BALANCING 417 
and B, and the arms to which they are attached projected on to them. 
_ By the preceding Article the forces A, and A, acting in the planes 1 and 
2 which will balance the centrifugal force of the mass A are determined. 
Also by the same Article the forces B, and B, acting in the planes 
1 and 2 which will balance the centrifugal force of the mass B are 
F determined. By the triangle of forces (Figs. 684 and 686) R,, the 
resultant of A, and B,, and R,, the resultant of A, and B,, are determined, 
_. and R, and R, are the centrifugal forces of the masses P and Q respec- 
tively. Hence if the radii at which the masses P and Q act are fixed, 
the weights of P and Q can be found. 
Tf there are more than two given masses to be balanced the pro- 
cedure is the same, but instead of the triangle of forces, the polygon of 
forces will be used to find R, and R,. 
In solving this problem it is desirable to tabulate the working, as 
shown in the form below :— 
; Centri Balancing Forces 
foals Distance when w*=g. 
Weight : of Mass 
Mass. | of Mass. — ber from 
w=g Plane 2. | In Plane 1. | In Plane 2. 
Wr ¢ F,= Wt F,=Wr- Fy. 
A 
B 
Cc 
at ase Sra meee? 0: ten eg ee Se dl ei es we heh Ce lerece 0: 6; Meteiece eie.« 
P w, =R,/r r" wr, = R, U R, ears 
Q We = Ro/r T; Wo = Ro 0 petals Re : 
Care must be taken to give the proper signs to the forces in the last 
two columns. When the mass in the first column lies between planes 1 
and 2, then F’, and F, have both the same sign ; but when either plane is 
between the mass and the other plane, F, and F, have opposite signs. 
R, and R, are obtained from the polygon of forces in planes 1 and 2 
respectively. 
If the planes 1 and 2 (Fig. 685) are the central transverse planes of 
the bearings of the shaft carrying the given masses, then, when these 
masses are unbalanced by other revolving masses, the centrifugal forces 
of the masses P and Q, determined as above, will be the forces exerted 
by the bearings on the shaft due to the centrifugal forces of the given 
Masses. 
Exercises XXVIa. 
7 1. Three masses, A, B, and C, revolve in the same plane about an axis which 
cuts the plane of revolution at O. The centres of gravity of A, B, and C are 15 
_ inches, 18 inches, and 20 inches respectively from O, and the angle AOB is 90°. 
_ The weights of A and B are 80 Ibs. and 50 Ibs. respectively. Find the weight 
of a the angle BOC in order that the revolving masses may balance one 
another. 
__ 2, Two masses, of 10 lbs, and 20 lbs, respectively, are attached to a balanced 
disc at an angular distance apart of 90°, and at radii 2 feet and 3 feet respec- 
2D 
rN Pee 
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