3.1 What is a Rigid Body?
.
Figure 3 – 1: An interesting Rigid Body.
Figure 3 – 2: A lever is a simple rigid body.
This chapter deals with rigid body statics problems.
As a starting point, this section discusses the differences
between a particle and a rigid body and how these differences
affect the problem solving process.
There are an infinite number of combinations of F and
P that satisfy Eq. (3 – 1), each of which maintains the lever in
translational equilibrium. Although translational equilibrium is
satisfied, the lever is not necessarily balanced. For example,
when no force F is applied, the solution F = 0, P = 100 lb
satisfies Eq. (3 – 1) yet the system is clearly unbalanced. The
problem is that rotational equilibrium has not been satisfied.
Degrees-of-Freedom
Consider the lever shown in Fig. 3 – 2. Assume that a
resistance force of R =100 lb acts on the right end of the lever b
= 1 ft from fulcrum A. A downward force F is applied on the
other end of the lever a = 2 ft from the fulcrum to keep the
lever in equilibrium. The equilibrium of the lever is maintained
by the applied force F and an upward force P at the fulcrum.
Let’s first regard the lever as a particle and then as a rigid body.
In order to maintain the lever in rotational equilibrium,
an additional relationship is needed. The additional relationship
is
When the lever is regarded as a particle, the three
forces can be thought of as acting concurrently (See Figure 3 –
3). Translational equilibrium in the vertical direction is
maintained by setting the resultant force in the vertical direction
to zero, that is
In ancient times, Eq. (3 – 2) was known as the Law of the
Lever. You’ll see in Example 3 – 1 and later in Section 3.4 that
Eq. (3 – 2) follows from Newton’s First and Third Laws. After
learning about moments, you’ll discover in Section 3.4 that Eq.
(3 – 2) merely expresses the statement that the sum of the
moments acting on the lever is zero.
(3 – 1)
(3 – 2)
0 = – R – F + P.
aF = bR .
Equation (3 – 1) is one equation expressed in terms of
the two unknown forces F and P (Recall that R is known).
Figure 3 – 3: Freebody diagram of a
lever treated as a
particle
Chapter Objectives
Section
3.1 What is a Rigid Body?
3.2 Moments in a Plane
3.3 Moments in Space
3.4 Equilibrium of a Rigid Body
3.5 Equivalent Systems
3.6 System Behavior
Figure 3 – 4: Free-body diagram of a lever treated as a rigid body
Objective
To describe the type of engineering problem commonly referred to as a rigid body problem
To become proficient at mathematically manipulating moments in a plane
To become proficient at mathematically manipulating moments in space
To show how to solve rigid body problems in space
To show how to reduce complexity through the use of equivalent systems
To describe the different types of behavior of static systems
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Equation (3 – 1) is the governing equation associated with
the translational degree-of-freedom in the vertical direction.
Equation (3 – 2) is the governing equation associated with the
rotational degree-of-freedom about the z axis (perpendicular to
the x-y plane). Substituting Eq. (3 – 2) into (3 – 1) yields
b
1
R  100  50 lb,
a
2
P  R  F  100  50  150 lb.
F
This solution maintains the lever in a state of equilibrium – both
translational and rotational.
In planar problems, like in the lever
problem just discussed, there are
generally three governing equations.
Two of them are associated with the
two translational degrees-of-freedom
and one is associated with the rotational
degree-of-freedom. (In the lever
problem just discussed we didn’t need
to look at translational equilibrium in
the horizontal direction.) In the most
general situation, when the forces lie in
any of the three directions of space,
there are six governing equations. Three
of them are associated with the three
translational degrees-of-freedom and
the other three are associated with the
three rotational degrees of freedom.
The Set-up Step and the
Transition Step
In the transition step, where a word statement is
converted into a mathematical statement, unimportant
information is discarded, leaving only the information that is
absolutely essential to the analysis. In this step, transition
diagrams are drawn. The most important transition diagram is
the free body diagram. To draw the free body diagrams of the
rigid body in a system, the bodies in the system are cut away
from each other. The influences that the surrounding bodies
have on a given rigid body are replaced with forces and
moments acting on the rigid body. The free-body diagram of a
rigid body will differ from the free-body diagram of a particle
in two significant ways. First, for a rigid body, the locations of
the forces will be important whereas for a particle the
Table 1: The Particle versus the Rigid
Body
Particle
Mass is idealized
to be located at
a point.
Forces are
idealized to be
located at a
point.
Forces maintain
translational
equilibrium.
To maintain
equilibrium,
F = 0.
Recall that the problem solving process begins with the
set-up step and the transition step. In the set-up step the system
is divided into bodies and each body is identified as either a
particle or as a rigid body. Recall that the decision to regard the
body as a particle or as a rigid body does not depend on the size
of the body, as much as on what equilibrium conditions you
want to impose. In general, there are six equilibrium conditions
associated with the rigid body (three translational and three
rotational), so up to six conditions can be imposed. If you
impose only translational equilibrium conditions then the body
is called a particle and the forces can be thought to act
concurrently. If you impose any of the rotational equilibrium
conditions then the locations of the forces become important
and the body is called a rigid body.
Rigid Body
Size and shape of
the body are
influential.
Locations of
forces are
important. Forces
create moments
acting on the
body.
Forces maintain
translational and
rotational
equilibrium.
To maintain
equilibrium,
F = 0, MA = 0.
locations of the forces were not
important. Secondly, the free-body
diagram for a rigid body will contain
moments (which have not yet been
defined) whereas the free-body diagram
of a particle did not contain moments.
After the transition step, the governing
equations are listed in the equation step,
the equations are solved in the answer
step, and finally an understanding of the
behavior of the system is gained in the
knowledge step. However, let’s not
jump ahead of ourselves. We need to
first understand what a moment is. The
next two sections develop the moment.
Key Terms
Degree-of-Freedom, Law of the Lever, Lever, Moment,
Rotational Equilibrium, Translational Equilibrium
Review Questions
1. At most, how many degrees-of-freedom does a particle
have? At most, how many degrees-of-freedom does a rigid
body have?
2. When a body is regarded as a rigid body, is it important to
know where on the body the forces are applied?
3. In what two ways does a free-body diagram of a rigid body
differ from a free-body diagram of a particle?
4. State in words the Law of the Lever.
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