Let r1, r2 be the radii drawn from C to A1, A2, and let w1, w2 be the
components of v1, v2, perpendicular to these radii, making angles
[beta] and [alpha] with v1, v2. Then
v1 = w1 sec [beta]; v2 = w2 sec [alpha]
p1 = r1 cos [beta]; p2 = r2 cos [alpha],
.: M = (W/g) (w2r2 - w1r1), (3)
where the moment of the couple is expressed in terms of the radii
drawn to the positions of the body at the beginning and end of a
second, and the tangential components of its velocity at those points.
Now the water flowing through a turbine enters at the admission
surface and leaves at the discharge surface of the wheel, with its
angular momentum relatively to the axis of the wheel changed. It
therefore exerts a couple -M tending to rotate the wheel, equal and
opposite to the couple M which the wheel exerts on the water. Let Q
cub. ft. enter and leave the wheel per second, and let w1, w2 be the
tangential components of the velocity of the water at the receiving
and discharging surfaces of the wheel, r1, r2 the radii of those
surfaces. By the principle above,
-M = (GQ/g)(w2r2 - w1r1). (4)
If [alpha] is the angular velocity of the wheel, the work done by the
water on the wheel is