BASIC CONCEPTS: RECAP

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BASIC CONCEPTS RECAP

• BY
• GP CAPT NC CHATTOPADHYAY

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• Rigid body
• Definition An idealized extended solid whose
  size and shape are definitely fixed and remain
  unaltered when forces are applied.
• Treatment of the motion of a rigid body in terms
  of Newton's laws of motion leads to an
  understanding of certain
• important aspects of the translational and
  rotational motion of real bodies without the
  necessity of considering the
• complications involved when changes in size and
  shape occur.
• Many of the principles used to treat the motion
  of rigid bodies apply in good approximation to
  the motion of real elastic solids

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RIGID BODY CONTD

• In physics, a rigid body is an idealization of a
  solid body of finite size in which deformation is
  neglected.
• In other words, the distance between any two
  given points of a rigid body remains constant in
  time regardless of external forces exerted on it.
  
• Even though such an object cannot physically
  exist due to actual deformation, objects can
  normally be assumed to be perfectly rigid if they
  are not moving near the speed of light

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CONCEPT OF RIGID BODY
The position of a rigid body is determined by the
position of its center of mass and by its
attitude at least six parameters in total
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CONCEPT OF RIGID BODY
HELICOPTER FRAME AS A RIGID BODY

• An un deformed body is a rigid body
• No object is absolutely rigid

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RECAP..
A force is a vector quantity that, when applied
to some rigid body, has a tendency to produce
translation (movement in a straight line) or
translation and rotation of body. When problems
are given, a force may also be referred to as a
load or weight. Characteristics of force are the
magnitude, direction(orientation) and point of
application.
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RECAP..
Concurrent Force Systems
A concurrent force system contains forces whose
lines-of action meet at some one point. Forces
may be tensile (pulling)
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RECAP..
Concurrent Force Systems
A concurrent force system contains forces whose
lines-of action meet at some one point. Forces
may be compressive (pushing)
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TYPES OF FORCES(LOADS)

1. Point loads - concentrated forces exerted at
   point or location
2. Distributed loads - a force applied along a
   length or over an area. The distribution can be
   uniform or non-uniform.

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TWO EFFECTS OF FORCE
Force exerted on a body has two effects

• The external effect, which is tendency to change
  the motion of the body
• The internal effect, which is the tendency to
  deform the body.

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EQUILLIBRIUM
If the force system acting on a body produces no
external effect, the forces are said to be in
balance and the body experiences no change in
motion with no unbalanced moments. Hence the
body is said to be in equilibrium.
SUFFICIENT CONDITIONS ARE ?FX 0, ?FY 0 ,
?FZ 0 ?MX 0 ?MY 0 , ?MZ 0
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RECAP..
Scalar Quantity has magnitude only (not
direction) and can be indicated by a point on a
scale. Examples are temperature, mass, time and
dollars. Vector Quantities have magnitude and
direction. Examples are wind velocity, distance
between to points on a map and forces.
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RECAP..
The process of reducing a force system to a
simpler equivalent stem is called a reduction.
The process of expanding a force or a force
system into a less simple equivalent system is
called a resolution.
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RECAP..
Collinear If several forces lie along the same
line-of action, they are said to be
collinear. Coplanar When all forces acting on a
body are in the same plane, the forces are
coplanar.
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TYPE OF VECTORS
Free Vector - is vector which may be freely moved
in space. Direction and line of action can be
changed Sliding Vector - action of a force on a
rigid body is represented by vectors which may
move or slid along their line of action. Bound
Vector or Fixed Vector - can not be moved without
modifying the conditions of the problem.
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PRINCIPLE OF TRANSMISSIBILITY
The principle of transmissibility states that the
condition of equilibrium or of motion of a rigid
body will remain unchanged if a force F acting
at a given point of the rigid body is replaced by
a force F of the same magnitude and the same
direction, but acting at a different point,
provided that the two forces have the same line
of action.
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PRINCIPLE OF TRANSMISSIBILITY
Line of action

• This principle leads to use of sliding vectors
• Only point of application is changed without
  altering magnitude direction

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RECAP
Resultant Forces
If two forces P and Q acting on a particle A may
be replaced by a single force R, which has the
same effect on the particle.
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RECAP
Resultant Forces
This force is called the resultant of the forces
P and Q and may be obtained by constructing a
parallelogram, using P and Q as two sides of the
parallelogram. The diagonal that pass through A
represents the resultant.
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RECAP
Resultant Forces
This is known as the parallelogram law for the
addition of two forces. This law is based on
experimental evidence and can be proved or
derived mathematically. Rv(P2 Q22PQ.COS ?)
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RECAP
Resultant Forces
For multiple forces action on a point, the forces
can be broken into the components of x and y.
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Example Problems

1. Determine the magnitude and direction of the
   resultant of the two forces.

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Example Problems

1. Two structural members B and C are riveted to the
   bracket A. Knowing that the tension in member B
   is 6 kN and the tension in C is 10 kN, determine
   the magnitude and direction of the resultant
   force acting on the bracket.

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Example Problems

1. Determine the magnitude and direction of P so
   that the resultant of P and the 900-N force is a
   vertical force of 2700-N directed downward.

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Example Problems

1. A cylinder is to be lifted by two cables.
   Knowing that the tension in one cable is 600 N,
   determine the magnitude and direction of the
   force so that the resultant of the vertical force
   of 900 N.

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Example Problems

1. Determine the force in each supporting wire.

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Example Problems

1. The stoplight is supported by two wires. The
   light weighs 75-lb and the wires make an angle of
   10o with the horizontal. What is the force in
   each wire?

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Example Problems

• In a ship-unloading operation, a 3500-lb
  automobile is supported by a cable. A rope is
  tied to the cable at A and pulled in order to
  center the automobile over its intended position.
  The angle between the cable and the vertical is
  2o, while the angle between the rope and the
  horizontal is 30o. What is the tension in the
  rope?

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Example Problems

1. The barge B is pulled by two tugboats A and C.
   At a given instant the tension in cable AB is
   4500-lb and the tension in cable BC is 2000-lb.
   Determine the magnitude and direction of the
   resultant of the two forces applied at B at that
   instant.

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Example Problems

1. Determine the resultant of the forces on the bolt.

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Example Problems

1. Determine which set of force system is in
   equilibrium. For those force systems that are not
   in equilibrium, determine the balancing force
   required to place the body in equilibrium.

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Example Problems

1. Two forces P and Q of magnitude P1000-lb and
   Q1200-lb are applied to the aircraft connection.
   Knowing that the connection is in equilibrium,
   determine the tensions T1 and T2.

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Example Problems

1. Determine the forces in each of the four wires.

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Example Problems

1. The blocks are at rest on a frictionless
   incline. Solve for the forces F1 and F2 required
   for equilibrium.

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Example Problems

1. Length A 5 m, and length B 10 m and angle a
   30o. Determine the angle b of the incline in
   order to maintain equilibrium.

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Example Problems

1. Solve for the resisting force at pin A to
   maintain equilibrium.

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MOMENT

• Moment of force (often just moment) is the
  tendency of a force to twist or rotate an object
  similar to torque . This is an important, basic
  concept in engineering and physics.
• (Note In mechanical and civil engineering,
  "moment" and "torque" have different meanings,
  while in physics they are synonyms. Moment arm is
  a quantity used when calculating moments of
  force.
• The Principle of moments is if an object is
  balanced then the sum of the clockwise moments
  about a pivot is equal to the sum of the
  anticlockwise moments about the same pivot.
• A pure moment is a special type of moment of
  force.

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MOMENT

• Moment of a vector is a generalization of the
  moment of force. The moment M of a vector B about
  the point A is
• where
• B is the vector from point A to the position
  where quantity B is applied.
• represents the cross product of the vectors.
• Thus M can be referred to as "the moment M with
  respect to the axis that goes through the point
  A", or simply "the moment M around A". If A is
  the origin, or, informally, if the axis involved
  is clear from context, one often omits A and says
  simply moment.
• When B is the force, the moment of force is the
  torque as defined above (ROTATING BODY)

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MOMENT OF A FORCE
MOMENT M r X F
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VARIGNONS THEOREM
The Principle of Moments, also known as
Varignon's Theorem, states that the moment of any
force is equal to the algebraic sum of the
moments of the components of that force. It is a
very important principle that is often used in
conjunction with the Principle of
Transmissibility in order to solve systems of
forces that are acting upon and/or within a
structure
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Varignons Theorem a theorem in mechanics that
establishes the dependence between moments of
forces of a given system and the moment of their
resultant force. This theorem was first
formulated and proved by the French scientist P.
Varignon. According to Varignons theorem, if a
system of forces Fi has a resultant force R, then
the moment M0(R) of the resultant force relative
to any center O (or z-axis) is equal to the sum
of the moments M0(Fi) of the component forces
relative to the same center O (or the same
z-axis). Mathematically, Varignons theorem is
expressed by the formulas M0(R) SM0(Fi) or Mz(R)
SMz(Fi) Varignons theorem is used for solving a
series of problems in mechanics (especially
statics), resistance of materials, construction
theories, and other areas.
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COUPLE

• TWO EQUAL AND OPPOSITE FORCES FORM A COUPLE
• COUPLE CREATES A TURNING ACTION i.e MOMENT
• MOMENT OF A COUPLE IS MC LX F
• A SINGLE FORCE DOES NOT FORM A COUPLE
• MOMENT MAY BE CLOCKWISE OR ANTICLOCKWISE

F
COUPLE ARM L
F
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NOT THIS COUPLE.
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CHARACTERISTICS OF A COUPLE

• ALGEBRIC SUM OF FORCES FORMING COUPLE IS ZERO
• ALGEBRIC SUM OF MOMENTS OF ALL FORCES
  CONSTITUTING A COUPLE IS EQUAL OF THE MOMENT OF
  THE COUPLE ITSELF
• COUPLE IS BALANCED BY A COUPLE AND NOT BY A FORCE
• NUMBER OF COPLANER COUPLES CAN BE REDUCED TO A
  SINGLE COUPLE AND ITS MAGNITUDE ME ?MCi

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LETS SCRATCH.. OUR BRAIN
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NUMERICAL
RAlt350N
RBlt 350N
3m
X?
A
B
200N
400N