Transcript Example Problems

BASIC CONCEPTS: RECAP
BY
GP CAPT NC CHATTOPADHYAY
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
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
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 ]
CONCEPT OF RIGID BODY
HELICOPTER FRAME AS A RIGID BODY
• An un deformed body is a rigid body
• No object is absolutely rigid
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.
RECAP…..
Concurrent Force Systems:
A concurrent force system contains forces whose
lines-of action meet at some one point.
Forces may be tensile (pulling)
RECAP…..
Concurrent Force Systems:
A concurrent force system contains forces whose
lines-of action meet at some one point.
Forces may be compressive (pushing)
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.
TWO EFFECTS OF FORCE
Force exerted on a body has two effects:
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
The external effect, which is tendency
to change the motion of the body
The internal effect, which is the
tendency to deform the body.
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
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.
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.
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.
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.
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.
PRINCIPLE OF
TRANSMISSIBILITY
Line of action
• This principle leads to use of sliding vectors
•Only point of application is changed without altering magnitude direction
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.
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.
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.
R=√(P2+ Q2+2PQ.COS θ)
RECAP…
Resultant Forces
For multiple forces action on a point, the forces can
be broken into the components of x and y.
Example Problems
1.
Determine the magnitude and direction of the
resultant of the two forces.
Example Problems
2.
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.
Example Problems
3.
Determine the magnitude and direction of P so that the
resultant of P and the 900-N force is a vertical force of 2700N directed downward.
Example Problems
4.
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.
Example Problems
5.
Determine the force in each supporting wire.
Example Problems
6.
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?
Example Problems
7.
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?
Example Problems
8.
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.
Example Problems
9.
Determine the resultant of the forces on the bolt.
Example Problems
10. 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.
Example Problems
11. Two forces P and Q of magnitude P=1000-lb and Q=1200-lb are applied to
the aircraft connection. Knowing that the connection is in equilibrium,
determine the tensions T1 and T2.
Example Problems
12. Determine the forces in each of the four wires.
Example Problems
13. The blocks are at rest on a frictionless incline. Solve for the forces F1 and F2
required for equilibrium.
Example Problems
14. 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.
Example Problems
15. Solve for the resisting force at pin A to maintain equilibrium.
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.
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)
MOMENT OF A FORCE
MOMENT
M= r X F
VARIGNON’S 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
Varignon’s 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 Varignon’s
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,
Varignon’s theorem is expressed by the formulas
M0(R) =ΣM0(Fi)
or
Mz(R) =ΣMz(Fi)
Varignon’s theorem is used for solving a series of problems in
mechanics (especially statics), resistance of materials, construction
theories, and other areas.
COUPLE
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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
NOT THIS COUPLE….
CHARACTERISTICS OF A COUPLE

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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 IT’S MAGNITUDE ME= ∑MCi
LET’S SCRATCH….. OUR BRAIN
NUMERICAL
RA<350N
A
X=?
RB< 350N
3m
B
200N
400N