Overview of Mechanical Engineering
for Non-MEs
Part 1: Statics
3
Rigid Bodies I:Equivalent Systems ofForces
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Contents
Introduction
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Introduction
•Treatment of a body as a single particle is not always possible. Ingeneral, the size of the body and the specific points of application of theforces must be considered.
•Most bodies in elementary mechanics are assumed to be rigid, i.e., theactual deformations are small and do not affect the conditions ofequilibrium or motion of the body (deformable body will be consideredin Chapter ??).
•Current chapter describes the effect of forces exerted on a rigid body andhow to replace a given system of forces with a simpler equivalent system.
•moment of a force about a point
•moment of a force about an axis
•moment due to a couple
•Any system of forces acting on a rigid body can be replaced by anequivalent system consisting of one force acting at a given point and onecouple.
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External and Internal Forces
•Forces acting on rigid bodies aredivided into two groups:
-External forces
-Internal forces
•External forces are shown in afree-body diagram.
•If unopposed, each external forcecan impart a motion oftranslation or rotation, or both.
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Principle of Transmissibility: Equivalent Forces
•Principle of Transmissibility -Conditions of equilibrium or motion arenot affected by transmitting a forcealong its line of action.NOTE: F and F’ are equivalent forces.
•Moving the point of application ofthe force F to the rear bumperdoes not affect the motion or theother forces acting on the truck.
•Principle of transmissibility mayNOT always apply in determininginternal forces and deformations.
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Vector Product of Two Vectors
•Concept of the moment of a force about a point ismore easily understood through applications ofthe vector product or cross product.
•Vector product of two vectors P and Q is definedas the vector V which satisfies the followingconditions:
1.Line of action of V is perpendicular to planecontaining P and Q.
2.Magnitude of V is
3.Direction of V is obtained from the right-handrule.
•Vector products:
-are not commutative,
-are distributive,
-are not associative,
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Vector Products: Rectangular Components
•Vector products of Cartesian unit vectors,
•Vector products in terms of rectangularcoordinates
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Moment of a Force About a Point
•A force vector is defined by its magnitude anddirection. Its effect on the rigid body also dependson it point of application.
•The moment of F about O is defined as
•The moment vector MO is perpendicular to theplane containing O and the force F.
•Any force F’ that has the same magnitude anddirection as F, is equivalent if it also has the same lineof action and therefore, produces the same moment.
•Magnitude of MO measures the tendency of the forceto cause rotation of the body about an axis along MO.
The sense of the moment may be determined by theright-hand rule.
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Moment of a Force About a Point
•Two-dimensional structures have length and breadth butnegligible depth and are subjected to forces contained inthe plane of the structure.
•The plane of the structure contains the point O and theforce F. MO, the moment of the force about O isperpendicular to the plane.
•If the force tends to rotate the structure clockwise, thesense of the moment vector is out of the plane of thestructure and the magnitude of the moment is positive.
•If the force tends to rotate the structure counterclockwise,the sense of the moment vector is into the plane of thestructure and the magnitude of the moment is negative.
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Varignon’s Theorem
•The moment about a give point O of theresultant of several concurrent forces is equalto the sum of the moments of the variousmoments about the same point O.
•Varigon’s Theorem makes it possible toreplace the direct determination of themoment of a force F by the moments of twoor more component forces of F.
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Rectangular Components of the Moment of a Force
The moment of F about O,
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Rectangular Components of the Moment of a Force
The moment of F about B,
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Rectangular Components of the Moment of a Force
For two-dimensional structures,
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Sample Problem 3.1
A 100-lb vertical force is applied to the end of alever which is attached to a shaft at O.
Determine:
a)moment about O,
b)horizontal force at A which creates the samemoment,
c)smallest force at A which produces the samemoment,
d)location for a 240-lb vertical force to producethe same moment,
e)whether any of the forces from b, c, and d isequivalent to the original force.
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Sample Problem 3.1
a) Moment about O is equal to the product of theforce and the perpendicular distance between theline of action of the force and O. Since the forcetends to rotate the lever clockwise, the momentvector is into the plane of the paper.
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Sample Problem 3.1
c) Horizontal force at A that produces the samemoment,
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Sample Problem 3.1
c) The smallest force A to produce the same momentoccurs when the perpendicular distance is amaximum or when F is perpendicular to OA.
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Sample Problem 3.1
d) To determine the point of application of a 240 lbforce to produce the same moment,
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Sample Problem 3.1
e)Although each of the forces in parts b), c), and d)produces the same moment as the 100 lb force, noneare of the same magnitude and sense, or on the sameline of action. None of the forces is equivalent to the100 lb force.
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Sample Problem 3.4
The rectangular plate is supported bythe brackets at A and B and by a wireCD. Knowing that the tension in thewire is 200 N, determine the momentabout A of the force exerted by thewire at C.
SOLUTION:
The moment MA of the force F exertedby the wire is obtained by evaluatingthe vector product,
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Scalar Product of Two Vectors
•The scalar product or dot product betweentwo vectors P and Q is defined as
•Scalar products:
-are commutative,
-are distributive,
-are not associative,
•Scalar products with Cartesian unit components,
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Scalar Product of Two Vectors: Applications
•Angle between two vectors:
•Projection of a vector on a given axis:
•For an axis defined by a unit vector:
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Mixed Triple Product of Three Vectors
•Mixed triple product of three vectors,
•The six mixed triple products formed from S, P, andQ have equal magnitudes but not the same sign,
•Evaluating the mixed triple product,
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Moment of a Force About a Given Axis
•Moment MO of a force F applied at the point Aabout a point O,
•Scalar moment MOL about an axis OL is theprojection of the moment vector MO onto theaxis,
•Moments of F about the coordinate axes,
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Moment of a Force About a Given Axis
•Moment of a force about an arbitrary axis,
•The result is independent of the point Balong the given axis.
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Sample Problem 3.5
a)about A
b)about the edge AB and
c)about the diagonal AG of the cube.
d)Determine the perpendicular distancebetween AG and FC.
A cube is acted on by a force P asshown. Determine the moment of P
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Sample Problem 3.5
•Moment of P about A,
•Moment of P about AB,
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Sample Problem 3.5
•Perpendicular distance between AG and FC,
Therefore, P is perpendicular to AG.
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Moment of a Couple
•Two forces F and -F having the same magnitude,parallel lines of action, and opposite sense are saidto form a couple.
•Moment of the couple,
•The moment vector of the couple isindependent of the choice of the origin of thecoordinate axes, i.e., it is a free vector that canbe applied at any point with the same effect.
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Moment of a Couple
Two couples will have equal moments if
•
•the two couples lie in parallel planes, and
•the two couples have the same sense orthe tendency to cause rotation in the samedirection.
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Addition of Couples
•Consider two intersecting planes P1 andP2 with each containing a couple
•Resultants of the vectors also form acouple
•By Varigon’s theorem
•Sum of two couples is also a couple that is equalto the vector sum of the two couples
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Couples Can Be Represented by Vectors
•A couple can be represented by a vector with magnitudeand direction equal to the moment of the couple.
•Couple vectors obey the law of addition of vectors.
•Couple vectors are free vectors, i.e., the point of applicationis not significant.
•Couple vectors may be resolved into component vectors.