CHAPTER XXVI 
BALANCING 
344. Centrifugal Force of Revolving Mass—Equivalent Mass at 
Different Radius.—If a mass A (Fig. 678) of weight W be attached to 
a straight arm OA at a distance 7 from an axis O about which the arm 
revolves with an angular velocity , the centrifugal force of A is bie 
(Art. 31, p. 19). If the mass A be 
removed and another mass B of weight 
W, be attached to the arm at a distance 
r, from the same axis O about which it 
revolves with the same angular velocity 
w, then the centrifugal force of B is 
W077, 
~~ \ 
~ 
- ue at 
actly | 
If the centrifugal force of B 
Fig. 678. 
g 
is equal to that of A, then 
2 2 
We ae Wet; , that is, W,7, = Wr, and when this relation holds, the mass 
g ; 
g : 
5B at the radius 7, is said to be equivalent to the mass A at the radius 7. 
345. Balancing one Revolving Mass by Another.—Referring - to 
Fig. 678, the centrifugal force of the mass A revolving about the axis O 
causes a tension in the arm OA, and this force will be transmitted to the 
bearings of the axle carrying the arm. As the arm revolves the direction 
of the forces on the bearings due to the centrifugal force of A will be 
continually changing, with the result that serious vibrations may be set 
up in the framing carrying the bearings, and through the framing the 
vibrations will extend to the foundations. If, however, a mass ©. of 
weight W, be placed on the arm produced beyond the axis O; and at 
a distance 7, from O, so that A and C are on opposite sides of O, and if 
W,r, = Wr, then the centrifugal force of C will cause a pull at O equal 
and opposite to the pull caused by the centrifugal force of A. The 
revolving masses will then balance one another, and there will be no 
straining actions on the bearings of the axle or the frame carrying them 
due to the centrifugal forces. 
346. Balancing any Number of Revolving Masses by means of 
one Mass, all the Masses being in the same Plane of Revolution.— 
Let A, B, C, ete. (Fig. 679), be a number of masses of weights W,, W., 
W3;, etc., respectively at distances 7, 73, 73, etc., respectively. from an 
axis O about which they revolve in the same plane with angular velocity 
; it is required to balance these by one mass in the plane of the others. 
Let W denote the weight of a mass X at a radius 7 which will 
balance the given masses. The centrifugal forces of the given masses 
; 414 
—S—S Oe 
