Now that is determined, you can use any of the equations to determine the magnitude, of the force. Solving the system of equations by the substitution method (substitute for ), you can write, Leading to the following system of equations,

The Cartesian axes set the positive direction for the equilibrium equations. The following examples present the approach of assuming the direction of an unknown force. In such a case, the sense of direction (arrowhead) of has to be switched. Solving a system of equations with as its unknowns may lead to a negative value for. Therefore, the unknowns of the equilibrium equations are (the force magnitude) and. To this end, an (unknown) angle,, from an axis quantifies the direction of the force. If the direction of a force is unknown (in addition to its magnitude), the direction, including the sense of direction, of the force should be assumed. In other problems, directions can also be the unknowns. angles of the cables) and nature of the cables (can only be in tension). Note that the directions of the forces were already known by the geometrical setting (i.e. In the previous example, the unknowns were the magnitudes of two internal cable forces. 5.2) for a system of forces acting on a particle in equilibrium, indicating there can be maximum two unknowns. In two-dimensional problems, there are two equations of equilibrium (Eq. Solve the system of equations for the unknowns. Where the non-bold capital letters are magnitudes and hence non negative.Īpply the equilibrium equation in each direction. Determine the magnitudes of the components of the (known and unknown) forces by decomposing them into their Cartesian components. However, if the particle (ring) is in equilibrium, the equations of equilibrium are utilized to determine the unknown forces. For example, The cable forces in the case of the ring shown in Fig.

The equations of equilibrium can be used to determine unknown internal forces or support reactions in a problem. Accordingly, the magnitudes of force components in the negative direction of their corresponding Cartesian axes appear with minus signs in the sum.Įquations 5.3 can be referred to as the scalar Cartesian equations of equilibrium. Thereby, the magnitudes of force components in the positive direction of their corresponding Cartesian axes appear with positive signs in the sum. The arrows and indicate the positive direction of the x and y axes respectively. Where the sum is over all forces acting on the particle, and each and is the magnitude of a force component. Based on the magnitude-based notation and scalar formulation introduced in the beginning of Section 3.2, Eq. Therefore, they can be written in terms of the magnitudes of the force components. Meaning that equilibrium of a particle subjected to forces is maintained if the sum of the force components in each direction is zero.Įquations 5.2 state that the magnitude of the resultant force is zero. 5.1 for the particle in equilibrium is written as, If each force is resolved into two perpendicular components in the x and y directions of a Cartesian coordinate system, Eq. Let be a system of coplanar forces acting on a particle. Equations of equilibrium for coplanar force systems (two dimensions) The equation of equilibrium can be separately considered for coplanar and spatial force systems. In Statics, mainly non-moving (at-rest) bodies are considered in other words, static equilibrium of a stationary body or structure is concerned. The equation of equilibrium includes the necessary and sufficient conditions for a particle to be in equilibrium.

According to Newton’s first law of motion, if a particle is in equilibrium, the resultant forces of all the force acting on it must be zero, expressed as the equation of equilibrium (of a particle), Relationships between Load, Shear, and MomentsĪ particle is said to be in equilibrium if it was originally at rest or moving along a straight line with constant velocity, and remains so.Shear and moment equations and their diagrams.Conditions for two dimensional rigid-body equilibrium.Equilibrium of Particles and Rigid Bodies.Simplification of force and couple systems.Vector operations using Cartesian vector notation.Vector operations using the parallelogram rule and trigonometry.

Statics equilibrium 3d problem