3.5 : Coplanar Forces

Consider an object upon which multiple forces are acting. If the lines of action of each force lie within the same plane, the system can be considered coplanar. The Cartesian vector form can be used to resolve each force into its respective components. For a coplanar system, the system will be in equilibrium if each component of the resultant force equals zero and the resultant force on the system is zero. If the sum of the forces is not equal to zero, then the object will not be in equilibrium and will accelerate in the direction of the net force.

To solve problems involving coplanar forces and equilibrium, all the forces acting on the object, along with their magnitudes and directions, must be identified. These forces can then be resolved into their x and y components using vector algebra. The equations of equilibrium are used to calculate the unknown forces and check whether the object is in equilibrium or not.

Consider a box in a state of equilibrium suspended by three strings as shown in the figure

This is a coplanar system because the forces are acting in a single plane. Two of the diagonal strings, connected to the plank, have equal angles with the plank, and the tension force in these two strings is known. In order to determine the magnitude of the tension force in the third vertical string, the equilibrium equations can be applied. Resolving the known tension forces of the two strings into their components shows that the horizontal components counterbalance each other. Therefore, the tension in the third string can be determined by rearranging the equation for the vertical components. Once the tension in the third string has been determined, the equilibrium equations can be used to check that the system is, indeed, in a state of equilibrium.