| = R = 2rb sin | φ | × | Gt | · | v2 | = 2 | G | btv2 sin | φ | , |
| 2 | g | r | g | 2 |
which is independent of the radius of curvature. It may be inferred that the resultant pressure is the same for any curved surface of the same projected area, which deviates the water through the same angle.
Case 2. Cylindrical Surface moving in the Direction AC with Velocity u.—The relative velocity = v − u. The final velocity BF (fig. 162) is found by combining the relative velocity BD = v − u tangential to the surface with the velocity BE = u of the surface. The intensity of normal pressure, as in the last case, is (G/g) t (v − u)2/r. The resultant normal pressure R = 2(G/g) bt (v − u)2 sin 1⁄2φ. This resultant pressure may be resolved into two components P and L, one parallel and the other perpendicular to the direction of the vane’s motion. The former is an effort doing work on the vane. The latter is a lateral force which does no work.
P = R sin 1⁄2φ = (G/g) bt (v − u)2 (1 − cos φ);
L = R cos 1⁄2φ = (G/g) bt (v − u)2 sin φ.
| Fig. 162. |
| Fig. 163. |
The work done by the jet on the vane is Pu = (G/g) btu (v − u)2(1 − cos φ), which is a maximum when u = 1⁄3v. This result can also be obtained by considering that the work done on the plane must be equal to the energy lost by the water, when friction is neglected.
If φ = 180°, cos φ = −1, 1 − cos φ = 2; then P = 2(G/g) bt (v − u)2, the same result as for a concave cup.
§ 161. Position which a Movable Plane takes in Flowing Water.—When a rectangular plane, movable about an axis parallel to one of its sides, is placed in an indefinite current of fluid, it takes a position such that the resultant of the normal pressures on the two sides of the axis passes through the axis. If, therefore, planes pivoted so that the ratio a/b (fig. 163) is varied are placed in water, and the angle they make with the direction of the stream is observed, the position of the resultant of the pressures on the plane is determined for different angular positions. Experiments of this kind have been made by Hagen. Some of his results are given in the following table:—
| Larger plane. | Smaller Plane. | |
| a/b = 1.0 | φ = ... | φ = 90° |
| 0.9 | 75° | 721⁄2° |
| 0.8 | 60° | 57° |
| 0.7 | 48° | 43° |
| 0.6 | 25° | 29° |
| 0.5 | 13° | 13° |
| 0.4 | 8° | 61⁄2° |
| 0.3 | 6° | .. |
| 0.2 | 4° | .. |