In this section, we will develop both the necessary and sufficient conditions for the equilibrium of the rigid body. As shown, the body is subjected to an external force and couple moment system that is the result of the effects of gravitational, electrical, magnetic, or contact forces caused by adjacent bodies.
This type of force and couple system is often referred to as a two-dimensional or coplanar force system. For example, the airplane has a plane of symmetry through its center axis, and so the loads acting on the airplane are symmetrical with respect to this plane.
Thus, each of the two wing tires will support the same load $\mathbf{T}$, which is represented on the side (two-dimensional) view of the plane as $2\mathbf{T}$.
Successful application of the equations of equilibrium requires a complete specification of all the known and unknown external forces that act on the body. The best way to account for these forces is to draw a free-body diagram. This diagram is a sketch of the outlined shape of the body, which represents it as being isolated or "free" from its surroundings, i.e., a "free body." On this sketch it is necessary to show all the forces and couple moments that the surroundings exert on the body so that these effects can be accounted for when the equations of equilibrium are applied. A thorough understanding of how to draw a free-body diagram is of primary importance for solving problems in mechanics.
We will first consider the various types of reactions that occur at supports and points of contact between bodies subjected to coplanar force systems. As a general rule,
Let us consider three ways in which a horizontal member, such as a beam, is supported at its end. One method consists of a roller or cylinder. Since this support only prevents the beam from translating in the vertical direction, the roller will only exert a force on the beam in this direction.
The beam can be supported in a more restrictive manner by using a pin. The pin passes through a hole in the beam and two leaves which are fixed to the ground. Here the pin can prevent translation of the beam in any direction $\phi$, and so the pin must exert a force $\mathbf{F}$ on the beam in the opposite direction. For purposes of analysis, it is generally easier to represent this resultant force $\mathbf{F}$ by its two rectangular components $\mathbf{F}_x$ and $\mathbf{F}_y$. If $F_x$ and $F_y$ are known, then $F$ and $\phi$ can be calculated.
The most restrictive way to support the beam would be to use a fixed support. This support will prevent both translation and rotation of the beam. To do this a force and couple moment must be developed on the beam at its point of connection. As in the case of the pin, the force is usually represented by its rectangular components $\mathbf{F}_x$ and $\mathbf{F}_y$.
When a body is within a gravitational field, then each of its particles has a specified weight. A system of forces can be reduced to a single resultant force acting through a specified point. We refer to this force resultant as the weight $\mathbf{W}$ of the body and to the location of its point of application as the center of gravity. In the examples and problems that follow, if the weight of the body is important for the analysis, this force will be reported in the problem statement.
Also, when the body is uniform or made from the same material, the center of gravity will be located at the body’s geometric center or centroid; however, if the body consists of a nonuniform distribution of material, or has an unusual shape, then the location of its center of gravity $G$ will be given.
We developed the two equations which are both necessary and sufficient for the equilibrium of a rigid body, namely, $\Sigma\mathbf{F} = 0$ and $\Sigma\mathbf{M}_O = 0$. When the body is subjected to a system of forces, which all lie in the $x$–$y$ plane, then the forces can be resolved into their x and y components. Consequently, the conditions for equilibrium in two dimensions are
