Find the general resultant of a system of coplanar forces

Principles of Coplanar Forces

Any system of coplanar forces can be reduced to a single resultant force, which will have the same external effect as all the separate forces acting together.
To find the resultant of a system of coplanar forces, first we have to resolve each force into components in any two perpendicular directions. Commonly, we use the horizontal (x) and vertical (y) directions.

Resolving a Force into Components

To resolve a force into its components, we need to use trigonometry. The x-component is calculated by multiplying the force magnitude by cos(θ), and the y-component by sin(θ), where θ is the angle between the force and the x-axis.
Once we have done this for all the forces, we add up all the x-components to get the resultant force in the x-direction (R_x), and all the y-components to get the resultant force in the y-direction (R_y).

Calculating the Resultant Force

The resultant force (R) is then found using Pythagoras’ theorem: R = √(R_x^2 + R_y^2).
The direction of the resultant force can be calculated by finding the angle to the x-axis using the tan inverse function: θ = tan^-1(R_y/R_x).
Remember that units of force should be consistent when doing these calculations.
Also pay attention to the direction of the forces when resolving them into components, use negative values for forces acting in opposite directions.

Practical Applications

The method of resolving forces is essential for understanding how forces distribute in rigid bodies under equilibrium or not.
It’s also applicable in many engineering and physics problems, such as analyzing forces in bridges and trusses, vehicle dynamics, fluid motion and many more.

When making progress in understanding force systems in two dimensions, be encouraged to seek practice problems and past paper questions. Developing accuracy in mathematical calculation, as well as fluency in physical interpretation of results, are the main goals of mastering this crucial topic in Applied Mathematics.