GENERAL PRINCIPLES OF HYDRAULICS 445 
th a8 
k= a » and therefore k= 3h = ae » which shows that the depth of 
A, the bottom of the cup below the original level of the liquid, is pro- 
‘portional to the square of the angular 
velocity. : 
__ When the top of the cup reaches the top 
of the cylinder, as shown by the dotted 
parabola in Fig. 725, h,=2k,, where /, is 
“the depth of the original level of the liquid 
_ below the top of the cylinder. 
Suppose now that the top of the cylinder 
is closed, and that the angular velocity 
is still further increased. The cup will 
still be a paraboloid. ‘Let the total depth Fie. 724. 
_ of the cup be now y,, and its greatest radius ; 
z,; then, since the volume of the cup of height h, must be the same as 
2 
i that of the cup of height y,, hr? =y,27, or 2} = ae , but y, = ay xi , therefore 
1 
4 hy . : ; 
y= , and ¥,=,7 Dy? where o, is the angular velocity. This 
shows that after the cup touches the top of the cylinder its total depth is 
- directly proportional to the angular velocity. 
The preceding investigation gives the 
theory of a well-known instrument for 
indicating the speed of revolution of a 
shaft at any instant, The cylinder contain- 
ing the liquid is made of glass, and the 
_ spindle upon which it is mounted is geared 
to the shaft whose speed is required. A 
uated scale placed beside the cylinder 
shows the speed at a glance by indicating 
the position of the vertex of the paraboloid. 
From the level CD to the level EF (Fig. 725) 
the graduations of the scale are unequal, but below the level EF they 
are equal. 
Another problem on whirling liquids may be considered here. A tube 
AB (Fig. 726), of length 27, closed at both ends and full of liquid, revolves 
with its axis in a horizontal plane about an 
wth r? 
2g 
axis bisecting the axis of the tube with an aX, 
angular velocity w. It is required to find the Al IL___JB 
pressure exerted by the liquid on the ends of the x I 
tube. Let a be the area of the cross section of FIG. 726. 
the tube, and wthe weight of a unit of volume : 
of the liquid. Consider a small portion of the liquid between two planes 
oN Ai ie to the axis of a tube and ata distance apart equal to dz, and 
t the distance of this portion of liquid from the axis of revolution be @. 
i . wadrw* x 
he centrifugal force exerted by this position of liquid is : a and 
