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MECHANICAL PHILOSOPHY. HYDRODYNAMICS. [FUJID EHUSTAKCI. 



. ..r a; 



. r a, '_' : 



fc V 



- or 2 : 1 



If tlio quantity Q be the whole content* of the vessel 

 that U, if V be zero then the times of discharging the 

 Yeuelful of fluid in the two cases are as 2 to 1, as already 

 inferred at pagr 



In order to test the accuracy of the theory by experi- 

 ment, (1) is the most suitable formula for the purpose, 

 since the time occupied by the surface in descending 

 f i-. .in one level to another, can be correctly observed ; 

 but the instant when the last portion of fluid escapes 

 cannot be accurately noted ; for when the vessel is 

 exhaustion, the fluid ceases to fill the orifice, and is dis- 

 charged in drops from the edges of the hole. The time 

 occupied by the surface descending through a certain 

 space, is found to agree very closely with the theoretical 

 expression for it. 



THE CLErSYDKA. The clepsydra, or water-clock, 

 is a contrivance for measuring time by the descent of the 

 surface of water in a vessel, the water flowing through 

 an orifice at the bottom. The most convenient form for 

 the vessel is that in which the surface descends through 

 equal vertical spaces in equal times. This form may be 

 determined as follows : 



Let the altitude of the surface at any instant be x, 

 and the radius of the surface that U, the correspond :;; 

 ordiuate of tlio gencratini; curve y. Then the velocity 

 at the orifice at that instant is J2gx (page 764), ami tin: 

 area of the descending surface iry 1 , where jr=3'1416. 

 The velocity V of the descending surface is found by the 

 theorem at page 7C3, which gives 



where a is the area of the orifice. 



Since the surface is to descend uniformly, V must be ft 

 constant quantity, which can be suitably assumed in 

 reference to the whole time of emptying : putting e for 

 this constant, we have 



. 



This, therefore, is the equation of the curve, tho 

 rotation of which about its vertical axis x will generate 

 the surface of the vessel. This generating curve is a 

 parabola of the fourth order. 



But the surface need not be a surface of revolution, or 

 one having all its horizontal sections circles. They may 

 all be rectangles. Let the sides of the rectangular section, 

 at the altitude x from the base, be y and p, then the 

 area of the section is JHJ, and we have 



o that if p be the same for every section that is, if 

 the rectangular sections have all the same breadth the 

 curve bounding the vessel towards the lengths of these 

 rectangles will be the common parabola. 



\M i: <iK FLUIDS. All bodies 

 moving in a fluid are impeded in thuir progress by thu 

 resistance of the fluid, which must itself be moved, in 

 order that motion may take place in the immersed body. 

 The resistance to motion in the fluid is of course mainly 

 dne to it* inertia ; friction and the tenacity of the fluid, 

 add somewhat to the resistance ; but the retarding in- 

 fluence of these is too inconsiderable to ren.ler the con- 

 sideration of them of much practical consequence. 



It may safely be assumed that the resistance on a plane 

 surface, moving with a given velocity through stagnant 

 water, U the same as when, the plane being at rest in- 

 stead of the water, a stream moving with the same 

 velocity act* against it. 



If a stream act perpendicularly against a plane surface, 



or if the surface act against the quiescent fluid by moving 

 perpendicular to its surface through the fluid, the re- 

 sistance will be as the area of the plane, the density of 

 the fluid, and the square of the velocity conjointly. 



For the velocity v of the stream being regarded as 

 uniform throughout the whole depth of the plane, the 

 resistance is the same at every point of t : and 



therefore, other circumstances being the same, it is pro- 

 portional to the area A of the plane. 



But the quantity of fluid matter striking against the 

 plane, in a given time, is proportional to the density and 

 velocity of the fluid conjointly ; and this quantity of 

 matter strikes with a velocity r, and therefore with a 

 momentum equal to 



quantity of matter X 



Hence, D being the density of the fluid, the resistance 

 of the area A varies as A DP*. 



If a body were to fall by the force of gravity till it 

 acquired the velocity v of the stream, the space fallen 



through would be -J- (DYNAMICS, page 736); this, 



9 



therefore, would vary as the square of the velocity ; so 

 does the resistance A Do 2 ; hence the resistance varies as 

 the weight of a column of the fluid, whose base is the 

 area of the plane, and the altitude equals the space 

 through which a heavy body must fall to acquire the 



: v of the stream or of the moving plane. 

 The force with which a stream acts perpendicularly 

 upon a plane opposed to it obliquely, varies as tlie square 

 of the sine of the inclination of the plane to the stream. 

 Let AB (Fig. 189) and LN be sections of the plane 

 Fig. 189. 



and stream, moving in the direction of the latter ; draw 

 A C in that direction, meeting B C perpendicular to A B 

 in C ; draw also B D perpendicular to A C. 



Now the quantity of fluid acting against A Bis the 

 same as that which acts perpendicularly against D 15 ; 

 it varies, therefore, as the length of B D ; that is, as the 

 sine of the angle A. If the velocity with which thin 

 quantity moves in the direction of the stream be repre- 

 sented by A C, then B C will represent the velocity with 

 which it moves perpendicular to A U ; an;l just as BD 

 varies as the sine of the angle A, so does B C vary as the 

 sine of the angle A ; hence the force with which the stream 

 acts perpendicularly upon the plane varies as sin. * A. 



The force impelling tho plane in the direction of the 

 stream, varies as tho cube of the sine of the inclination 

 of the plane to that direction. 



Let B C (Fig. 190), perpendicular to A B, represent 



Fig. 190. 



tho force on AB perpendicular to it ; this may be re- 



! into two forces BD. DC, the la ' the 



direction of the stream, and the former BU perpend. 



