1) The document discusses the concepts of resultant forces, including the resultant of two concurrent coplanar forces using the parallelogram law, and the resultant of several concurrent coplanar forces using the polygon law.
2) It also covers parallel coplanar forces and how to determine the resultant and point of application using the principle of moments and Varignon's theorem.
3) Methods to reduce a system of forces to a resultant force and couple are presented, including reducing multiple forces to a single resultant force using the summation of moments.
moments couples and force couple systems by ahmad khanSelf-employed
To determine the resultant force acting at the top of the tower (point D), I would:
1. Resolve each cable force into its x and y components.
2. Use the parallelogram law of forces to combine the x-components of each cable force into a single x-component force. Do the same for the y-components.
3. The x and y component forces obtained from step 2 are the x and y components of the resultant force acting at D.
4. Use the Pythagorean theorem to determine the magnitude of the resultant force from its x and y components.
5. Use trigonometry to determine the direction of the resultant force relative to the x-axis
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
The document discusses basic principles of statics and structural design. It covers:
1) Statics deals with forces on bodies at rest, while dynamics deals with moving bodies. Statics is used to analyze structural systems and ensure strength, stiffness, and stability.
2) Structural design involves preliminary design stages using experience and intuition, followed by detailed analysis and load estimations based on statics principles.
3) Static equilibrium equations must be satisfied for coplanar forces. Systems can be determinate, allowing determination of specific unknowns, or indeterminate.
Free body diagrams show the relative magnitude and direction of all forces acting upon an object by isolating it from its surroundings. The document provides examples of free body diagrams for the Statue of Liberty, a sitting gorilla, a wooden swing, a bungee jumper's bucket, a traffic light, and the pin at point A of a truss bridge. Forces are shown as vectors with arrows indicating direction and labels providing magnitudes. Diagrams for static systems will sum the vertical and horizontal forces to zero, indicating equilibrium.
Free body diagrams show the relative magnitude and direction of all forces acting on an object. They include only physical forces touching the object like gravity, applied forces, friction, and reactions, drawn as arrows from a dot representing the object. To analyze motion, forces are resolved into horizontal and vertical components and Newton's second law is applied to each direction separately. For example, with an applied force at an angle on a block, the horizontal force component gives acceleration along the plane while the vertical forces sum to zero for no jump.
Definition of force,types of forces,law of forces,system of forces, moment of a force, couple,moment of a couple,types of moments,features of couple and principle of moments.
moments couples and force couple systems by ahmad khanSelf-employed
To determine the resultant force acting at the top of the tower (point D), I would:
1. Resolve each cable force into its x and y components.
2. Use the parallelogram law of forces to combine the x-components of each cable force into a single x-component force. Do the same for the y-components.
3. The x and y component forces obtained from step 2 are the x and y components of the resultant force acting at D.
4. Use the Pythagorean theorem to determine the magnitude of the resultant force from its x and y components.
5. Use trigonometry to determine the direction of the resultant force relative to the x-axis
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
The document discusses basic principles of statics and structural design. It covers:
1) Statics deals with forces on bodies at rest, while dynamics deals with moving bodies. Statics is used to analyze structural systems and ensure strength, stiffness, and stability.
2) Structural design involves preliminary design stages using experience and intuition, followed by detailed analysis and load estimations based on statics principles.
3) Static equilibrium equations must be satisfied for coplanar forces. Systems can be determinate, allowing determination of specific unknowns, or indeterminate.
Free body diagrams show the relative magnitude and direction of all forces acting upon an object by isolating it from its surroundings. The document provides examples of free body diagrams for the Statue of Liberty, a sitting gorilla, a wooden swing, a bungee jumper's bucket, a traffic light, and the pin at point A of a truss bridge. Forces are shown as vectors with arrows indicating direction and labels providing magnitudes. Diagrams for static systems will sum the vertical and horizontal forces to zero, indicating equilibrium.
Free body diagrams show the relative magnitude and direction of all forces acting on an object. They include only physical forces touching the object like gravity, applied forces, friction, and reactions, drawn as arrows from a dot representing the object. To analyze motion, forces are resolved into horizontal and vertical components and Newton's second law is applied to each direction separately. For example, with an applied force at an angle on a block, the horizontal force component gives acceleration along the plane while the vertical forces sum to zero for no jump.
Definition of force,types of forces,law of forces,system of forces, moment of a force, couple,moment of a couple,types of moments,features of couple and principle of moments.
The document discusses systems of forces acting on a body. It defines different types of force systems including coplanar forces that act in the same plane, and non-coplanar forces whose lines of action are not in the same plane. Within these categories, forces can be further classified as collinear if they act along the same line, concurrent if they intersect at a single point, parallel if their lines of action are parallel, and like or unlike based on direction. Examples are provided of different force system configurations such as concurrent coplanar forces intersecting in a plane, and non-concurrent non-coplanar forces that do not intersect and act in different planes.
A system of forces is said to exist when there is more than one force acting on an object or at a point simultaneously. It is important to acquaint ourselves with the various systems of forces. Copy the link given below and paste it in new browser window to get more information on System of Force:-
http://www.transtutors.com/homework-help/civil-engineering/fundamental-concepts/system-of-force.aspx
A force is an external agent acting on another body. This force may moves or tends to move the body in the direction of its action. The force is a vector quantity since it is represented by its magnitude and direction. The force may be of pulling or pushing type. Copy the link given below and paste it in new browser window to get more information on Principle Of Transmissibility:-
http://www.transtutors.com/homework-help/mechanical-engineering/force-systems-and-analysis/principle-of-transmissibility.aspx
This document contains notes from a prestressed concrete design course taught by Munshi Galib Muktadir. It defines torque, angular acceleration, and moment of inertia. It explains that moment of inertia is a geometric property that reflects how an area's points are distributed around an axis. It also describes area moment of inertia and polar moment of inertia, noting that larger values mean a beam will bend or twist less.
This document discusses concepts related to static equilibrium of rigid bodies, including:
- Conditions for static equilibrium are that the net force and net moment are both zero
- Free body diagrams show all forces acting on a body in isolation
- Types of supports (fixed, hinge, roller) and the reactions they provide are described
- Concepts like two-force and three-force members, Lami's theorem, and finding equilibrant forces to balance unbalanced systems are explained
- Several example problems are provided to illustrate applying concepts to determine reactions and tensions in static systems
The force is defined as the action of a body about another body and it is a vector quantity. The vector quantity, the force, has four characteristic: magnitude, direction, sense and point of application.
The document discusses mechanics and dynamics. It begins by defining mechanics as a branch of physics dealing with the behavior of physical bodies under forces or displacements. Dynamics is identified as the branch of mechanics concerned with the effects of forces on motion, especially external forces. The document goes on to provide information on internal forces, types of fundamental forces, Newton's laws of motion, and concepts such as inertia, mass, and equilibrium. It includes examples of applying dynamics concepts to problems involving forces.
This document contains a student's acknowledgements and summary of basic mechanics concepts including: statics, dynamics, kinematics, kinetics, hydromechanics, hydrostatics, hydrodynamics, the parallelogram law of forces, Lami's theorem, resultants of force systems, and resolving forces into components. The student thanks God, their teacher, parents, and friends for their support and contributions to completing their project.
This document provides information about the ME 101 Engineering Mechanics course offered by the Department of Civil Engineering at Indian Institute of Technology Guwahati. It includes the lecture and tutorial schedule, syllabus, textbook references, assessment details, and tutorial group assignments. The course covers fundamental concepts of mechanics including forces, equilibrium, structures, friction, moments of inertia, kinematics, and kinetics applied to both particles and rigid bodies using Newton's laws of motion and the law of universal gravitation.
The document discusses various types of forces including contact forces, body forces, point forces, distributed forces, frictional forces, wind forces, and cohesive and adhesive forces. It also describes characteristics of forces such as magnitude, direction, and point of application. Additionally, it covers concepts like Newton's third law of motion, systems of forces, resolution of forces, and fundamental principles of mechanics including transmissibility, the parallelogram law of forces, gravitation, and superposition.
This document introduces systems of forces and their components. It begins by stating the objectives of understanding different force systems, resolving forces into components, and calculating moments and resultants. Key concepts explained include defining a force as a vector, resolving forces into horizontal and vertical components, and using the parallelogram and polygon laws to determine the resultant of multiple forces. Examples are provided to demonstrate resolving forces and finding resultants.
In Engineering Mechanics the static problems are classified as two types: Concurrent and Non-Concurrent force systems. The presentation discloses a methodology to solve the problems of Concurrent and Non-Concurrent force systems.
1. The document discusses concepts related to engineering mechanics and strength of materials including force, force systems, equations of equilibrium, free body diagrams, and laws of forces.
2. Key topics covered include defining force as a vector quantity, types of forces and force systems, using scalar equations to represent equilibrium conditions, drawing free body diagrams to isolate external forces, and laws of forces including the parallelogram and triangle laws.
3. Engineering mechanics concepts are presented including stress, strain, elasticity, bending moments, shear forces, buckling, and material properties.
Composition of forces refers to finding the resultant force of multiple forces acting on a body. Resolution of a force is splitting a force into components without changing its effect. There are two main methods to find the resultant force - analytical using parallelogram law or triangle law of forces, and graphical using vector resolution. The polygon law of forces states the sides of a polygon representing forces in magnitude and direction will have a resultant equal to the closing side in opposite direction.
A free body diagram is a sketch that shows an isolated body and all the external forces acting on it. It does not show internal forces or the body's environment. Forces are drawn as vectors at the point where they are applied. Common forces shown include weight, normal force, friction, and tension. Free body diagrams are used to write force balance equations for mechanical systems. Examples of free body diagrams include a block on a ramp, a book on a table, an object in projectile motion, and an object slowing down due to friction.
Principle of Virtual Work in structural analysisMahdi Damghani
The document provides an overview of the principle of virtual work (PVW) for structural analysis. Some key points:
1) PVW is based on the concept of work and energy methods. It states that for a structure in equilibrium under applied forces, the total virtual work done by these forces due to a small arbitrary displacement is zero.
2) PVW can be used to determine unknown internal forces or displacements in statically indeterminate structures by applying virtual displacements or forces.
3) Examples demonstrate using PVW to calculate the bending moment at a point in a beam and the force in a member of an indeterminate truss by equating the external virtual work to internal virtual work.
Lecture 1 Introduction to statics Engineering Mechanics hibbeler 14th editionaxmedbaasaay
This document discusses mechanics, specifically rigid body mechanics. It defines mechanics as dealing with forces and motion of bodies. Rigid body mechanics examines objects that do not deform under applied forces. This field is divided into statics, which considers motionless bodies, and dynamics, which examines moving bodies. Rigid body mechanics forms the basis for understanding deformable bodies and fluid mechanics. Key concepts introduced are particles, rigid bodies, forces, and Newton's laws of motion.
The document defines key concepts related to moments and forces, including:
1) The moment of a force is the tendency of a force to produce rotation about an axis, and is equal to the force magnitude times the perpendicular distance to the axis (moment arm).
2) Varignon's theorem states the algebraic summation of force components about any point equals the moment of the original force.
3) Two parallel forces of equal magnitude but opposite direction form a couple, which causes rotation about an axis perpendicular to its plane.
4) Forces can be resolved and moved to different points using the principle that the moment of any added couple must equal the original force times the distance of movement.
1) The document discusses engineering mechanics concepts related to moments including how to calculate the moment of a force using the cross product of the force and perpendicular distance from the axis.
2) It also covers parallel force systems and how to calculate the resultant force of coplanar forces that are parallel, unlike, equal or unequal.
3) The key properties of moments and couples are defined including how couples can only be balanced by another couple of the opposite sense.
The document discusses systems of forces acting on a body. It defines different types of force systems including coplanar forces that act in the same plane, and non-coplanar forces whose lines of action are not in the same plane. Within these categories, forces can be further classified as collinear if they act along the same line, concurrent if they intersect at a single point, parallel if their lines of action are parallel, and like or unlike based on direction. Examples are provided of different force system configurations such as concurrent coplanar forces intersecting in a plane, and non-concurrent non-coplanar forces that do not intersect and act in different planes.
A system of forces is said to exist when there is more than one force acting on an object or at a point simultaneously. It is important to acquaint ourselves with the various systems of forces. Copy the link given below and paste it in new browser window to get more information on System of Force:-
http://www.transtutors.com/homework-help/civil-engineering/fundamental-concepts/system-of-force.aspx
A force is an external agent acting on another body. This force may moves or tends to move the body in the direction of its action. The force is a vector quantity since it is represented by its magnitude and direction. The force may be of pulling or pushing type. Copy the link given below and paste it in new browser window to get more information on Principle Of Transmissibility:-
http://www.transtutors.com/homework-help/mechanical-engineering/force-systems-and-analysis/principle-of-transmissibility.aspx
This document contains notes from a prestressed concrete design course taught by Munshi Galib Muktadir. It defines torque, angular acceleration, and moment of inertia. It explains that moment of inertia is a geometric property that reflects how an area's points are distributed around an axis. It also describes area moment of inertia and polar moment of inertia, noting that larger values mean a beam will bend or twist less.
This document discusses concepts related to static equilibrium of rigid bodies, including:
- Conditions for static equilibrium are that the net force and net moment are both zero
- Free body diagrams show all forces acting on a body in isolation
- Types of supports (fixed, hinge, roller) and the reactions they provide are described
- Concepts like two-force and three-force members, Lami's theorem, and finding equilibrant forces to balance unbalanced systems are explained
- Several example problems are provided to illustrate applying concepts to determine reactions and tensions in static systems
The force is defined as the action of a body about another body and it is a vector quantity. The vector quantity, the force, has four characteristic: magnitude, direction, sense and point of application.
The document discusses mechanics and dynamics. It begins by defining mechanics as a branch of physics dealing with the behavior of physical bodies under forces or displacements. Dynamics is identified as the branch of mechanics concerned with the effects of forces on motion, especially external forces. The document goes on to provide information on internal forces, types of fundamental forces, Newton's laws of motion, and concepts such as inertia, mass, and equilibrium. It includes examples of applying dynamics concepts to problems involving forces.
This document contains a student's acknowledgements and summary of basic mechanics concepts including: statics, dynamics, kinematics, kinetics, hydromechanics, hydrostatics, hydrodynamics, the parallelogram law of forces, Lami's theorem, resultants of force systems, and resolving forces into components. The student thanks God, their teacher, parents, and friends for their support and contributions to completing their project.
This document provides information about the ME 101 Engineering Mechanics course offered by the Department of Civil Engineering at Indian Institute of Technology Guwahati. It includes the lecture and tutorial schedule, syllabus, textbook references, assessment details, and tutorial group assignments. The course covers fundamental concepts of mechanics including forces, equilibrium, structures, friction, moments of inertia, kinematics, and kinetics applied to both particles and rigid bodies using Newton's laws of motion and the law of universal gravitation.
The document discusses various types of forces including contact forces, body forces, point forces, distributed forces, frictional forces, wind forces, and cohesive and adhesive forces. It also describes characteristics of forces such as magnitude, direction, and point of application. Additionally, it covers concepts like Newton's third law of motion, systems of forces, resolution of forces, and fundamental principles of mechanics including transmissibility, the parallelogram law of forces, gravitation, and superposition.
This document introduces systems of forces and their components. It begins by stating the objectives of understanding different force systems, resolving forces into components, and calculating moments and resultants. Key concepts explained include defining a force as a vector, resolving forces into horizontal and vertical components, and using the parallelogram and polygon laws to determine the resultant of multiple forces. Examples are provided to demonstrate resolving forces and finding resultants.
In Engineering Mechanics the static problems are classified as two types: Concurrent and Non-Concurrent force systems. The presentation discloses a methodology to solve the problems of Concurrent and Non-Concurrent force systems.
1. The document discusses concepts related to engineering mechanics and strength of materials including force, force systems, equations of equilibrium, free body diagrams, and laws of forces.
2. Key topics covered include defining force as a vector quantity, types of forces and force systems, using scalar equations to represent equilibrium conditions, drawing free body diagrams to isolate external forces, and laws of forces including the parallelogram and triangle laws.
3. Engineering mechanics concepts are presented including stress, strain, elasticity, bending moments, shear forces, buckling, and material properties.
Composition of forces refers to finding the resultant force of multiple forces acting on a body. Resolution of a force is splitting a force into components without changing its effect. There are two main methods to find the resultant force - analytical using parallelogram law or triangle law of forces, and graphical using vector resolution. The polygon law of forces states the sides of a polygon representing forces in magnitude and direction will have a resultant equal to the closing side in opposite direction.
A free body diagram is a sketch that shows an isolated body and all the external forces acting on it. It does not show internal forces or the body's environment. Forces are drawn as vectors at the point where they are applied. Common forces shown include weight, normal force, friction, and tension. Free body diagrams are used to write force balance equations for mechanical systems. Examples of free body diagrams include a block on a ramp, a book on a table, an object in projectile motion, and an object slowing down due to friction.
Principle of Virtual Work in structural analysisMahdi Damghani
The document provides an overview of the principle of virtual work (PVW) for structural analysis. Some key points:
1) PVW is based on the concept of work and energy methods. It states that for a structure in equilibrium under applied forces, the total virtual work done by these forces due to a small arbitrary displacement is zero.
2) PVW can be used to determine unknown internal forces or displacements in statically indeterminate structures by applying virtual displacements or forces.
3) Examples demonstrate using PVW to calculate the bending moment at a point in a beam and the force in a member of an indeterminate truss by equating the external virtual work to internal virtual work.
Lecture 1 Introduction to statics Engineering Mechanics hibbeler 14th editionaxmedbaasaay
This document discusses mechanics, specifically rigid body mechanics. It defines mechanics as dealing with forces and motion of bodies. Rigid body mechanics examines objects that do not deform under applied forces. This field is divided into statics, which considers motionless bodies, and dynamics, which examines moving bodies. Rigid body mechanics forms the basis for understanding deformable bodies and fluid mechanics. Key concepts introduced are particles, rigid bodies, forces, and Newton's laws of motion.
The document defines key concepts related to moments and forces, including:
1) The moment of a force is the tendency of a force to produce rotation about an axis, and is equal to the force magnitude times the perpendicular distance to the axis (moment arm).
2) Varignon's theorem states the algebraic summation of force components about any point equals the moment of the original force.
3) Two parallel forces of equal magnitude but opposite direction form a couple, which causes rotation about an axis perpendicular to its plane.
4) Forces can be resolved and moved to different points using the principle that the moment of any added couple must equal the original force times the distance of movement.
1) The document discusses engineering mechanics concepts related to moments including how to calculate the moment of a force using the cross product of the force and perpendicular distance from the axis.
2) It also covers parallel force systems and how to calculate the resultant force of coplanar forces that are parallel, unlike, equal or unequal.
3) The key properties of moments and couples are defined including how couples can only be balanced by another couple of the opposite sense.
Here are the key differences between a particle and a rigid body in mechanics:
Particle:
- Has no size or internal structure, it is considered a point object.
- Cannot transfer or support moments/torques. Only forces can act on a particle.
Rigid Body:
- Has size, shape and internal structure. It is an extended object.
- Can transfer and support both forces and moments/torques at its different points.
Other differences:
- Equations of equilibrium for a particle involve only forces. Equations for a rigid body involve both forces and moments.
- Deformations are not considered for a particle as it has no internal structure. Deformations may need to be
This document discusses engineering mechanics concepts including:
1. Mechanics involves how bodies work together due to applied forces and is divided into statics, dynamics, kinematics, and kinetics.
2. Basic concepts like length, mass, time, scalars, and vectors are introduced. Newton's laws of motion, the law of transmissibility of force, and the parallelogram law for adding forces are covered.
3. Forces are defined and the concepts of resultant, force systems, composition and resolution of forces, moments, Varignon's theorem, and couples are explained.
This document provides an overview of mechanics of solids, including fundamentals of statics such as laws for analyzing coplanar concurrent force systems using the parallelogram, triangle, and polygon laws. It also discusses analytical methods for resolving and composing concurrent coplanar forces using components, as well as graphical methods. Additional topics covered include coplanar non-concurrent forces, moments, couples, Varignon's theorem, and equilibrium conditions.
This document provides an introduction to analyzing forces acting on rigid bodies. It defines key terms like moment of a force, vector and scalar products, and equivalent force systems. The document also presents methods for determining the moment of a force about a point using vector products and rectangular components. Sample problems demonstrate calculating moments and locating equivalent forces. In summary, the document outlines techniques for representing and evaluating systems of forces on rigid bodies in mechanics.
The document discusses non-concurrent forces and how to find their resultant. It defines non-concurrent forces as those whose lines of action do not meet at a single point. It provides examples of such forces, like those on a ladder leaning against a wall. The document discusses using graphical and algebraic methods to resolve non-parallel, non-concurrent forces into components. It also addresses calculating the total moment of such force systems to find the resultant force and its location.
This document discusses coplanar non-concurrent forces and the conditions for equilibrium. It defines key concepts like moment, couple, equivalent couples, and Varignon's principle of moments. It also explains that for a body to be in equilibrium under coplanar non-concurrent forces, the net force components and net moment must all be zero. Specifically, the sum of moments about any point must equal the moment of the resultant force about that point.
This document provides an introduction to analyzing forces acting on rigid bodies. It defines key terms like moment of a force, vector and scalar products, and equivalent force systems. It also presents the principles of transmissibility and Varignon's theorem. Sample problems demonstrate calculating the moment of a force about a point and determining equivalent forces that produce the same moment. The document aims to describe how to represent and simplify systems of forces exerted on rigid bodies.
This document outlines the course content for Mechanics of Solids. It is divided into two parts: Mechanics of Rigid Bodies and Mechanics of Deformable Bodies. The first part covers topics like forces, moments, couples, and equilibrium of force systems for rigid bodies. The second part covers stresses, strains, indeterminate problems, and shear and bending moment diagrams for deformable bodies. The document also lists several recommended reference books and provides an overview of the first lecture which introduces concepts of rigid bodies, forces, and force composition and resolution.
This document provides an overview of forces and force systems in engineering. It introduces the concept of a force vector and its components. Key points covered include:
- A force vector depends on both magnitude and direction. Most bodies are treated as rigid.
- Any system of forces on a rigid body can be replaced by a single force and couple. The principle of transmissibility allows treating forces as "sliding vectors".
- Forces are classified as contact or body forces, and as concentrated or distributed. Weight is treated as a concentrated force through the center of gravity.
- Methods for adding concurrent forces include the parallelogram and triangle laws. Forces can be resolved into rectangular components.
This document provides information about moments of forces from a textbook on vector mechanics for engineers. It defines the moment of a force about a point and describes how to find the moment vector. It also discusses couples, which are two forces of equal magnitude and opposite direction, and how to calculate the moment of a couple. The document explains how to resolve a force into equivalent force and couple components at a given point using vector algebra. It provides examples of calculating the equivalent force and couple for systems of forces.
This document discusses various types of machine balancing. It begins by defining static and dynamic balancing. Static balancing deals with balancing forces when a machine is at rest, while dynamic balancing deals with balancing forces during motion. It then discusses balancing of single and multiple rotating masses, as well as reciprocating masses. Methods for analytically and graphically balancing multiple masses are provided. The document also covers balancing of engines with different cylinder configurations, including inline, V-shaped, radial, and locomotive engines. Partial balancing techniques are discussed for reducing unbalanced forces in locomotives.
This document discusses structural analysis and beam analysis. It covers topics such as support reactions, shear force and bending moment diagrams, equilibrium of forces in structures, and determinacy. Key points include:
1. To design structures, it is necessary to know bending moments, torsion moments, shear forces, and axial forces in each member.
2. Equilibrium requires that the downward forces from gravity have equal and opposite upward reaction forces.
3. Shear force and bending moment diagrams show the variation of these internal forces along a beam under different loading conditions.
This document discusses the principles of static equilibrium for rigid bodies in 2D and 3D. It states that for a body to be in static equilibrium, the sum of the external forces and moments/torques must equal zero. Free body diagrams are used to identify all external forces on a body. For a 2D rigid body, the 3 equations of static equilibrium involve summing forces in the x, y, and z directions. For a 3D rigid body, 6 equations are required to account for summing forces and moments in 3 dimensions. Free body diagrams are illustrated for 2D and 3D examples.
The document defines moment as the turning effect of a force about a point, and provides the mathematical formula for moment as M = P x l, where P is the force and l is the perpendicular distance to the point. It then explains Varignon's Principle of Moments, which states that the algebraic sum of moments of all forces about any point equals the moment of the resultant force about the same point. The document goes on to discuss parallel forces and couples, defining a couple as two equal and parallel forces acting in opposite directions, and provides examples of couples in real life. It compares torque and moment to a couple. Finally, it provides example problems calculating resultants and moments.
MECHANICS OF SOLIDS(coplanar concurrent forces)Parthivpal17
This document discusses coplanar concurrent forces and methods for finding their resultant. It defines coplanar concurrent forces as multiple forces acting at the same point in the same plane. Two conditions must be satisfied for equilibrium: the sum of forces along two perpendicular directions must equal zero. Methods to find the resultant include the parallelogram law, resolving forces into perpendicular components, and the triangle law of forces. The polygon law of forces states that closing sides of a polygon represent the resultant when sides are taken in opposite order to the forces. Lami's theorem specifies that each force is proportional to the sine of the angle between the other two forces.
The document discusses coplanar non-concurrent force systems where multiple forces act in the same plane but at different points. It defines moment as the product of a force and its perpendicular distance from the point of interest. A couple is formed by two equal and opposite forces that rotate a body without translating it. Varignon's principle states that the sum of the moments of individual forces equals the moment of the resultant force about any point. Conditions for equilibrium of coplanar non-concurrent forces on a body are also presented. An example problem finds the resultant and location of a non-concurrent force system.
This document outlines the course CM 154 Statics of Rigid Bodies taught by Dr. Kofi Agyekum. It will cover topics related to engineering mechanics including types of force systems, resolution of forces, and determining the resultant of concurrent coplanar forces using various methods like the triangle law and parallelogram law. Students will learn to analyze rigid bodies that are either at rest or in static equilibrium by applying the principles of static equilibrium. Recommended textbooks are also listed.
The document discusses the objectives and syllabus for the course AR7202 Mechanics of Structures I. It aims to make students aware of how structural resolutions are important in architectural design and teach basic properties of solids and sections. The syllabus covers principles of statics including types of forces, force systems, and equilibrium. It also addresses analysis of plane trusses, properties of cross-sections, elastic properties of solids, and elastic constants. The course aims to teach students to apply equilibrium concepts and understand material behavior under forces.
This document discusses different types of beams and beam loadings. It defines beams as members that support loads perpendicular to their longitudinal axis. It describes simply supported beams, cantilever beams, overhanging beams, propped cantilevers, continuous beams, and beams with one end hinged and the other on rollers. The document also discusses concentrated loads, uniformly distributed loads, uniformly varying loads, general loadings, and external moments on beams. It provides examples of how to represent these loads for structural analysis.
This document provides information about simple machines. It discusses different types of simple machines like the wheel and axle, screw, lever, pulley, and their uses. Simple machines make work easier by changing the amount or direction of force. They allow us to lift heavy loads using less effort. Common examples mentioned are a screwdriver and wrench, which act as a wheel and axle to make unscrewing easier. The document also covers compound machines, lifting machines, and defines terms related to simple machines like mechanical advantage and efficiency.
The document discusses the concept of friction. It defines friction as the resisting force along the surfaces of contact that opposes the motion of one body moving over another. It states that the magnitude of the frictional force depends on factors like the materials of the surfaces in contact, the roughness of the surfaces, and the pressure between them. It also distinguishes between different types of friction like static friction, dynamic friction, sliding friction, and rolling friction.
The document discusses the concepts of centroid and centre of gravity. The centroid, also known as the center of gravity, is the point location of an object's average position of weight. For symmetrical objects, the centroid will be at the exact center, but for irregularly shaped objects it depends on the weight distribution. Various methods are described for calculating the centroid of standard shapes, composite objects, and through integration. The centroid is important for balancing objects and determining their stability.
The document discusses frames and trusses, which are structures consisting of bars, rods, angles, and channels pinned or fastened together to support loads and transmit them to supports. Trusses contain only two-force members that experience either tension or compression, while frames can contain multi-force members and experience transverse forces as well. Common truss configurations include pinned, gusset plate, and bolted or welded joints. Trusses are analyzed using methods of joints or sections to determine member forces.
The document provides an introduction to the topic of applied mechanics. It defines mechanics as a branch of science dealing with bodies at rest or in motion under the influence of forces. Mechanics is divided into statics, dynamics, and the mechanics of rigid bodies, deformable bodies, fluids, and their compressible and incompressible types. Fundamental concepts such as particles, bodies, rigid bodies, deformable bodies, space, time, force, and systems of units are also defined. Several fundamental principles are outlined, including Newton's laws of motion and the laws of gravitation, the parallelogram of forces, and Lami's theorem. Finally, the document instructs students to form groups of five and design a simple
Essential Tools for Modern PR Business .pptxPragencyuk
Discover the essential tools and strategies for modern PR business success. Learn how to craft compelling news releases, leverage press release sites and news wires, stay updated with PR news, and integrate effective PR practices to enhance your brand's visibility and credibility. Elevate your PR efforts with our comprehensive guide.
केरल उच्च न्यायालय ने 11 जून, 2024 को मंडला पूजा में भाग लेने की अनुमति मांगने वाली 10 वर्षीय लड़की की रिट याचिका को खारिज कर दिया, जिसमें सर्वोच्च न्यायालय की एक बड़ी पीठ के समक्ष इस मुद्दे की लंबित प्रकृति पर जोर दिया गया। यह आदेश न्यायमूर्ति अनिल के. नरेंद्रन और न्यायमूर्ति हरिशंकर वी. मेनन की खंडपीठ द्वारा पारित किया गया
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Youngest c m in India- Pema Khandu BiographyVoterMood
Pema Khandu, born on August 21, 1979, is an Indian politician and the Chief Minister of Arunachal Pradesh. He is the son of former Chief Minister of Arunachal Pradesh, Dorjee Khandu. Pema Khandu assumed office as the Chief Minister in July 2016, making him one of the youngest Chief Ministers in India at that time.
2. Introduction
• The objective for the current chapter is to investigate the effects of forces
on particles:
- replacing multiple forces acting on a particle with a single
equivalent or resultant force,
- relations between forces acting on a particle that is in a
state of equilibrium.
2-2
3. Resultant of Two Forces
• force: action of one body on another;
characterized by its point of application,
magnitude, line of action, and sense.
• Experimental evidence shows that the
combined effect of two forces may be
represented by a single resultant force.
• The resultant is equivalent to the diagonal of
a parallelogram which contains the two
forces in adjacent legs.
• Force is a vector quantity.
2-3
4. Vectors
• Vector: parameters possessing magnitude and direction
which add according to the parallelogram law. Examples:
displacements, velocities, accelerations.
• Scalar: parameters possessing magnitude but not
direction. Examples: mass, volume, temperature
• Vector classifications:
- Free vectors may be freely moved in space without
changing their effect on an analysis.
- Sliding vectors may be applied anywhere along their
line of action without affecting an analysis.
• Equal vectors have the same magnitude and direction.
• Negative vector of a given vector has the same magnitude
and the opposite direction.
2-4
5. External and Internal Forces
• Forces acting on rigid bodies are
divided into two groups:
- External forces
- Internal forces
• External forces are shown in a
free-body diagram.
• If unopposed, each external force can impart a motion of
translation or rotation, or both.
2-5
6. Principle of Transmissibility: Equivalent Forces
• Principle of Transmissibility -
Conditions of equilibrium or motion are
not affected by transmitting a force
along its line of action.
NOTE: F and F’ are equivalent forces.
• Moving the point of application of
the force F to the rear bumper
does not affect the motion or the
other forces acting on the truck.
2-6
9. Resultant of Several Forces
• When a number of coplanar forces are acting on a rigid body
then these forces can be replaced by a single force which has
the same effect on the rigid body as that of all the forces acting
together, then this single force is known as the resultant of
several forces.
2-9
10. Resultant of Several Forces
• Definition:
• A single force which can replace a number of forces acting one a rigid body,
without causing any change in the external effects on the body is known as
the resultant force.
2 - 10
11. Resultant of Coplanar Forces
• The resultant of coplanar forces may be determined by
following two method:
1. Analytical method
2. Graphical method
2 - 11
12. Resultant of Collinear Coplanar Forces
• Analytical Method
• The resultant is obtained by adding all
the forces if they are acting in the same
direction.
• If any one of the forces is acting in the
opposite direction, then resulting is
obtained by subtracting that forces.
R= F1 + F2 - F3
2 - 12
13. Resultant of Collinear Coplanar Forces
• Graphical Method
• Some suitable scale is chosen and vectors are drawn to the
chosen scale.
• These vectors are added/or subtracted to find the resultant.
F1 F2 F3
a b c d
R= F1 + F2 + F3
2 - 13
14. Resultant of Concurrent Coplanar Forces
• Concurrent coplanar forces are those forces which act in the
same plane and they intersect or meet at a common point.
• Following two cases are consider:
i. When two forces act at a point.
ii. When more than two forces act at a point.
2 - 14
15. Resultant of Concurrent Coplanar Forces
• When two forces act at a point.
• (a) Analytical Method:
– When two forces act at a point,
their resultant is found by the law
of parallelogram of forces.
• The magnitude of Resultant force R
• The direction of Resultant force R with the force P
Q sin α
θ = tan
P + Q cos α
−1
2 - 15
16. Resultant of Concurrent Coplanar Forces
• (b) Graphical Method
i. Choose a convenient scale to represent the forces P and
Q.
ii.From point O, draw a vector OA= P.
iii.Now from point O, draw another vector OB= Q and at
an angle of α as shown in fig.
iv.Complete the parallelogram by drawing lines AC║ to
OB and BC ║ to OA.
v. Measure the length OC.
vi.Resultant R = OC x Chosen Scale
• The Direction of resultant is given by angle θ.
• Measure the angle θ.
2 - 16
17. Resultant of Concurrent Coplanar Forces
• When more than two forces act at a point.
• Analytical Method
R = (∑ H ) 2 + (∑ V ) 2 tan θ =
∑V
∑H
2 - 17
18. Resultant of Concurrent Coplanar Forces
• (b) Graphical Method
• The resultant of several forces acting at a point is found graphically with the
help of Polygon law of forces.
• Polygon law of forces
– “if a number of coplanar forces are acting at a point such that they can be
represented in magnitude and direction by the sides of a polygon taken in
the same order, then their resultant is represented in magnitude and
direction by the closing side of the polygon taken in the opposite order.”
F2 c
F1 F3
d F2
o F4 b
F1
F3 e
F4 R a
2 - 18
20. Coplanar Parallel Forces
• A parallel coplanar force system consists of two
or more forces whose lines of action are parallel
to each other.
• Two parallel forces will not intersect at a point.
• The line of action of forces are parallel so that
for finding the resultant of two parallel forces,
the parallelogram cannot be drawn.
• The resultant of such forces can be determined
by applying the principle of moments.
2 - 20
21. Coplanar Parallel Forces
• Moment of Forces
• The tendency of a force to produce
rotation of body about some axis or point
is called the moment of a force.
• moment of a force about a point
• moment of a force about an axis
• moment due to a couple
The moment (m) of the force F about O is given by,
M=F x d
Unit: force x distance =F*L = N-m, kN-m (SI unit)
2 - 21
24. Moment of Forces
• The tendency of moment is to rotate about the body
in the clockwise direction about O is called
clockwise moment.
• If the tendency of a moment is to rotate the body in
anti clockwise direction, then that moment is
known as anti clock wise moment.
• Sign conv. : clockwise (-ve),
counterclockwise (+ve)
2 - 24
25. Resultant Moment of Forces
• The resultant moment of F1, F2, and F3 about O
= -F1 x r1 – F2 x r2 + F3 x r3
2 - 25
26. Resultant Moment of Forces
20N
30N D C
• For Example:
• Find the resultant moment about point A.
• Soln:
• Forces at point A and B passes through Point 2m
A.
• So perpendicular distances from A on the line
of action of these forces will be zero.
• Hence their moments about point A will be
zero. A 2m B 10N
• Moment of force at C about point A:
20 x 2 =40N(CCW) 40N
• Moment of force at D about point A :
30 x 2= 60N(CCW)
So resultant moment at point A = 40 + 60 = 100N(CCW)
2 - 26
27. Principle of Moments
• The Principle of Moments, also known as Varignon's Theorem, states that
the moment of any force about any point is equal to the algebraic sum of the
moments of the its components of that force about that point.
• As with the summation of force combining to get resultant force
u uu uu
r r r uu
r
R = F1 + F2 + K + Fn
• Similar resultant comes from the addition of moments
uuu
r u r uu
r uu
r uu
r
M 0 = R d R = F1 d1 + F2 d 2 + K + Fn d n
2 - 27
29. Types of Parallel Forces
• Two important types of parallel forces
1. Like parallel forces
2. Unlike parallel forces
• Like Parallel forces
• Two parallel forces which are acting in the same direction
are known as like parallel forces.
• The magnitude of a forces may be equal or unequal.
• Unlike Parallel forces
• Two parallel forces which are acting in the opposite
direction are known as like unparallel forces.
• The magnitude of a forces may be equal or unequal.
2 - 29
30. Resultant of Two Parallel forces
• The resultant of following two parallel forces will be considered:
– Two parallel forces are like.
– Two parallel forces are unlike and are unequal in magnitude.
– Two parallel forces are unlike but equal in magnitude.
2 - 30
31. Resultant of Two Parallel forces
• Two parallel forces are like
• Suppose that two like but unequal parallel
forces act on a body at position A and B as
shown in figure.
• We have to calculate the resultant force acting on the body and
its position.
• From condition of static equilibrium;
R = F1 + F2 ...(1)
2 - 31
32. Resultant of Two Parallel forces
• The position of R can be obtained by using Varignon’s theorem. To use the
theorem consider a point O along the line AB, such that
• Algebraic sum of moments of F1 and F2 about O = Moment of resultant about O
• Now,
Moment of F1 about O = F1 × AO (clockwise)
Moment of F2 about O = F2 × BO (anti-clockwise)
Moment of R about O = R × CO (anti-clockwise)
– F1 × AO + F2 × BO = + R × CO …(2)
– F1 × AO + F2 × BO = (F1 + F2) × CO
F1 (AO + CO) = F2 (BO – CO)
F1 × AC = F2 × BC
F1 / F2 = BC / AC
• Therefore, it can be observed that R acts at a point C which divides the length
AB in the ratio inversely proportional to the magnitudes of F1 and F2.
2 - 32
33. Resultant of Two Parallel forces
• Two parallel forces are unlike and are unequal in magnitude
• Suppose that two unlike and unequal
parallel forces act on a body at position A
and B as shown in figure.
• We have to calculate the resultant force
acting on the body and its position.
• From condition of static equilibrium,
R = F1 – F2 …(1)
2 - 33
34. Resultant of Two Parallel forces
• Once again, the position of R can be obtained by using Varigonon’s theorem.
Consider a point O along the line AB, such that
• Algebraic sum of moments of F1 and F2 about O = Moment of resultant about O
• Now, Moment of F1 about O = F1 × AO (clockwise)
• Moment F2 about O = F2 × BO (anti-clockwise)
• Moment of R about O = R × CO (anti-clockwise)
⇒ – F1 × AO – F2 × BO = – R × CO …(2)
⇒ F1 × AO + F2 × BO = R × CO
⇒ F1 × AO + F2 × BO = (F1 – F2) × CO
⇒ F2 (BO + CO) = F1 (CO – AO)
⇒ F1 / F2 = BC / AC …(3)
• Since F1 > F2, BC will be greater than AC. Hence point C will lie outside AB on
the same side of F1. Thus, it can be observed that R acts at C which externally
divide length AB in the ratio inversely proportional to the magnitude of F1 and
F2 .
2 - 34
35. Moment of a Couple
• Two parallel forces are unlike but equal in magnitude
• Two parallel forces having different lines of action, equal in
magnitude, but opposite in sense constitute a couple.
• A couple causes rotation about an axis
perpendicular to its plane.
• The perpendicular distance between the
parallel forces is known as arm of the couple.
M=F*a
Unit: Nm
2 - 35
36. Moment of a Couple
• Two couples are equivalent if they cause the same moment:
2 - 36
37. Resolution of a Force into a Force and a Couple
• A force, F, acting at point B can be replaced by the force, F,
and a moment, MA, acting at point A.
F F F F
B A B A B A
= d F = MA
MA = d F
2 - 37
38. Replace a Force-Couple System with Just Forces
F F
A A
F2
d2
MA =
F2
C C
d2 F2 = MA
2 - 38
39. Reducing a System of Forces to a Resultant Force-
Couple System (at a Chosen Point)
F1
R
r1
r2 F2
A
r3 = MA
r r
F3 R = ∑F
r r r
MA = ∑ r ×F
( )
2 - 39
40. Reducing a System of Forces to a Resultant Force-
Couple System (at a Chosen Point)
2 - 40
41. Reduce a System of Forces to a Single Resultant Force
F1
R R R
r1
r2 F2 B
A
r3 = MA
= MA
F3 R R
Using method B
from prior slide =
2 - 41
42. Reduce a System of Forces to a Single Resultant Force
R
R
Ry
B B
A
A Rx Rx
MA Ry
R
R
dx
dx R = –MA
2 - 42
43. General Case of parallel forces in a plane
• R1= Resultant of (F1, F2, F4) and R2= Resultant of (F3, F5)
• The resultant R1 and R2 are acting in opposite direction and parallel to each
other.
• Two important case are possible.
• 1. R1 may not be equal to R2.
– Then two unequal parallel forces acting in opposite direction.
– The resultant R= R1-R2
– The point of application easily found with the help of Varignon's
Theorem or moments of forces.
2 - 43
44. General Case of parallel forces in a plane
• 2. R1 is equal to R2.
– Then two equal parallel forces acting in opposite direction.
– The resultant R= R1-R2=0
– Now the system may be reduce to a couple or a system is in equilibrium.
– The algebraic sum moment of all forces(F1, F2,…, F5) taken about any
point.
– If
∑ M = 0 then system is in equilibrium .
– If ∑ M ≠ 0 the system reduce to a resultant couple.
– And the calculated moment gives the moment of that couple.
2 - 44
45. Equivalent System
• Two force systems that produce the same external effects on a rigid body are
said to be equivalent.
• An equivalent system for a given system of coplanar forces, is a
combination of a force passing through a given point and a moment about
that point.
• The force is the resultant of all forces acting on the body.
• The moment is the sum of all the moments about that point.
• Equivalent system consists of :
• (1) a single force R passing through the given point P
• (2) a single moment MR
2 - 45
46. Equivalent System
• For Examples:
• Determined the equivalent system through
point O.
• These means find:
• (1) a single resultant force, R
• (2) a single moment through, O
2 - 46
47. Difference between moment and couple
Moment Couple
• Moment = force x perpendicular • Two equal and opposite forces whose lines of
distance M = Fd action are different from a couple
• It is produced by a single force not • It is produced by the two equal and opposite
passing through Centre of gravity of parallel, non collinear forces.
the body.
• The force move the body in the • Resultant force of couple is zero. Hence,
direction of force and rotate the body. body does not move, but rotate only.
It is the resultant force.
• To balance the force causing moment, • Couple cannot be balanced by a single force,
equal and opposite force is required. it can be balanced by a couple only.
• For example, • For example,
• To tight the nut by spanner • To rotate the key in lock
• To open or close the door • To open or close the wheel valve of water line
• To rotate the steering wheel of car.
2 - 47
48. Equilibrium of Rigid Bodies
External forces Body start moving or rotating.
• If the body does not start moving and also does not start rotating about any
point, then body is said to be in equilibrium.
• For a rigid body in static equilibrium, the external forces and moments are
balanced and will impart no translational or rotational motion to the body.
2 - 48
49. Equilibrium of Rigid Bodies
• Principle of Equilibrium:
•
∑F = 0 …………(1)
• ∑M = 0 ………....(2)
• Eq. (1) is known as the force law of equilibrium and Eq. (2) is known as the
moment law of equilibrium.
• The forces are generally resolved into horizontal and vertical components.
∑ Fx = 0 ∑ Fy = 0
2 - 49
50. Equilibrium of Rigid Bodies
• Equilibrium of non-concurrent forces system:
• A non-concurrent forces system will be in equilibrium if the resultant of all
forces and moment is zero.
∑ Fx = 0 ∑ Fy = 0 ∑ M = 0
• Equilibrium of concurrent forces system:
• For the concurrent forces, the line of actions of all forces meet at a point, and
hence the moment of those forces about that point will be zero
automatically.
∑ Fx = 0 ∑ Fy = 0
2 - 50
51. Equilibrium of Rigid Bodies
• Force Law of Equilibrium:
• There are three main force Law of Equilibrium:
• Two force system
• Three force system
• Four or more force system
2 - 51
52. Equilibrium of Rigid Bodies
(1) Two force system:
• According to this principle, if a body is in equilibrium under the action of
two forces, then they must be equal, opposite and collinear.
• If the two forces acting on a body are equal and
opposite but are parallel, as shown in fig., then the
body will not be in equilibrium.
• Two condition is satisfied:
• (1) ∑ Fx = 0 (2) ∑ Fy = 0 as F1 = F2
• Third condition is not satisfied:
• (3) ∑ M ≠0 MA = -F2 X AB
• A body will not be in equilibrium under the action of two equal and
opposite parallel forces.
• Two equal and opposite parallel forces produce a couple.
2 - 52
53. Equilibrium of Rigid Bodies
• (2) Three force system:
• According to this principle, if a body is in equilibrium under the action of
three forces then the resultant of any two forces must be equal, opposite and
collinear with the third force.
• Three forces acting on a body either concurrent or parallel
• Case (a) When three forces are concurrent
• The resultant of F1 and F2 is given by R.
• If the force F3 is collinear equal, opposite to the
resultant R, then the body will be in equilibrium.
• The force F3 which is equal and opposite to resultant R is known as
equilibrant.
• Hence for three concurrent forces acting on a body when the body is in
equilibrium, the resultant of the two forces should be equal and opposite to
the third force.
2 - 53
54. Equilibrium of Rigid Bodies
• Case (2): When three forces are parallel
• If the three parallel forces F1, F2, and F3 are acting in the same direction,
then there will be a resultant R= F1 + F2 + F3 and body will not be in
equilibrium.
• If the three forces are acting in opposite direction and their magnitude is so
adjusted that there is no resultant forces and body is in equilibrium.
• Apply the three condition of equilibrium:
• (1) Σ Fx = 0,(No horizontal forces) (2) Σ Fy = 0, (F1+ F3=F2)
• (3) Σ M = 0 about any point.
Σ MA= -F2 X AB + F3 X AC
• For equilibrium Σ MA should be zero.
-F2 X AB + F3 X AC= 0
• If the distance AB and AC are such that the above equation
is satisfied, then the body will be in equilibrium under the action of three
parallel forces.
2 - 54
55. Equilibrium of Rigid Bodies
• (3) Four or more force system:
• According to this principle, if a body is in equilibrium under the action of
four forces then the resultant of any two forces must be equal, opposite and
collinear with the resultant of the other two forces.
Σ Fx = 0, Σ Fy = 0, Σ M = 0
2 - 55
56. Equilibrium of Rigid Bodies
• Two moment equations.
• Σ Fy = 0
• Σ MA = 0,
• Σ MB = 0
• where A and B are any two points in the xy-plane, provided that the line AB
is not parallel to the y-axis.
2 - 56
57. Equilibrium of Rigid Bodies
• Free Body Diagram of a Body:
• The first step in equilibrium analysis is to identify all the forces that act on
the body. This is accomplished by means of a free-body diagram.
• The free-body diagram (FBD) of a body is a sketch of the body showing all
forces that act on it. The term free implies that all supports have been
removed and replaced by the forces (reactions) that they exert on the body.
• Free-body diagrams are fundamental to all engineering disciplines that are
concerned with the effects that forces have on bodies.
• The construction of an FBD is the key step that translates a physical problem
into a form that can be analyzed mathematically.
2 - 57
58. Equilibrium of Rigid Bodies
• Forces that act on a body can be divided into two general categories—
Reactive forces (or, simply, reactions) and
Applied forces (action)
• Reactions are those forces that are exerted on a body by the supports to
which it is attached.
• Forces acting on a body that are not provided by the supports are called
applied forces.
2 - 58
59. Equilibrium of Rigid Bodies
• The following is the general procedure for constructing a free-body
diagram.
1. A sketch of the body is drawn assuming that all supports (surfaces of
contact, supporting cables, etc.) have been removed.
2. All applied forces are drawn and labeled on the sketch. The weight of the
body is considered to be an applied force acting at the center of gravity.
3. The support reactions are drawn and labeled on the sketch. If the sense of a
reaction is unknown, it should be assumed. The solution will determine the
correct sense: A positive result indicates that the assumed sense is correct,
whereas a negative result means that the correct sense is opposite to the
assumed sense.
4. All relevant angles and dimensions are shown on the sketch.
2 - 59
60. Equilibrium of Rigid Bodies
• The most difficult step to master in the construction of FBDs is the
determination of the support reactions.
• Flexible Cable (Negligible Weight).
• A flexible cable exerts a pull, or tensile force, in the direction of the cable.
With the weight of the cable neglected, the cable forms a straight line. If its
direction is known, removal of the cable introduces one unknown in a free-
body diagram—the magnitude of the force exerted by the cable.
Support Reaction(s) Description Number of
of reaction(s) unknowns
Tension of unknown one
magnitude T in the
direction of the
cable
2 - 60
61. Equilibrium of Rigid Bodies
• Frictionless Surface: Single Point of Contact.
• When a body is in contact with a frictionless surface at only one point, the
reaction is a force that is perpendicular to the surface, acting at the point of
contact.
• This reaction is often referred to simply as the normal force.
• Therefore, removing such a surface introduces one unknown in a free-body
diagram—the magnitude of the normal force.
Support Reaction(s) Description Number of
of reaction(s) unknowns
Force of unknown one
magnitude N
directed normal to
the surface
2 - 61
62. Equilibrium of Rigid Bodies
• Roller Support.
• A roller support is equivalent to a frictionless surface: It can only exert a
force that is perpendicular to the supporting surface.
• The magnitude of the force is thus the only unknown introduced in a free-
body diagram when the support is removed.
Support Reaction(s) Description Number of
of reaction(s) unknowns
Force of unknown one
magnitude N normal
to the surface
supporting the roller
2 - 62
63. Equilibrium of Rigid Bodies
• Surface with Friction: Single Point of Contact.
• A friction surface can exert a force that acts at an angle to the surface.
• The unknowns may be taken to be the magnitude and direction of the force.
• However, it is usually advantageous to represent the unknowns as N and F,
the components that are perpendicular and parallel to the surface,
respectively.
• The component N is called the normal force, and F is known as the friction
force.
Support Reaction(s) Description Number of
of reaction(s) unknowns
Force of unknown
magnitude N normal Two
to the surface and a
friction force of
unknown magnitude
F parallel to the
surface
2 - 63
64. Equilibrium of Rigid Bodies
• Pin Support.
• Neglecting friction, the pin can only exert a force that is normal to the
contact surface, shown as R in Fig.(b).
• A pin support thus introduces two unknowns: the magnitude of R and the
angle α that specifies the direction of R (α is unknown because the point
where the pin contacts the surface of the hole is not known).
2 - 64
65. Equilibrium of Rigid Bodies
• Built-in (Cantilever) Support.
• A built-in support, also known as a cantilever support, prevents all motion
of the body at the support. Translation (horizontal or vertical movement) is
prevented by a force, and a couple prohibits rotation.
• Therefore, a built-in support introduces three unknowns in a free-body
diagram:
– The magnitude and direction of the reactive force R (these unknowns are
commonly chosen to be two components of R, such as Rx and Ry )
– The magnitude C of the reactive couple.
Support Reaction(s) Description Number of
of reaction(s) unknowns
Unknown force R
and a couple of Three
unknown magnitude
C
2 - 65
66. Equilibrium of Rigid Bodies
• You should keep the following points in mind when you are drawing
free-body diagrams.
1. Be neat. Because the equilibrium equations will be derived directly from the
free-body diagram, it is essential that the diagram be readable.
2. Clearly label all forces, angles, and distances with values (if known) or
symbols (if the values are not known).
3. Show only forces that are external to the body (this includes support
reactions and the weight). Internal forces occur in equal and opposite pairs
and thus will not appear on free-body diagrams.
2 - 66
67. Equilibrium of Rigid Bodies
• Sample Problem:
The mass of the bar is 50 kg. Take g = 9.81 m/s2
2 - 67
68. Equilibrium of Rigid Bodies
• Sample Problem:
• Neglecting the weights of the members.
2 - 68
72. Equilibrium of Rigid Bodies
• Equilibrium analysis of a body
• The three steps in the equilibrium analysis of a body are:
• Step 1: Draw a free-body diagram (FBD) of the body that shows all of the
forces and couples that act on the body.
• Step 2: Write the equilibrium equations in terms of the forces and couples
that appear on the free-body diagram.
• Step 3: Solve the equilibrium equations for the unknowns.
2 - 72
73. Equilibrium of Rigid Bodies
• Statically determinate and Statically indeterminate
• The force system that holds a body in equilibrium is said to be statically
determinate if the number of independent equilibrium equations equals the
number of unknowns that appear on its free-body diagram
• If the number of unknowns exceeds the number of independent equilibrium
equations, the problem is called statically indeterminate.
• The solution of statically indeterminate problems requires the use of
additional principles.
• When the support forces are sufficient to resist translation in both the x and
y directions as well as rotational tendencies about any point, the rigid body
is said to be completely constrained, otherwise the rigid body is unstable or
partially constrained.
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74. Equilibrium of Rigid Bodies
• Statically indeterminate and Improper Constraints
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