Introduction to statics

• Space - geometrical area in which physical or mechanical processes i.e. the movement of the body, occur. In classical non-relativistic mechanics, the space is homogenous and isotropic. It is considered to be absolute, i.e. to exist independently of the material bodies that occupy it.
• Time - is a measure which determines the sequence of events. In classical mechanics, the vortex is absolute and homogeneous, ie two equal processes last equally, regardless of the beginning of the process.
• Matter - or substance occupies some space. The body is a matter which is bounded by a closed surface, i.e. it's surface.
• Inertia - it is the property of a substance to resist a change of motion i.e. acceleration.
• Mass - is a measure of inertia, also called sluggish mass or inertial mass. Mass is also a property that causes the body to attract each other. This mass is called gravitational mass or heavy mass. As part of previous research in physics, heavy mass is equal to slow mass.
• Rigid body - is an idealization of a real body. Parts of a rigid body or its particles always retain the same position with each other. A rigid body cannot deform or break.
• Particle or material point - is a rigid body whose dimensions do not affect motion. And the motion of a very large body can sometimes be thought of as the motion of a particle. One such example is the movement of planets around the Sun. As soon as the rotation of a body is considered, it cannot be considered as a particle no matter how small it may be, e.g., atomic particles.
• Force - is the interaction of material bodies that seeks to change the state of motion of the body. The force seeks to accelerate the body on which it acts in the direction of its action. In addition to acceleration, the force also tends to deform the body. Force shown in Figure 1.1, is a directed or vector quantity that is determined by the direction of action, grip, and magnitude. The length of the arrow is proportional to the magnitude of the force. Figure 1.1 - Display of force by means of an arrow. Two bodies, shown in Figure 1.2 are in contact and often act on each other with forces distributed over a very small area.These forces are replaced by their resultant \(\vec{F}\) which is further considered as a concentric force. As the interaction of two bodies, forces always appear in pairs.
  However, for the sake of simplicity, only one force is often shown in drawings and sketches. Figure 1.2 - (a) Two bodies in contact, (b) Contact replaced by resultant \(\vec{F}\), (c) Only one body shown with force \(\vec{F}\) acting on the body.
• Force system A group of forces that balance or move a body, machine, or structure is called a system of forces. If the system of forces acts on an isolated particle, it is in equilibrium if the particle is stationary or if it moves uniformly in the direction. A system is a force in equilibrium if its action on a rigid body does not change the state of that body (state of rest, ie state of motion). Two force systems are by definition equivalent if by acting on the same rigid body they cause an equal change in its state.

Figure 1.3 - Vector representation of a force \(\vec{F}\)
In Cartesian coordinate system the force F can be expresed in the form: $$ \begin{eqnarray} \vec{F} &=& F_x\vec{e}_x + F_y \vec{e}_y + F_z \vec{e}_z \nonumber\\ &=& F_x\vec{i} + F_y\vec{j}+ F_z\vec{k} \nonumber\\ &=&(F_x \cos \alpha) \vec{e}_x + (F \cos \beta) \vec{e}_y + (F \cos \gamma) \vec{e}_y \nonumber\\ &=& (F_x \cos \alpha) \vec{i} + (F \cos \beta) \vec{j} + (F \cos \gamma) \vec{k} \end{eqnarray}$$ where: \(F = |F| = \sqrt{F_x^2+F_y^2+F_z^2.}\)
The cosines are defied as
$$ \begin{eqnarray} \cos \alpha = \cos \theta_x &=& \frac{F_x}{F},\nonumber \\ \cos \beta = \cos \theta_y &=& \frac{F_y}{F},\nonumber \\ \cos \gamma = \cos \theta_z &=& \frac{F_z}{F}. \end{eqnarray} $$

Axioms of statics

There are total of five axioms of statics and these are:

1. First Axiom - If two forces act on a rigid body as shown in Figure 1.4, it will be in equilibrium if these forces are collinear, equal in magnitude, and directed in the opposite direction. Collinear forces are those forces that lie in the same direction. Figure 1.4 - the rigid body in equilibrium under the action of two forces.
2. Second Axiom- The resultant of two forces acting at the same point of a rigid body is determined by using parallelogram law of addition. A triangle of forces can be used instead of a parallelogram of forces. Two explaim how parallelogram law of addition and triangle rule works the operations with vectors in general must be explained.
   Scalar - any positive or negative physical quantity that is completely defined by its magnitude. The most common examples of scalars are length, mass, and time.
   Vector - as seen in the previous graphical and mathematical representation of the force vector, the vector is a physical quantity that requires magnitude and a direction for its complete description. The most common examples of the vectors are force, position and the moment.
   
3. Third Axiom - The balance or motion of a rigid body will not change if the body is released from the bond and forces equal to the bond reactions are added instead (Figure 1.5). Figure 1.5 - The system released of bonds (a) and forces equal to the bond reactions are added instead (b).
4. Fourth Axiom - The state of equilibrium or motion of a rigid body will not change if a balanced system of forces is added to or subtracted from the body. Figure 1.6 - The state of a rigid body will not change if a new balanced system of forces is added to a given force system.
5. Fifth Axiom - This axiom is often called the principle of solidification or the principle of stiffening. This principle reads: If a deformable body under the action of forces occupies a deformed equilibrium position, the balance will not be disturbed if the deformed body is considered as an ideally rigid body. The principle of solidification also applies to the movement of a deformable body.