Rigid Body Equilibrium

6
Rigid Body Equilibrium
6.１ The condition for Equilibrium and Torque
The dimension or shape of a body is changed because the work of a force and will affect
the motion of that body. The motion of a body can be considered as all the motion of that body
itself, the translation motion, and the rotation too, if there. Generally, only a force that worked at
a body can cause a motion changed, its translation motion and its rotation as well. Nevertheless,
if there are a few of forces at once, maybe the force that worked canceled each other, so the
translation motion and the rotation do not change at all. If like that, the body called in
equilibrium. These mean that:
1. All of the body is motionless or move in regular rectilinear motion, and

2. The body does not rotate at all, or rotate by constant velocity.
When a body in equilibrium state, the resultant of all the forces that work on the body is
zero, its mean:
Where,
In addition, the direction of the resultant force is
(6-1)
The equation called the first condition for equilibrium, and the body is on equilibrium
translation. If a few forces work on a body on a plane, the force can be reducing in two forces. If
these two forces is equivalent, in opposite direction, and on an axis, so the torque on this body is
zero (Figure 6.1). In that state, the body does not rotate or rotate in constant velocity. Because of
that, the second condition for equilibrium can be express as below:
( at various axis) (6.2)
This equation called as the second condition for equilibrium, and the body considered in
rotation equilibrium.
Figure 6.1
A body considered in stable equilibrium if:
F = 0 and ∑ M0 = 0
Where M0 is the torque of F of O in everywhere point. If the force is:
The vector of the force is
and the torque is

So,
Where,
The magnitude of torque M is
The effect of force F is the rotational clockwise reversing due the axis at O point, usually
it is positive sign, whereas the effect of F2 is the rotational in clockwise and it is in negative sign.
̅1 of force 𝐹̅1 is
So the magnitude of torque 𝑀
̅2 of force 𝐹̅2 is
and the magnitude of torque 𝑀
The unit of torque is Newton meter (Nm) or lbft.
If the force 𝐹̅1 and 𝐹̅2 in parallel and not coincide, the force work in couple called
coupling. The common example of coupling is the forces of compass needle in magnetic earth
field. The force, which worked at north, and south poles of the needle are equivalent, one goes
to the north and the other goes to south.
Figure 6.2 illustrate a coupling which consist with 2 force, F1 and F2 is equivalent each
other and apart with perpendicular distant l.
The magnitude of these force are
The magnitude of resultant force 𝑅̿ is zero, that mean a coupling does not affect the
translation motion one whole. This coupling torque only causing rotation motion.
Figure 6.2
The resultant torque of the coupling at any point O is
In this way, coupling 𝐶̅ is a perpendicular vector due a plane which through both of that
forces.
The magnitude of coupling 𝐶̅ is

In this case the distance x1 and x2 is not include in calculation, so we can conclude that
the torque of this coupling due a plane in which the forces work and form that coupling, the
magnitude is equal and equal with the magnitude product of the force to perpendicular distance
of the forces’ lines.
A body in which worked a coupling will in balance if only there is another coupling
which worked in that body with same magnitude but in opposite direction.
6.2 Coplanar Force

Coplanar forces are forces at a plane, a concurrent system that are consist with
intersection forces at a point, which called intersection point. A parallel system that is consisting
forces which intersect at infinity point. An intersection point and nonparallel is consisting of
non-intersection and nonparallel forces.
Intersection forces are forces that their forces lines are intersecting at a point. Resultant
forces 𝑅̅ of the intersection forces may be a forces that though an intersection point or initial
point.
The magnitudes of resultant forces 𝑅̅ are
With their direction are
A body called at equilibrium state if it affected by intersection forces, when:

(a) That body is at rest and keep at rest (called static equilibrium), or
(b) That body moves with constant velocity (called translational equilibrium).
This condition for first equilibrium required:
Parallel forces are forces that intersect at infinity point. The resultant of parallel forces
has uniform direction with those initial forces direction and the magnitude equal with total
magnitude of the forces. The resultant forces are probably:
(a) a single-force 𝑅̅ , which parallel with the system.
(b) a coupling
(c) zero. If this parallel system parallel due y-axis, we find:
In this case, 𝑥̅ is a perpendicular distant at the center torque O to the resultant 𝑅̅ and the
magnitude of 𝑥̅ is
If ∑ F = 0, the magnitude of resultant coupling will equal with
Non-intersectional nonparallel forces are the forces with their forces lines that do not
intersecting at a point and not parallel.
The resultant forces of the system are probably:
(a) a single-force ̅𝑅 .
(b) A coupling is in a plane system or in parallel planes.
(c) Zero.
algebraically, the resultant force is
R = √(∑𝐹𝑥 )2 + (∑𝐹𝑦 )2
and
in which 𝜃𝑥 is an angle of resultant 𝑅̅ due positive x-axis.
Work line of resultant force 𝑅̅ is given by the equation:
With 𝑎̅ is perpendicular distance of the center torque O due resultant force 𝑅̅ .

Generally, force system of rigid body is not intersection and not parallel.
The condition of rigid body equilibrium under effect of coplanar force is
or
and
6.3 Center Mass of A Body

At many-particle systems ( a system which consist more than a particle), every
component of this system have their mass, so the mass of many-particle system is the sum of the
mass of all the part of the system and the location of the total mass is the center mass of the body.
The center of mass is a confection point of the resultant weights of each particles of the system, I
where the sum torque of this confection point (center of mass) is zero. It said that center of the
mass is a point at the system when a force applied in this point, the body will move with
translation motion only. Every particle had gravitational force w=mg, it called weight, the
direction of this force always lead up to the center of the earth, this weight ( wi ) will across at
miles, we can consider the direction as a parallel.
So,
or we can write the equation above by its component:
(xp,m, yp,m, zp,m) are the coordinate of the center of mass.
Consider this equation:

For the rigid body, the equation be lim ∑∆𝑚𝑖 = ∫ 𝑑𝑚 which compound of much of
∆𝑚𝑖 →0 𝑖=1
masses points.
The coordinate of center of mass of rigid body is
With:
Ρ = mass per unit volume
σ = mass per unit area
λ = mass per unit length
The coordinate of center of the mass can be written by:

If the homogeneous rigid body has symmetric form, the center of mass will be coincide to
the center of its symmetry etc.: sphere, parallel epicedium (beam), cube,

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6

Rigid Body Equilibrium

6.１ The condition for Equilibrium and Torque

1. All of the body is motionless or move in regular rectilinear motion, and

In addition, the direction of the resultant force is

( at various axis) (6.2)

A body considered in stable equilibrium if:

Where M0 is the torque of F of O in everywhere point. If the force is:

The vector of the force is

and the torque is

The magnitude of torque M is

The unit of torque is Newton meter (Nm) or lbft.

The magnitude of these force are

The resultant torque of the coupling at any point O is

The magnitude of coupling 𝐶̅ is

6.2 Coplanar Force

With their direction are

A body called at equilibrium state if it affected by intersection forces, when:

This condition for first equilibrium required:

If ∑ F = 0, the magnitude of resultant coupling will equal with

algebraically, the resultant force is

in which 𝜃𝑥 is an angle of resultant 𝑅̅ due positive x-axis.

Work line of resultant force 𝑅̅ is given by the equation:

With 𝑎̅ is perpendicular distance of the center torque O due resultant force 𝑅̅ .

6.3 Center Mass of A Body

or we can write the equation above by its component:

(xp,m, yp,m, zp,m) are the coordinate of the center of mass.

Consider this equation:

The coordinate of center of mass of rigid body is

Ρ = mass per unit volume

σ = mass per unit area

λ = mass per unit length

The coordinate of center of the mass can be written by:

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