If the value of the couple U is constant, it is seen that w varies inversely with w1. That is, if w is large w, is small, and if w is small w1 is large. This was pointed out in Art. 122 and illustrated in the case of the top when it is dying down. As the spin decreases in such cases, the precession increases.

127. Precessional Moment; General Case. The precessional moment for any body when OZ is perpendicular to OY may be obtained from a consideration of the moment equation (Art. 126) by retaining the dM. The equation may then be written

be

changed slightly. To make the ideas clear, let the original axis OZ be drawn as well as the inclined axis OZ' making the angle & with OY, as in Fig. 147. Let p the distance of an elementary dM from OZ, as before, and P1 the distance of dM from OZ. Then p = P1 sind, and the expression for the velocity, v=pw1 = rw1 sin a becomes

therefore

or

dP1 = a1dM= dмrww sin & cos a;

U-SdP (rcos a)=SdMau sin &r coa = 0, sin & SdM(r cos a),

The gyroscopic action of the fly wheel of an automobile has much to do in causing the machine to overturn when rounding sharp curves at high speeds. Even when the machine is not overturned, the gyroscopic moment due to the rotation of the wheel causes an extra pressure on the bearings. This pressure is shown by the wear on the

bearings. It is left as an exercise to determine the overturning moment due to the action of an automobile fly It will also interest the

wheel under assumed conditions.

student to know that a German torpedo boat 116 ft. long, and 56 tons displacement, was held upright in a heavy sea by an 1100 lb. disk rotating 1600 revolutions per

minute.

Problem 189. A locomotive is going at the rate of 40 mi. per hour around a curve of 600 ft. radius. The diameter of the drivers is 80 in., and a pair of drivers and axle have a moment of inertia about an axis midway between the wheels and perpendicular to the axle of 3000. What is the magnitude of the couple introduced by the precessional motion of this pair of wheels? Give the direction in which it acts. Does it tend to make the locomotive tip inward or outward?

Problem 190. A car pulled by the locomotive in the preceding problem has four pairs of wheels. The moment of inertia of each pair of wheels and their connecting axle, with respect to an axis midway between the wheels and perpendicular to the axle, is 320 (see problem 87). What is the magnitude of the precessional couple acting upon the whole car?

Problem 191. The fly wheel of an engine on board a ship makes 300 revolutions per minute. The rim has the following dimensions: outside radius 4 ft., inside radius 3 ft., width 12 in. The ship rolls with an angular velocity of a radian per second; find the torque acting on the ship due to the gyrostatic action of the fly wheel.

Problem 192. A conical top is made of wood and is spinning about its axis with a velocity of 20 revolutions per second. The cone has a base of 2 in. and a height of 2 in., and spins on the apex. While spinning steadily with its axis vertical (sleeping), it is disturbed by a blow so that its axis is inclined at an angle of 30° with the vertical. Find the velocity of precession and the torque U that tends to keep the top from falling. See Fig. 143.

128. Car on Single Rail. An interesting application of the gyroscope has been made recently in England. A car (see Fig. 148) is run upon a single rail, and is held upright by means of rapidly rotating fly wheels. Each car contains two of these wheels rotating in opposite directions, at the rate of 8000 revolutions per minute.

Any tendency of the car to tip over, either when running or standing at the station, is righted by the gyroscopic action of the fly wheels. The experimental car

FIG. 148

was so successful in operation that it maintained itself in an upright position even when loaded eccentrically. The action of the fly wheels is such as to place the center of gravity of the car and load directly over the rails.

(NOTE. See Engineering, June 7, 1907.)

CHAPTER XIII

WORK AND ENERGY

129. Definitions. When the forces acting upon a body cause a motion of that body, work is done. We define the work done by a force as the magnitude of the force times the distance through which the body, upon which it acts, moves along its line of action.

This definition may be less exactly stated by saying that a force acting on a body that moves through a distance does work. This brings to mind the forces considered as acting in Chapters II, III, and VI, where no motion was produced; that is, where the point of application did not move. Such forces produce no work according to our definition. To make the idea of work clearer, suppose the body C (Fig. 149) to be acted upon by a force P, and

A

FIG. 149

that the body is moved until the point at A is finally at B. The work done by P is P times AB. Suppose the plane upon which moves is rough, so that it offers a resistance F. In passing over the distance AB, the force F does a work of resistance equal to F times AB.

229