A resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body. The defining feature of a resultant force, or resultant force-torque, is that it has the same effect on the rigid body as the original system of forces.
The point of application of the resultant force determines its associated torque. The term resultant force should be understood to refer to both the forces and torques acting on a rigid body, which is why some use the term resultant force-torque.
The diagram illustrates simple graphical methods for finding the line of application of the resultant force of simple planar systems.
A force applied to a body has a point of application. The effect of the force is different for different points of application. For this reason a force is called a bound vector, which means that it is bound to its point of application.
Forces applied at the same point can be added together to obtain the same effect on the body. However, forces with different points of application cannot be added together and maintain the same effect on the body.
It is a simple matter to change the point of application of a force by introducing equal and opposite forces at two different points of application that produce a pure torque on the body. In this way, all of the forces acting on a body can be moved to the same point of application with associated torques.
A system of forces on a rigid body are combined by moving the forces to the same point of application and computing the associated torques. The sum of these forces and torques yields the resultant force-torque.
If a point R is selected as the point of application of the resultant force F of a system of n forces Fi then the associated torque T is determined from the formulas
It is useful to note that the point of application R of the resultant force may be anywhere along the line of action of F without changing the value of the associated torque. To see this add the vector kF to the point of application R in the calculation of the associated torque,
The right side of this equation can be separated into the original;formula for T plus the additional term including kF,
because the second term is zero. To see this notice that F is the sum of the vectors Fi which yields
thus the value of the associated torque is unchanged.
It is useful to consider whether there is a point of application R such that the associated torque is zero. This point is defined by the property
where F is resultant force and Fi form the system of forces.
Notice that this equation for R has a solution only if the sum of the individual torques on the right side yield a vector that is perpendicular to F. Thus, the condition that a system of forces has a torque-free resultant can be written as
If this condition is satisfied then there is a point of application for the resultant which results in a pure force. If this condition is not satisfied, then the system of forces includes a pure torque for every point of application.
The forces and torques acting on a rigid body can be assembled into the pair of vectors called a wrench.If a system of forces and torques has a net resultant force F and a net resultant torque T, then the entire system can be replaced by a force F and an arbitrarily located couple that yields a torque of T. In general, if F and T are orthogonal, it is possible to derive a radial vector R such that , meaning that the single force F, acting at displacement R, can replace the system. If the system is zero-force (torque only), it is termed a screw and is mathematically formulated as screw theory.
The resultant force and torque on a rigid body obtained from a system of forces Fi i=1,...,n, is simply the sum of the individual wrenches Wi, that is
Notice that the case of two equal but opposite forces F and -F acting at points A and B respectively, yields the resultant W=(F-F, A×F - B× F) = (0, (A-B)×F). This shows that wrenches of the form W=(0, T) can be interpreted as pure torques.