20 APPLIED MECHANICS 
32. Resultant Centrifugal Force of Two Small Revolving Masses. 
—tLet A and B (Fig. 17) be two small masses of weight w, and w, — 
respectively, revolving in the same 
plane with uniform angular velocity 
about the centre O. The centrifugal 
2, 
ww, . a 
1-1 in the direc- 
force of A is F,= 
tion OA. The centrifugal force of 
2 
B is Fo in the direction OB. 
Make OC = F,, and draw CD parallel to 
OB and equal to F,. Join OD, then OD will be the direction and 
magnitude of F, the resultant centrifugal force.of A and B. Join AB, 
cutting OD at E. Draw EH perpendicular to OA, EK perpendicular to 
OB, and EL parallel to CD or OB. Since E is a point in the resultant 
of the forces F, and F,, the moment of F, about E must be equal to the 
moment of I", about E, 
; 2, 2 
therefore  —11. BH = 2" "2. EK, 
g 
ae w, 1,:EK area of triangle OBE BE 
w, 1,°EH area of triangle OAE AE’ 
Therefore E is the centre of gravity of A and B. 
ale CO BE. aay - Wr; 
Again, OA = BA = w, + w, therefore OL = w,+u, + Wy, 
Also, OD _OC _Fy(w, + wy) _ wyw*ry (07, +) _ (0, + 9)u? - 
OE OL Wy"; J"; g 
2 
But OD=F, and if OB=r; then FoC1 "That ia, Phe 
g 
resultant centrifugal force of the two masses A and B is the centrifugal 
force of the sum of the masses concentrated at their centre of gravity. 
33. Centrifugal Force of a Thin Plate revolving about an Axis 
Perpendicular to its Plane.—If the plate be divided into a large 
number of small parts of weights w,, w., ws, etc., then by the preceding 
Article the resultant centrifugal force of the parts w, and w, is the same 
as if these parts were concentrated at their centre of gravity. Again, 
the resultant centrifugal force of w, and w, (at their centre of gravity) 
and w, will be the same as if w,, w,, and w, were concentrated at their 
centre of gravity. Proceeding in this way until all the masses have been 
included, it is evident that the centrifugal force of the whole plate will 
be the same as the centrifugal force of the whole mass concentrated at 
its centre of gravity. 
34. Extension of the Foregoing to Certain Solids.—If a solid can 
be built up of a number of thin plates, the centres of gravity of which 
all lie on a line parallel to the axis of revolution, then it is easy to see 
that the centrifugal force of the whole solid is the same as if the whole 
mass were concentrated at its centre of gravity. . 
35. Moment of a Force.—The moment of a force about a point, or 
about an axis perpendicular to it, is the product of the magnitude of 
the force and its perpendicular distance from the point or axis. This 
moment is called a torque. If the distance is measured in inches and 
