40 APPLIED MECHANICS 
: 10 : 
Therefore, sin 0= 50°75 * 098481 = 0°4746, 
and 0 = 28° 20’. 
(2) The lines of action of four forces, P, Q, 8, and T, are as shown at 
(a), Fig. 37. The magnitudes of P and Q are 10 and 20 units respec- 
tively, and the four forces are 
in equilibrium. It is required 
to find the magnitudes and 
senses of S and T. 
Drawing the polygon of 
forces shown at (0), the senses 
of S and T are seen at once, 
Projecting the sides of 
the polygon on to the hori- 
zontal, it is evident that the 
projection of T is equal to 
the projection of S minus the projection of P plus the projection of Q, 
or T cos 30° = 8 cos 45° — 10 cos 60° + 20 cos 30°... (A) 
Projecting the sides of the polygon on to the vertical, it is evident 
that the projection of T plus the projection of Q is equal to the projec- 
tion of S plus the projection of P, : 
or — Tsin 30° + 20sin 30°=Ssin 45°+10sin60°.. . . (ii) 
Solving the equations (i) and (ii), 
T=10(24+.,/3) =37°32, and S = 20,/2 = 28°28. 
(3) P,,P.,P3, ete., are forces acting at a 
point O (Fig. 38), and their lines of action 
are inclined to a horizontal axis OX at angles 
0,, 9, 93, etc., respectively. Produce XO to Xx 
X,, and draw the vertical axis YOY, Re- 
solve each force into two components, one 
along the horizontal axis and the other 
along the vertical axis. The horizontal Yi 
components are, P, cos 6,, P, cos 6,, P; cos 9; Fig. 38. 
etc. ; and the vertical components are, P, sin 6, P, sin @,, P, sin 0g, ete. 
The resultant of the horizontal components is, 
P, cos 6,+ P,cos 0,+P, cos O,+etc. . . .=2(P cos 6). 
The resultant of the vertical components is, 
P, sin 6,+P, sin 6,+ P; sin #,+ etc. . . .=2(P sin 0). 
If R is the resultant of all the forces, then _ 
R?= {>(P cos 6)? + {2(P sin 6)!2, and the line of action of R makes an 
: _ &(P sin 0) 
angle @ with OX such that tan O= rca | 
If the forces P,,P,,P,, etc.,are in equilibrium,then R =0, 2(P cos @) = 0, 
and 2(P sin #)=0. 
In applying the foregoing equations to numerical examples care must 
be taken to give the proper algebraical sign (+ or —) to each quantity. 
