possible effect from 

 the current. If we take a particular case, and try various 

 relations, we may find the most advantageous bearing 

 in mind that the result to be obtained is the maximum 

 of power or useful effect ; that is to say, the pressure on 

 any float multiplied by the velocity of its motion, which 

 product gives the power of that float as a mover of 

 machinery. If we take the velocity of a current as 6 

 feet per second, and form a table of velocities of float 

 from feet per second up to 6 feet per second, of pres- 

 sures due to the excess of the stream's velocity over that 

 of the float, and of the powers or products of those pres- 

 sures multiplied by the corresponding velocities of 11 

 we shall find the power greatest when the velocity of 

 the float is exactly 2 feet per second, or Jrd of that of 

 the stream. Thus 



Velocity of 



Telocity of 



liu.lt. 

 



1 



2 

 S 

 4 

 5 

 6 



Excess of 



stream over 



float. 



6 



6 



4 



3 



2 



1 







Pressure or 



excess 



squared. 



36 



H 



16 

 9 



4 

 1 

 



Power or 

 pitas, xreloe. 





 23 

 32 



27 

 16 



5 







Were we to take any other velocity of stream wo should 

 find the same result, that the velocity of the float should 

 be Jrd of that of the stream, in order to attain the maxi- 

 mum effect. J If this rule be adhered to, the excess of 

 the stream's velocity over that of the float is Srds of itself ; 

 and the pressure per square foot would be the square of 



jrds of the stream's velocity, or Jths of the square of 

 the velocity, since J X $ = 4- The power would depend 

 on the number of square feet pressed upon, aud the 



velocity of the float, and would therefore be found by 

 multiplying the surface of the float by Jtlis of the square 

 of the stream's velocity, and the product by the float's 

 velocity, or Jrd of the stream's velocity. As there are 

 generally several floats immersed in the water, it may 

 appear that the surface acted on is considerably larger 

 than that of one float ; but when it is remembered that 

 the volume of water contained between any two of tlie 

 floats, if it press the one forward by its direct action, 

 must equally press the other backward by its reaction, 

 we cannot safely estimate more than the surface of one 

 float as really effective. The velocities of currents are 

 generally reckoned in feet per second, while the velo- 

 cities of moving parts of machinery, in estimating 



J If V = velocity of stream, e = velocity of float, p = pressure 

 per square foot = (V t)* and P = power = pv = V^ 7)"ti, 

 to find t>, so that P may be a maximum. 



P = V t>V = V"e 2\V + r'. 



rf / = V 4Vr + 3* = O. 

 oc 



.-. ,' -Jv. = - V 3 ' and ._! V. + Jv = Jv. 



Hence v |V = JV, or r= V, or JV. 



</ 2 P 



= 4V + Gr, and substituting for t each of its values. 



dv 3 



4V -f- GV = 2V positive, gives P a minimum when r = V. 

 4V + 2V= 2V negative, gives P a maximum when c = i V. 



