GOVERNORS 341 
For the Porter governor, H,K, (Fig. 521) is made equal to my 4. w, 
where m is the ratio of the motion of the sleeve to the vertical motion of 
the balls. 
The ratio of the vertical motion of the balls to that of the sleeve is 
easily found from a diagram such as that shown in Fig. 530, where OA 
is the pendulum, and AC the link connecting it to 
the sleeve. Produce OA to meet the horizontal line 
through C at O,, then O, is the instantaneous centre 
for the link AC in the given position, and if V, =the 
velocity of A in the direction at right angles to OA, 
and V,=the velocity of C in the vertical direction, 
then vy = 0,4 . Draw the vertical line AD; then if V, 
is the vertical component of V,, it is easily proved that 
Vi 9.) tt the link Decupies the position A’C’. Fra. 680. 
Vz O,C 
where A’C’ is parallel to AC, then vi os ae x ° where V, now denotes 
the vertical velocity of C’. Pibicn 2 
Exercises XX. 
1. Plot on squared paper the height, in inches, and revolutions per minute, 
for a simple conical pendulum, from 30 to 120 revolutions per minute. Scales.— 
1 inch to 10 inches, and 1 inch to 20 revolutions per minute. 
2. If the arm of a simple conical pendulum is 15} inches long, what will be. 
its inclination to the axis when running at 50 revolutions per minute, and what 
will it be at 60 revolutions per minute? Also, what will its speed be, in revolu- 
tions per minute, when its inclination to the axis is 30°, and what will it be when 
the inclination is 45°? 
3. A simple conical pendulum is running at 60 revolutions per minute 5 
what is the decrease in height if the speed is increased 5 per cent., and what is 
the increase in height if the speed is decreased 5 per cent, ? 
4. Add to the diagram drawn in answer to Exercise 1 the curve showing 
the relation of height to speed for a conical pendulum when the weight of the 
arm is half the weight of the ball. Assume weight of arm per inch of length 
to be uniform. 
5. Referring to Figs. 511, 512, and 513, p. 329, the length of the arm AB is 
10 inches for Fig. 511, 8 inches for Fig. 512, and 12 inches for Fig. 5138. The 
distance of B from the axis OY is 1 inch for Figs. 512 and 513. Starting in each 
case from the position in which the’ arm is inclined at 30° to the axis, calculate 
the percentage increase in speed for a rise of 1 inch in level of the balls, Draw 
the figures half full size for each position. 
6. Find the answers to the preceding exercise when the weight of the arm is 
taken into account, The weight of the ball in each case is 6 Ibs., and the weight 
of the arm is 1°75 lbs. for Fig. 511, 1°5 Ibs. for Fig. 512, and 2 lbs. for Fig. 513, 
7. Draw the speed curves, as in Fig. 527, p. 335, for the pendulums of 
Exercise 5, (1) neglecting friction, (2) taking friction into account, the amount 
of the friction being equivalent to a vertical force of 1 lb. at the centre of each 
ball. The weight of each ball is 6 lbs. 
8. In a direct loaded governor (Fig. 521, p. 333) the arms are 10 inches long. 
Each ball weighs 4 lbs., and the load is 75 lbs. The sleeve is in its lowest 
ition when the arms are inclined at 27° to the axis. The lift of the sleeve is 
inch, What is the force of friction at the sleeve if the speed at the beginning 
