| Fig. 146. |
| Fig. 147. |
Suppose the apparatus arranged so that the disk x is at that level in the stream where the velocity is to be determined. The plane abcd is placed parallel to the direction of motion of the water. Then the disk x (acting as a rudder) will place itself parallel to the stream on the down stream side of the frame. The torsion rod will be unstrained, and the needle will be at zero on the graduated circle. If, then, the instrument is turned by pressing the needle, till the plane abcd of the disk and the zero of the graduated circle is at right angles to the stream, the torsion rod will be twisted through an angle which measures the normal impulse of the stream on the disk x. That angle will be given by the distance of the needle from zero. Observation shows that the velocity of the water at a given point is not constant. It varies between limits more or less wide. When the apparatus is nearly in its right position, the set screw at g is made to clamp the torsion spring. Then the needle is fixed, and the apparatus carrying the graduated circle oscillates. It is not, then, difficult to note the mean angle marked by the needle.
Let r be the radius of the torsion rod, l its length from the needle over ef to r, and α the observed torsion angle. Then the moment of the couple due to the molecular forces in the torsion rod is
M = EtIα / l;
where Et is the modulus of elasticity for torsion, and I the polar moment of inertia of the section of the rod. If the rod is of circular section, I = 1⁄2πr4. Let R be the radius of the disk, and b its leverage, or the distance of its centre from the axis of the torsion rod. The moment of the pressure of the water on the disk is
Fb = kb (G / 2g) πR2v2,
where G is the heaviness of water and k an experimental coefficient. Then
EtIα / l = kb (G / 2g) πR2v2.
For any given instrument,