Principle of Resolution
It states – “The algebraic sum of the resolved components of a number of forces, through a given path, is equal to their resultant resolved part in the same direction."
Referring to the figure, consider a couple of forces denoted as P and Q. Let the forces be represented in magnitude and direction through the sides OA and OB of the shown parallelogram respectively.
Here, it’s known that the diagonal OC represents the resultant of the two forces P and Q in magnitude and direction.
Let OX represent the direction needed for resolving the forces.
Drawing from the points A, B and C on OX, perpendicular lines AL, BM and CN, also from point A another perpendicular line AT on CN, we get two triangles OBM and ACT.
The two sides AC and OB are same in magnitude and are parallel. Also, OM is parallel to AT.
Therefore, OM = AT = LN.
The geometry of the drawing suggests, ON = OL + LN = OL + OM, since LN = OM.
However, ON is the part that’s resolved of the resultant R.
Similarly OL is the part resolved of P and OM of the force Q.
Therefore, finally we get the resolved part of OX and R = Resolved part of P along OX + Resolved part of Q along OX.
In the above example, for the sake of easier understanding, we took just a couple of forces; however the concept may be applied for any number of forces.
The magnitude and the direction of the resultant of the forces may be determined by:
- Analytical Method
- Graphical Method
Analytical Method – The following points identifies how the resultant force may be calculated using the analytical method:
Resolve the given forces vertically and calculate the algebraic total of all the vertical parts or Σ V.
Resolve the given forces horizontally and calculate the algebraic total of all the horizontal parts or Σ H.
The resultant force R of all the given forces can be expressed as:
R = (Σ V)^2 + (√ H)^2
The inclination of the above resultant force will be at an angle Ɵ with the horizontal component and may be expressed as:
tan Ɵ = √ΣV / √ΣH
For a given system of forces, the resultant may be identified graphically through any of the following methods:
- By Triangle Law of Forces,
- By Polygon Law of Forces,
- By the Vector Method