Resolution of a force means “finding the components of a given force in two given directions.”
Let a given force be R which makes an angle e with X-axis as shown in Fig. 1.21. It is required to find the components of the force R along X-axis and Y-axis.
Components of R along X-axis = R cos θ.
Components of R along Y-axis = R sin θ.
Hence, the resolution of forces is the process of finding components of forces in specified directions.
1.8.1. Resolution of a Number of Coplanar Forces. Let a number of coplanar forces (forces acting in one plane are called co-planar forces) R1′ R2 , R3, . . are acting at a point as shown in Fig. 1.22.
Let θ1 = Angle made by R1 with X-axis
θ2 =Angle made by R2 with X-axis
θ3 = Angle made by R3 with X-axis
H =Resultant component of all forces along X-axis
V = Resultant component of all forces along Y-axis
R =Resultant of all forces
θ = Angle made by resultant with X-axis.
Each force can be resolved into two components, one along X-axis and other along Y-axis.
Component of R1 along X-axis = R1 cos θ1
Component of R1 along Y-axis = R1 sin θ1.
Similarly, the components of R2 and R3 along X-axis and Y-axis are (R2 cos θ2, R2 sin θ2) and (R3 cos θ3, R3 sin θ3) respectively . .
Resultant components along X-axis
= Sum of components of all forces along X-axis.
H = R1 cos 91 + R2 cos θ2 + R3 cos θ3 + …
Resultant component along Y-axis.
= Sum of components of all forces along Y-axis.
V = R1 sin θ1 + R2 sin 92 + R 3 sin θ3 + …
Then resultant of all the forces, R =
The angle made b)l R with X-axis is given by, tan θ = V / H
1.8.2. Moment of a Force. The product of a force and the perpendicular distance of the line of action of the force from a point is known as moment of the force about that point.
P =A force acting on a body as shown in Fig. 1.27.
r = Perpendicular distance between the point 0 and line of action of the force P.
The moment of the force P about 0 = P x r
The tendency of the moment P x r is to rotate the body in the clockwise direction about 0.
Hence this moment is called clockwise moment. If the tendency of rotation is anti-clockwise, the moment is called anti-clockwise moments.
1.8.3. Units of Moment. In M.K.S. system the moment is expressed as kgf m whereas in S.I. system, moment is expressed as newton metre (N m).
1.8.4. Effect of Force and Moment on a Body. The force acting on a body causes linear displacement while moment causes an angular displacement. Hence a body when acted by a number of coplanar forces will be in equilibrium if:
(i) Resultant component of forces along any direction is zero i.e., resultant component of forces in the direction of x, in the direction of y and in the direction of z are zero.
(ii) Resultant moments of the forces about any point in the plane of the forces is zero or clockwise moment is equal to anti-clockwise moments.
Note. If the resultant component of forces acting on a body along any direction is zero but the net moment of the forces about any point is not zero the body will not be in equilibrium. The body will have the tendency to rotate about the point.