Coplanar concurrent forces are those which are lying in the same plane and acting at the same point. The forces can be represnted in scalar form (magnitude and angle) or in a cartesian form(components along x, y, z axes alongwith unit vectors).
Scalar Form
To find the resultant of the system of coplanar concurrent forces the following steps should be followed.
1. Resolve all the forces into their rectangular components along x-axis and y-axis.
Note: Component of a force F along x-axis is F cos θ, (where θ is the angle between the force and the positive x-axis), and the component of a force F along y-axis is F sin θ.
2. Take the algebraic sum of the components along x-axis and y-axis
Σ Fx and Σ Fy.
3. Resultant of the force system will be calculated as
Magnitude, R= sqrt{(Fx)(Fx)+(Fy)(Fy)}
Direction, Angle α =Tan(-1){(Σ Fy/Σ Fx )}
Cartesian form
The forces are commonly resprented in the cartesian form with their x, y and z components as;
F=fx i + fy j + fz k
where fx, fy and fz are the components of the force along x, y and z axis and i, j, k are the unit vectors in the direction of x, y,z axis respectively, e.g., F=2 i + 3 j + 4 k
To find the resultant in the cartesian form, follow the following steps.
1. Resultant R=(Σ fx) i + (Σ fy) j + (Σ fz) k
where Σ fx is the algebraic sum of the x-components of all the forces
Σ fy is the algebraic sum of the y-components of all the forces
Σ fz is the algebraic sum of the z-components of all the forces
2. The magnitude and direction of the resultant will be calculatd in the normal way as for any vector.
For more on Engineering Mechanics visit http://civilengineer.webinfolist.com/mechanics.htm
or refer to Engineering Mechanics - Statics (11th Edition) by Hibbeler.
