2. STATICS OF PARTICLES
2.1 Concurrent Forces. In this chapter we shall study the effect of forces on
particles. The use of the word particle does not imply that we shall restrict our
study to that of small corpuscles. It means that the size and shape of the bodies
under considerations will not affect the solution of problems treated in this
chapter and that all the forces acting on given body will be assumed to have the
same point of application. Forces whose lines of action intersect at one point are
said to be concurrent. A system of concurrent forces acting on a particle A can be
replaced by an equivalent force, i.e. by the resultant R (Fig.2.1)
The problem of determining the resultant of concurrent forces F1, F2 , …, Fn is
reduced, according to the 3rd principle of statics, to the composition of the given
forces, i.e.
R = F1 + F2 + … + Fn
(2.1)
2.2 Resolution of Forces. As we know from the chapter Elements of Vector
Algebra, the problem of resolution of a force, F, into components is
indeterminate and can be solved uniquely only if additional conditions are stated.
Two cases are of particular interest:
1. One of the two components, P, is known. The second component, Q, is
obtained by applying the triangle rule and joining the tip of P to the tip of F
(Fig.2.2a). Ones Q has been determined graphically or by trigonometry, both
components P and Q should be applied at A ;
2. The line of action of each component is known. Both components are obtained
by means of parallelogram law (Fig.2.2b). This process leads to well-defined
components, P and Q, which may be determined graphically or by applying of
the law of sines.
2.3 Analytical method of addition of concurrent forces. We shall determine
the resultant R of two or more forces in space by summing their orthogonal
components. Graphical or trigonometrical methods are generally not practical in
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the case of forces in space. Thus, consider n forces F1, F2 , …, Fn and suppose
that their orthogonal components are known. The resultant R is given by eq.
(2.1). Resolving each force in (2.1) into its orthogonal components, we write
(
)
r
r
r
r
r
r
r
r
r
R x i + R y j + R z k = ∑ Fix i + Fiy j + Fiz k = (∑ Fix )i + (∑ Fiy ) j + (∑ Fiz )k (2.2)
from which it follows that
R x = ∑ Fix ;
R y = ∑ Fiy ;
R z = ∑ Fiz
(2.3)
The magnitude of the resultant and the angles θx, θy, θz it forms with the axes of
coordinates are as follows:
R = R 2x + R 2y + R 2z
cos θ x =
Rx
;
R
cos θ y =
Ry
R
;
(2.4)
cos θ z =
Rz
R
(2.5)
2.4 Equilibrium of a particle. It follows from the laws of mechanics that a
particle subjected to the action of an external set of mutually balanced forces can
either be at rest or in inertial motion, i.e. a motion without acceleration. In such
cases the particle is said to be in equilibrium. For a system of concurrent forces
acting on a body to be in equilibrium it is necessary and sufficient for the
resultant of the forces to be zero. The conditions which the forces themselves
must satisfy can be expressed in either graphical or analytical form.
Analytical form of conditions of equilibrium .
To express algebraically the conditions for the equilibrium of a particle, we write
R = ∑F = 0
(2.6)
Resolving each force F into rectangular (=orthogonal) components, we conclude
that the necessary and sufficient conditions for the equilibrium of a particle are
∑ Fx
= 0,
∑ Fy = 0,
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∑ Fz
=0
(2.7)
Graphical form of conditions of equilibrium .
Since the resultant R of a system of concurrent forces is defined as the closing
side of a force polygon constructed with the given forces, it follows that R can be
zero only if the polygon is closed. Thus, for a system of concurrent forces to be in
equilibrium it is necessary and sufficient for the force polygon drawn with these
forces to be closed.
If all concurrent forces acting on a body lie in one plane, they form a coplanar
system of concurrent forces. Obviously, for such a force system only two
equations are required to express the conditions of equilibrium.
2.5 Solution of problems of static’s. Free-body diagram. In practice, a problem
in engineering mechanics is derived from an actual physical situation. A sketch
showing the physical conditions of the problem is known as a space diagram.
The methods of analysis discussed in this chapter apply to a system of forces
acting on a particle. A large number of problems involving actual structures,
however, may be reduced to problems concerning the equilibrium of a particle.
This is done by choosing a significant particle and drawing a separate diagram
showing this particle and all forces acting on it. Such diagram is called a freebody diagram.
A next stage of problem solution requires application of the equilibrium
conditions in either graphical or analytical form.
The final stage of problem solution relies upon the determination of unknown
quantities and analysis of results.
In practical problems where the equilibrium of either particles or rigid bodies
is considered, the reactions of the constraints are unknown quantities. Their
number depends on the number and type of the constraints. A problem can be,
however, solved only if the number of unknown reactions is not greater than the
number of equilibrium equations in which they are present. Such system as well
as the corresponding reactions are said to be statically determinate.
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Systems in which the number of unknown reactions of the constraints is
greater than the number of equilibrium equations in which they are present are
called statically indeterminate similarly as the corresponding reactions.
An example of statically indeterminate system is a weight hanging from three
strings lying in one plane (Fig.2.3). There are three unknown quantities (the
tensions T1, T2, T3 of the strings), but only two equations can be formed.
Bibliography
Beer F.P, Johnston E.R., Jr., Vector Mechanics for Engineers, McGraw-Hill
Shames I.H., Engineering Mechanics – Statics, Prentice-Hall, 1959
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