### Force. Equivalence of forces. Resultant force. Addition of forces. Decomposition of force.

The force is characterized by the point of its application to the body, its spatial direction and its numerical value, and this gives the basis for considering it as a vector argument.

But the force cannot be completely identified with such mathematical concept as a vector. A vector can be displaced in space parallel to itself and after that it definitely remains the same vector. This means that in mathematics we deal with so called *free vectors*. The operations with such vectors are studied in mathematical courses. The operation of the addition of two vectors according to the well-known parallelogram rule is one of the important operations.

However, let's try to transfer the force parallel to itself, that is, to displace the point of application of the force. You can see, that the character of the motion of the body will change. For example, let's try to drag the rope attached to one of the chair's legs, and after that try to drag the rope attached to another chair's leg with the force with identical modulus and direction.

Thus, the result of the force's action depends from the point of its application and the force is not a free vector. The question arises, how to work with forces and which mathematical operations on free vectors should be valid for forces? The answer to this question can be obtained only from the experiment.

Numerous experimental facts confirm the validity of the following statements: that *the point of application of the force can be displaced only along the line of its action to any point of the solid body*, and that *the two forces*
*
*
*and*
applied to the same point of the body and directed to some angle relative to each other, will have the same influence to the body as the single force, equal to their vector sum calculated according to the parallelogram rule and applied to the same point.

Let's recall that *the solid body* is the body with distances between its parts is not changed in the result of the force's action.

We shall call several forces applied to the solid body as *a system of forces*. If it is possible to change one system of forces to another without the change of the character of the body's motion, then such systems are called as *equivalent* ones. In particular, if it is possible to change the system of forces to the single force, such force is called the *resultant* force. Therefore, the resultant force has the similar influence to the body, as the system of forces, equivalent to it The resultant force is considered as equal to zero, if the forces applied to the body did not change the character of its motion.

It is shown in courses of theoretical mechanics how it is possible to change the arbitrary spatial system of forces, acting to the body, by more simple equivalent system, or, in some cases, by the single force, that is, the resultant force. It is turned out, that not every system of forces can be reduced to the resultant force, that is, not every system of forces has the resultant force. In most general case the spatial system of forces is reduced to the combination of the single force, which results in the motion of the body as a whole, and of so called pair of forces, which results in the rotation of the body.Two forces, with equal modules and opposite directions, are called the pair of forces, if they are not belong to the same straight line (see Figure 1).

The pair of forces is the simplest example of the system of forces, which has no resultant force. In fact, let's try to find in your mind the point of application of some force, which results in same motion, as one resulted from the pair of forces (see Figure 1). You will have no success.

The operation of the determination of the resultant force is called the addition of forces. Don't confuse the addition of forces with the addition of vectors. The result of the addition of vectors is a free vector, and the result of the addition of forces is the vector value, which has the point of application.

For the determination of the resultant force of two forces with their lines of actions crossed in the point O, it is necessary to displace forces along their lines of action and to apply them in the point O, and after that to add them according to the parallelogram rule.

For the determination of existence of the resultant force of several forces It is meaningful to try to find the resultant force. For this purpose it is necessary to find the resultant force of any two forces, after that to add this resultant force with third force, and so on, that is, to change the system of forces to more simple equivalent system. If the result of these consequent additions will be the single force, then this force should be the resultant force. From the proposed method of the determination of the resultant force the following statement is evident; if the resultant force of several forces exists, then it is equal to the vector sum of these forces.

The operation of the change of a single force by the equivalent system of several forces is called the *decomposition of the force*. In practice it is often necessary to decompose a single force(see Figure 2) along two directions 1 and 2, crossing the point C of the application of the force.In this case it is convenient to use* parallelogram *rule for the change of the single force to two ones. For this purpose we will draw two straight lines from the end of the vector which are parallel to the directions 1 and 2, and after that we will construct on the sides of the resulting parallelogram two vectors and , beginning in the point C. This is a method of the decomposition of the single forceto two force components and along the directions 1 and 2.

### INDEX

- §1. Introduction.
- §2. Force. Equivalence of forces. Resultant force. Addition of forces. Decomposition of force.
- §3. Equilibrium of a material point.
- §4. Equilibrium of a body in the absence of the rotation.
- §5. Equilibrium of a body with a fixed axis of the rotation in the planar case. Moment of the force.
- Test questions.
- Problems.