Mt = (W/g) (v2p2 − v1p1),
or, if the change of momentum is estimated for one second,
M = (W/g) (v2p2 − v1p1).
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 β and α with v1, v2. Then
v1 = w1 sec β; v2 = w2 sec α
p1 = r1 cos β; p2 = r2 cos α,
∴ 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,