GENERAL PRINCIPLES OF HYDRAULICS 483 
mtrance is found, and its direction determines the direction of the 
fangent to the vane at entrance, so that there shall be no shock there. 
At exit the relative velocity uv, is in the direction of the tangent to the 
vane at that point. v,, the absolute velocity of the water at exit, is the 
diagonal of the parallelogram, having c. and uw, for adjacent sides. The 
‘angle between v, and cy is 45. 
. 
_ Consider a small portion of the water of mass m. At entrance the 
velocity of this mass in the tangential direction c, is v, cos 9, ; this is the 
velocity of whirl at entrance. At exit the velocity of whirl is v, cos 4,. 
Taking moments about O, the angular momentum of the mass at entrance 
is mv,7, cos 9,, and at exit its angular momentum is mygr_cos A. Hence 
he loss of angular momentum of the mass in passing over the vane is 
m(v,7, cos 9, — v7". cos 0). If W is the weight of water impinging on 
the vane per second, then the angular momentum lost by the water per 
" ow ; : 
‘second is —(v,7, cos 9, — vgr,cos 6,). If there is a succession of vanes, 
then W is the weight of water supplied by the jet per second, and the 
turning moment on the wheel, due to the action of the water on the vanes, 
i -M=—(0,7; cos 4, — vyr cos 63), since the angular momentum gained 
by the wheel is equal to that lost by the water. In the foregoing discus- 
sion the effect of friction has been neglected. 
If @ is the angular velocity of the wheel, then the work imparted to 
the wheel per second is 
Mo= Cow cos 6, — Yg7Q cos A) = Hore cos 6; — V_Co COs 9). 
= Exercises XXVIIIc. 
1. A jet of water 2 inches in diameter, and having a velocity of 30 feet per 
‘second, impinges upon a fixed flat plate. Find the total pressure on the plate 
‘due to the impact of the jet, (2) when the plate is perpendicular to the axis of 
the jet, (6) when the plate is inclined at 30° to the axis of the jet. 
_ 2. A jet of water 3 inches in diameter, and having a velocity of 40 feet per 
‘second, strikes a flat vane which is perpendicular.to the axis of the jet. Deter- 
ee ne the total pressure on the vane, (a) when it is fixed, (b) when it is moving 
in the same direction as the jet with a velocity of 15 feet per second. 
_ 8. A fixed nozzle discharges 2 cubic feet of water per second. The jet, which 
has a cross section of 10 square inches, impinges on a flat vane which is moving 
in the same direction as the jet with a velocity of 10 feet per second. Find the 
_work done on the vane in horse-power. 
4. A series of flat vanes come in turn into a jet of water 4 inches in diameter. 
The vanes when in action are perpendicular to the axis of the jet, and they are 
driven forward by the jet with a velocity v, feet per second, The velocity of the 
jet is 50 feet per second. On a base representing values of v, from 0 to 50, plot 
the horse-power delivered to the vanes. State the value of the maximum horse- 
power, and the corresponding value of 1. 
5. A jet of water has a sectional area of 20 square inches, and delivers 1869 
gallons of water per minute. The jet impinges at right angles on a flat vane, 
which is driven in a direction inclined at 30° to the axis of the jet with a 
velocity of 12 feet per second. Find the work done on the vane in ft.-Ibs. 
per second, and the efficiency. 
_ 6. Taking the data of the preceding exercise, except that the jet impinges 
on the vane at an angle of 30° to its normal, find the work done on the vane in 
ft.-lbs. per second, and the efficiency. 
__ 7. Same as Exercise 5, except that there is a succession of vanes at equal 
distances apart. 
