Mechanics of Rigid Bodies Fundamental Concepts

Mechanics- a branch of physical sciences Mass- quantity of matter possessed by a

concerned with the state of rest or motion of body is called mass. The mass of a body
bodies subjected to the action of forces. can not change unless the body is damaged
- The physical science concerned with and part of it is physically separated
the behavior of bodies that are acted Length- concept to measure linear
upon by forces distances
- Oldest of the Physical sciences Time- a measure of succession of events.
3 types of Mechanics The successive event selected is the
1. Mechanics of Rigid Bodies rotation of the earth about its own axis and
- Statics this is called Day
- Dynamics Space- any geometric region in which the
2. Mechanics of Deformable Bodies study of a body has been done is called
3. Fluid Mechanics space
Rigid Bodies Displacement- defined as the distance
- A basic requirement for the study of moved by a body/particle in the specified
the mechanics of deformable bodies direction
and fluid mechanics Velocity- rate of change of displacement
- Essential for the design and analysis with respect to time
of many types of structural Acceleration- rate of change of velocity
members, mechanics, mechanical with respect to time
components, electrical devices, etc, Momentum- product of mass and velocity
encountered in Momentum= Mass x Velocity
A RIGID BODY DOES NOT DEFORM Particle- an object which has only mass
UNDER LOAD!!!!!!!!!!!! and no size
Statics- study which deals with the - Such a body cannot exist
condition of bodies in equilibrium subjected theoretically
to external forces - When we deal with the problem
- In other words, when the force involving distances considerably
system acting on a body is larger compared to the size of the
balanced, the system has no body, the body may be treated as
external effect on the body or moves particle
at constant velocity, the body is in Force- represents the action of one body to
equilibrium. another. Characterized by its magnitude,
Dynamics- a branch of mechanics in which direction of action and its point of
the forces and their effects on the bodies in application
motion are studies FORCE IS A VECTOR QUANTITY
Subdivided into two parts: Newtonian Mechanics- Length, Time,
1. Kinematics- deals with the Mass are absolute concepts independent of
geometry of motion of bodies without each other
and application of external forces Force is derived concept not
2. Kinetics- deals with the motion of independent of other fundamental concepts.
bodies with the application of Force acting on a body is related to mass of
external forces.
the body and the variation of its velocity with - A particle of mass m acted upon by
time. an unbalanced force F experiences
Force can also occur between bodies an acceleration a that has the same
that are physically separated (Ex. direction as the force and a
Gravitational, Electrical and magnetic magnitude that is directly
forces) proportional to the force
Mass- property of matter that does not Third Law: Law of Interaction
change form one location to another - For every action there is an equal
Weight- refers to the gravitational attraction and opposite reaction. The mutual
of the earth on a body or quantity of mass. forces of action and reaction
Its magnitude depends upon the elevation between two particles are equal,
at which the mass is located. opposite, and colinear.
Units- measurements are always made in
Mechanics Idealization comparison with certain standards
To simplify application of theory Four systemps of units used for the
Particle- body with mass but with measurement of physical quantities
dimensions that can be neglected 1. FPS (Foot Pound Second) System
- Size is negligible compared to 2. CGS (Centimitere Gram Second)
motion System
- All forces act through the center of 3. MKS (Meter Kilogram Second)
gravity System
- Neglect rotation about center of 4. SI (System International d’ unites the
gravity french name)
Rigid Bodies- points of application, and line - The fundamental units of the system
of action of forces are important are meter (m) for length, kilogram
- Rotation and moments about center (kg) for mass and second (s) for time
of gravity are important - The unit for force is newton (N). One
Concentrated Force- effect of a loading newton is the amount of force
assumed to act at a point (CG) on a body required to induce an acceleration of
- Provided the area over which the 1m/sec 2 on one kg mass.
load is applies is very small - Weight of a body (in N)= Mass of the
compared to the overall size of the body (in kg) x Acceleration due to
body. gravity (in m/sec2)
- W=MxG
Mechanics: Newton’s Three Laws of Dimensions- branch of mathematics
Motion dealing with dimensions of quantities is
Basis of formation of rigid body mechanics called dimensional analysis. There are two
First Law: Law of Inertia systems of dimensional analysis
- A particle at rest, or moving is a Absolute system (MLT system)- a system
straight line with constant velocity, of units defined on the basis of length, time,
tends to remain in this state provided and mass is referred to as an absolute
the particle is not subjected to an system
unbalanced force. Gravitation system ( FLT system)- a
Second Law: Law of Acceleration system of units defined on the basis of
length, time, and force is referred to as a
gravitational system
- In this system, force is measured in
a gravitational field. Thus, its
magnitude depends upon the
location where the measurement is
made. FLT system refers to the force
length time system
Sliding Vector- has a unique line of action
in space but not a unique point of
Scalars and Vectors
application
- Various quantities in engineering
- Ex. External force in a rigid body,
mechanics may be grouped into
Principle of Transmissibity, Imp in
scalars and vectors
rigid bodies
Scalar Quantity- completely defined by
magnitude alone
- Are
- Length
- Mass
- Moment of inertia
- Energy
- Fixed Vector- for which a unique
- Power
point of application is specified
- Volume
- Ex. Action of a force on a
- work
deformable body
Vector quantity- completely defined only
when its magnitude as well as direction are
specified
- Force
- Moment
- Momentum
- Displacement
- Velocity
- Acceleration
EQUIVALENT FORCE SYSTEM
Vectors Resolution of a force into components
Free vectors- action is not confined to or - Single force F acting on a particle
associated with a unique line in space may be replaced by two or more
- Ex. Movement of a body without forced that together have the same
rotation effect on the particle called
component of force
- Each forced F can be resolved into
an infinite number of possible sets of
components
Triangle rule appears as a point commonly called
- one of the two components, P, is the center of moments.
known. We obtain the second Varignon’s Theorem
component, Q, by applying the - The moment of a force is equivalent
triangle rule and joining the tip of P to the sum of the moments of its
to the tip of F. components
Parallelogram Method Couples
- the line of action of each - Couple is made up of two equal,
component is known. We obtain the parallel, oppositely directed forces
magnitude and sense of the - Their moment sum is constant and
components by applying the independent ofthe moment center.
parallelogram law and drawing lines - The moment of a couple C is equal
through the tip of F that are parallel to the product of one of the forces
to the given line of action. composing the couple multiplies by
Adding force by Components the perpendicular distance between
- adding forces using their their action line.
components, especially rectangular - C=Fxd
components. this method is often the
most convenient way to add forces EQUILIBRIUM
and in practice, is the most common - Remains at rest if originally at rest,
approach or has a constant velocity if originally
Parallelogram law for the addition of two in motion
forces - Satisfies newton’s first law of motion,
- method for finding the resultant which requires the resultant force
Parallel force system acting on a particle to be equal to
- one in which the action lines of all zero
the forces are parallel. - Follows from Newtin’s second law of
Moment of a Force motion
- The measure of its ability to produce - Force system and particle’s
turning or twisting about the axis. acceleration is equal to zer0
- The magnitude of the moment of a
force about an axis which is Equilibrium of a particle
perpendicular to a plane containing 1. When the resultant of all the forces
the line of action of the force is acting on a particle is zero, the
defined as the product of the force particle is in equilibrium
and the perpendicular distance from 2. A particle acted upon by two forces
the axis to rhe line of action of the is in equilibrium if the two forces
force (M=Fd). The distance d is have the same magnitude and the
frequently called the moment arm of same line of action but opposite
the force sense
Center of moments 3. A case of equilibriu, of a particle is
- the axis of moments which is represented in figure (a), where four
perpendicular to the plane of forces forces are shown acting in particle A
 - Frictionless pins in fitted
holes, hinges, rough
surfaces, Reactions of this
group involve two
unknowns and are usually
represented by their x and
y components

3. Reactionary equivalent to a force

and a couple
4. Resolving each force F into - Caused by fixed supports
rectangular components that oppose any motion of
5. We conclude that the necessary and the free body and thus
sufficient conditions for the constrain it completely.
equilibrium of a particle are scalar Reactions of this group
equations involve three unknowns
6. When the force and the couple are usually consisting of the two
both equal to zero, the external components of the force and
forces form a system equivalent to the moment of the couple
zero, and the rigid body is said to be
in equilibrium
7. The system of external forces
imparts no translational or rotational
motion to the body
8. In addition to the forces applied to a
structure, its supports exert reaction
on it. Specific reaction is associated
with each type of support
Reactions for a two-dimensional
structure
- The reactions exerted on a
two-dimensional structure fall into
three categories
1. Reactions Equivalent to a force
with known line of action
- Rollers, frictionless surfaces,
short links and cables, collars
on frictionless rods, and
frictoinless pins in slots. Each
of these supports and
connectoins can prevent
motion in one direction only

2. Reactions equivalent to a force of

unknown direction and magnitude
Statically Indeterminate Reactions and
Partial Constraints
1. Completely Constrained and
statically determinate
- The types of supports used
where such that the rigid
body could not possibly
move under the given loads
or under any other loading
conditions. The reactions
corresponding to these
supports involved three
unknowns and could be
determined by solving the
three equations available
2. Statically Indeterminate
- The number of unknowns
exceeds the number of
equilibrium equations
available
3. Partially Constrained
- The constraints provided by
the supports are not
sufficient to keep the object
from moving. In such a case,
the three equations of
equilibrium are not satisfied.
There are fewer unknowns
than equations
4. Improperly Constrained
- The supports (even though
the may provide sufficient
number of reactions) are
arranged in such a way that
the reactions must be either
concurrent or parallel.