APPLIED MECHANICS TEXTBOOK PDF
the present volume. Accordingly, the title. "Applied. Mechanics for Engineers " has been given to the book. The book is intended as a text-book for engineering. Reader Q&A. To ask other readers questions about A Textbook Of Applied Mechanics, please sign up. Popular Answered Questions. how to download in pdf?. PDF Drive is your search engine for PDF files. As of today we have 78,, eBooks for you to download for free. No annoying ads, no download limits, enjoy .
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entitled as A TEXTBOOK of APPLIED MECHANICS to the Students of Degree, Diploma and A.M.I.E. (I) classes. This object of this book is to present the subject . APPLIED MECHANICS A TEXTBOOK OF APPLIED MECHANICS (Including Laboratory Practicals) S.I. UNITS By R.K. RAJPUT M.E. (Heat Power Engg.) Hons . PDF | On Jul 10, , Mohamed Handawi Elmunafi and others published Applied Mechanics and Materials. Show abstract. Sustainable Construction: Green Building Design and Delivery. Book. Jan Charles Joseph.
Supplementary units. From the scientific point of view division of S. Nevertheless the General Conference, considering the advantages of a single, practical, world-wide system for international relations, for teaching and for scientific work, decided to base the international system on a choice of six well-defined units given in Table 1 below: Table 1. Base Units Quantity Name Symbol length metre m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K luminous intensity candela cd amount of substance mole mol The second class of S.
Several of these algebraic expressions in terms of base units can be replaced by special names and symbols can themselves be used to form other derived units. Derived units may, therefore, be classified under three headings.
Some of them are given in Tables 2, 3 and 4. Examples of S. Derived Units with Special Names S. A 2 magnetic flux weber Wb V. A —2 luminous flux lumen lm — cd. K —1 mol —1 The S. Refer Table 5 and Table 6.
Table 5. Supplementary Units S. Work, Energy or Heat: Specific heat: Thermal conductivity: Heat transfer co-efficient: Units and expressions 1. Value of g 0 9. Gas constant R Flow through nozzle-Exit Introduction to mechanics. Basic definitions. Rigid body. Scalar and vector quantities. Fundamental units and derived units.
Fundamentals have always been stressed in the learning of new skills. Similarly, the mechanics branch of science which deals with the forces and their effect on bodies on which they act is founded on basic concepts and forms the ground-work for further study in the design and analysis of machines and structures. Mechanics can be divided into two parts. In mechanics the term strength of materials refers to the ability of the individual parts of a machine or structure to resist loads.
It also permits the determination of dimensions to ensure sufficient strength of the various parts. Dynamics may be further into the following two groups: Engineering Mechanics Statics Dynamics Kinematics Kinetics Physical science of motion of points Physical science of bodies and the forces which are in motion Physical science of bodies in rest Fig.
The branch of science which deals with the study of different laws of mechanics as applied to solution of engineering problems is called Applied Mechanics. This term is applied to the linear dimensions of a straight or curved line. For example, the diameter of circle is the length of a straight line which divides the circle into two equal parts ; the circumference is the length of its curved perimeter. The two dimensional size of a shape or a surface is its area. The shape may be flat lie in a plane or curved, for example, the size of a plot of land, the surface of a fluorescent bulb, or the cross-sectional size of a shaft.
The three dimensional or cubic measure of the space occupied by a substance is known as its volume. This term is applied to any action on the body which tends to make it move, change its motion, or change its size and shape. A force is usually thought of a push or pull, such as a hand pushing against a wall or the pull of a rope fastened to a body. The external force per unit area, or the total force divided by the total area on which it acts, is known as pressure. Water pressure against the face of a dam, steam pressure in a boiler, or earth pressure against a retaining wall are some examples.
The amount of matter contained in a body is called its mass, and for most problems in mechanics, mass may be considered constant. The force with which a body is attracted towards the centre of earth by the gravitational pull is called its weight.
The weight of a unit volume of a body or substance is the density. This term is sometimes called weight density, to distinguish it from a similary definition mass density made in terms of mass. The tendency of a force to cause rotation about some point is known as a moment. The action of a force which causes rotation to take place is known as torque.
The action of a belt on a pulley causes the pulley to rotate because of torque. Also if you grasp a piece of chalk near each end and twist your hands in opposite directions, it is the developed torque that causes the chalk to twist and, perhaps, snap. It is the quantity of matter contained in a body. It is the force with which the body is attracted towards the centre of earth. It is constant at all places. It is different at different places. It resists motion in the body.
It produces motion in the body. It is a scalar quantity since it has magnitude only. It is a vector quantity since it has magnitude as well as direction. It can be measured by an ordinary balance. It is measured by a spring balance. It is never zero. It is zero at the centre of earth. It is measured in kilogram kg in M. It is measured in kilogram weight kg wt. The energy developed by a force acting through a distance against resistance is known as work. The distance may be along a straight line or along a curved path.
Common forms of work include a weight lifted through a height, a pressure pushing a volume of substance, and torque causing a shaft to rotate. The rate of doing work, or work done per unit time is called power. For example, a certain amount of work is required to raise an elevator to the top of its shaft. A 5 HP motor can raise the elevator, but a 20 HP motor can do the same job four times faster.
It differs from an elastic body in the sense that the latter undergoes deformation under the effect of forces acting on it and returns to its original shape and size on removal of the forces acting on the body. The rigidity of a body depends upon the fact that how far it undergoes deformation under the effect of forces acting on it.
In real sense no solid body is perfectly rigid because everybody changes it size and shape under the effect of forces acting on it. It actual practice, the deformation i. A scalar quantity is one that has magnitude only. Mass, volume, time and density. Vector quantity. A vector quantity is one that has magnitude as well as direction. Force, velocity, acceleration and moment etc.
A vector quantity is represented by a line carrying an arrow head at one end. The length of the line to convenient scale equals the magnitude of the vector. The line, together with its arrow head, defines the direction of the vector. Suppose a force of 60 N is applied to point A in Fig. The vector AB represents this force since its length equals 60 N to scale and its direction is proper.
If the vector BA is drawn to same scale Fig. The units of these quantities are called fundamental units and are developed by L, M and T respectively.
The units of all other quantities except above are derived with the help of fundamental units and thus they are known as derived units.
For example, units of velocity, acceleration, density etc. Foot-Pound-Second system F. Centimetre-Gram-Second system C. Metre-Kilogram-Second system M. International system of units S. Foot-Pound-Second system. In this system the units of fundamental quantities i. Centimetre-Gram-Second system.
In this system the value of length, mass and time are expressed as centimetre, gram and second respectively. Metre-Kilogram-Second system. In this system units of length mass and time are metre, kilogram and second respectively. International system of units. This system considers three more fundamental units of electric current, temperature and luminous intensity in addition to the fundamental units of length, mass and time.
Important M. System S. Length metre m metre m 2. Mass kilogram kg kilogram kg 3. Time second sec or s second s 4. Plane angle radians rad radians 6. Area square metre m 2 square metre m 2 7. Volume cubic metre m 3 cubic metre m 3 8. Force kilogram weight kgf newton N Moment of kilogram weight kgf m newton metre Nm force metre Moment of kilogram metre kg m 2 kilogram metre kg m 2 inertia squared squared Kinematics deals with the motion of bodies without any reference to the cause of motion.
Kinetics deals with the relationship between forces and the resulting motion of bodies on which they act. Mass is the amount of matter contained in a body. Weight is the force with which a body is attracted towards the centre of the earth by the gravitational pull.
Density is the weight of unit volume of a body or substance. Power is the rate of doing work. A rigid body is one which does not change its shape and size under the effect of forces acting over it. The basic quantities or fundamental quantities are those quantities which cannot be expressed in terms of one another.
System of units. The four system of units in use are: Fill in the Blanks: Answers 1. Define the following terms: Force, volume, pressure, work and power. Mass, force, volume, velocity, time, acceleration. Describe the various systems of units. Which system of units is being followed these days and why? Units of force. Characteristics of a force. Representation of forces. Classification of forces. Force systems. Free body diagrams. Transmissibility of a force.
Component of a force. Principle of resolved parts. Laws of forces. Resultant of several coplanar concurrent forces. Equilibrium conditions for coplanar concurrent forces. FORCE Force is some thing which changes or tends to change the state of rest or of uniform motion of a body in a straight line. Force is the direct or indirect action of one body on another. The bodies may be in direct contact with each other causing direct motion or separated by distance but subjected to gravitational effects.
There are different kinds of forces such as gravitational, frictional, magnetic, inertia or those cause by mass and acceleration. A static force is the one which is caused without relative acceleration of the bodies in question. The force has a magnitude and direction, therefore, it is vector. While the directions of the force is measured in absolute terms of angle relative to a co-ordinate system, the magnitude is measured in different units depending on the situation.
When a force acts on a body, the following effects may be produced in that body: Absolute units 2. Gravitational units. Absolute units. Because the mass and acceleration are measured differently in different systems of units, so the units of force are also different in the various systems as given below: In the F. In the C. In the M. Gravitational units are the units which are used by engineers for all practical purposes. These units depend upon the weight of a body i.
Now the weight of a body of mass m i. So the gravitational units of force in the three systems of units i. The relationship of units of force is given as under: Usually, kg, wt or kgf is written simply as kg. These are: Magnitude i. Direction or line of action angle relative to a co-ordinate system.
Sense or nature push or pull. Point of application. Vector representation 2. Vector representation. A force can be represented graphically by a vector as shown in Figs. It is a method of designating a force by writing two capital letters one on either side of the force a shown in Fig.
Some of the important classifications are given as under: According to the effect produced by the force: When a force is applied external to a body it is called external force. The resistance to deformation, or change of shape, exerted by the material of a body is called an internal force. An active force is one which causes a body to move or change its shape. A force which prevents the motion, deformation of a body is called a passive force. According to nature of the force: Whenever there are two bodies in contact, each exerts a force on the other.
Out of these forces one is called action and other is called reaction. Action and reaction are equal and opposite. These are actually non-contacting forces exerted by one body or another without any visible medium transmission such as magnetic forces.
When a body is dragged with a string the force communicated to the body by the string is called the tension while, if we push the body with a rod, the force exerted on the body is called a thrust. According to whether the force acts at a point or is distributed over a large area. The force whose point of application is so small that it may be considered as a point is called a concentrated force.
A distributed force is one whose place of application is area. According to whether the force acts at a distance or by contact. Magnetic, electrical and gravitational forces are examples of non-contacting forces or forces at a distance.
The pressure of steam in a cylinder and that of the wheels of a locomotive on the supporting rails are examples of contacting forces. According to the relative positions of the lines of action of the forces, the forces may be classified as follows: Coplanar concurrent collinear force system. It is the simplest force system and includes those forces whose vectors lie along the same straight line refer Fig.
Coplanar concurrent non-parallel force system. Forces whose lines of action pass through a common point are called concurrent forces. In this system lines of action of all the forces meet at a point but have different directions in the same plane as shown in Fig.
Coplanar non-concurrent parallel force system. In this system, the lines of action of all the forces lie in the same plane and are parallel to each other but may not have same direction as shown in Fig.
Coplanar non-concurrent non-parallel force system. Such a system exists where the lines of action of all forces lie in the same plane but do not pass through a common point. Non-coplanar concurrent force system.
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This system is evident where the lines of action of all forces do not lie in the same plane but do pass through a common point. An example of this force system is the forces in the legs of tripod support for camera Fig.
Non-coplanar non-concurrent force system. Where the lines of action of all forces do not lie in the same plane and do not pass through a common point, a non-coplanar non-concurrent system is present. Each element or support can be isolated from the rest of the system by incorporating the net effect of the remaining system through a set of forces. Free-body diagrams are useful in solving the forces and deformations of the system. In case of a body in Fig.
The Fig. Let us consider another case of a beam shown in Fig. The beam is supported on a hinge at the left end and on a roller at the right end. The hinge offers vertical and horizontal reaction whereas the roller offers vertical reaction. The beam can be isolated from the supports by setting equivalent forces of the supports.
Similarly, the free body diagrams of hinge and roller supports are shown in Figs. A force may be considered as acting at any point on its line of action so long as the direction and magnitude are not changed. Suppose a body Fig. The force P will have the same effect if it is applied at 1, 2, 3 Fig.
This property of force is called transmissibility.
In any problem of mechanics, when the applied forces have no tendency to rotate the body on which they act, the body may be considered as a particle. Forces acting on the particle are concurrent, the point through which they pass being the point representing the particle. It is fundamental principle of mechanics, demonstrated by experiment, that when a force acts on a body which is free to move, the motion of the body is in the direction of the force, and the distance travelled in a unit time depends on the magnitude of the force.
Then, for a system of concurrent forces acting on a body, the body will move in the direction of the resultant of that system, and the distance travelled in a unit time will depend on the magnitude of the resultant. COMPONENT OF A FORCE As two forces acting simultaneously on a particle acting along directions inclined to each other can be replaced by a single force which produces the same effect as the given force, similarly, a single force can be replaced by two forces acting in directions which will produce the same effect as the given force.
This breaking up of a force into two parts is called the resolution of a force. The force which is broken into two parts is called the resolved force and the parts are called component forces or the resolutes. Generally, a force is resolved into the following two types of components: Mutually perpendicular components 2. Non-perpendicular components. Mutually perpendicular components.
Let the force P to be resolved is represented in magnitude and direction by oc in Fig. Complete the rectangle oacb. Then the other component P y at right angle to P x will be represented by ob which is also equal to ac.
Refer Fig. Let oc represents the given force P in magnitude and direction to some scale. Through c draw ca parallel to ob and cb parallel to oa to complete the parallelogram oacb. Then the vectors oa and ob represent in magnitude and direction to the same scale the components P 1 and P 2 respectively. Let the two forces P and Q be represented by the sides oa and ob of the parallelogram oacb and the resultant R of these two forces is given by the diagonal oc in magnitude and direction.
Let ox is the given direction. Draw bf, ae, cd and ag perpendicular to cd. It may be noted that this principle holds good for any number of forces. The various laws used for the composition of forces are given as under: Parallelogram law of forces 2.
Triangle law of forces 3. Polygon law of forces. Parallelogram law of forces. It states as under: Let two forces P and Q acting simultaneously on a particle be represented in magnitude and direction by the adjacent sides oa and ob of a parallelogram oacb drawn from a point o, their resultant R will be represented in magnitude and direction by the diagonal oc of the parallelogram.
The value of R can be determined either graphically or analytically as explained below: Draw vectors oa and ob to represent to some convenient scale the forces P and Q in magnitude and direction. Complete the parallelogram oacb by drawing ac parallel to ob and bc parallel to oa.
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The vector oc measured to the same scale will represent the resultant force R. Analytical method. As shown in Fig. Now from the geometry of the figure. Triangle law of forces. The resultant force R in this case can be obtained with the help of the triangle law of forces both graphically and analytically as given below: Draw vectors oa and ac to represent the forces P and Q to some convenient scale in magnitude and direction.
Join oc which will represent the resultant force R in magnitude and direction to the same scale. From the geometry of triangle oac Fig. The law is actually an extension of triangle law of forces. This is so because ob is the resultant of oa and ab and therefore oc which is resultant of ob and bc is also the resultant of oa, ab and bc. Similarly, od is the resultant of oc and cd and therefore of ob, bc and cd and finally of oa, ab, bc and cd.
Graphical method Polygon law of forces 2. Analytical method Principle of resolved parts. Draw a vector ab equal to force P 1 to some suitable scale and parallel to the line of action of P 1. Join ad which gives the required resultant in magnitude and direction, the direction being a to d as shown in the vector diagram.
Resultant by analytical method. Following sign conventions may be kept in view: The force of 60 N being horizontal. Refer to Fig. Example 2. Determine the resultant in magnitude and direction, if a the forces are pulls ; and b the 15 N force is a push and 20 N force is a pull. Case a. Find the magnitude and direction of P.
Two forces of magnitudes 3P, 2P respectively acting at a point have a resultant R. If the first force is doubled, the magnitude of the resultant is doubled. Find the angle between the forces. Two forces P and Q acting at a point have a resultant R.
If Q be doubled, R is doubled. Again if the direction of Q is reversed, then R is doubled, show that P: Two equal weights of 10 N are attached to the ends of a thin string which passes over three smooth pegs in a wall arranged in the form of an equilateral triangle with one side horizontal. Find the pressure on each peg. Let A, B and C be three pegs. A particle is acted upon by the following forces: Find graphically the magnitude and direction of the resultant force.
Draw space diagram as in Fig. So the resultant is a Determine analytically the magnitude and direction of the resultant of the following four forces acting at a point: The various forces acting at a point are shown in Fig. The following forces all pull act at a point: Find the resultant force.
What angle does it make with East? Determine the resultant completely. Let us resolve the forces along AB and AE. Analytical and graphical conditions of equilibrium of coplanar concurrent forces are given as under: Analytical conditions: The algebraic sum of components of all the forces in any direction which may be taken as horizontal, in their plane must be zero. The algebraic sum of components of all the forces in a direction perpendicular to the first direction, which may be taken as vertical, in their plane, must be zero.
Graphical conditions. The force polygon, i. Let us first consider the two forces P and Q which are represented by the two sides oa and ob of a parallelogram oadb as shown in Fig. Then the resultant of these two forces will be given by od the diagonal of the parallelogram in magnitude and direction. This means od should be equal to R in magnitude but opposite in direction to oc as P, Q and R are in equilibrium. A wheel has five equally spaced radial spokes, all in tension.
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If the tensions of three consecutive spokes are N, N and N respectively, find the tensions in the other two spokes. Three forces keep a particle in equilibrium. One acts towards east, another towards north-west and the third towards south. If the first be 5 N, find the other two.
Let force P act towards north-west and Q towards south. A machine weighing N is supported by two chains attached to some point on the machine. Find the tensions in the two chains.
The machine is in equilibrium under the following forces: Find the tensions T 1 and T 2 in kN in the cords. If the length of the string is equal to the radius of sphere, find tensions in the string and reaction on the wall. The sphere is in equilibrium under the action of following forces: A string is tied to two point at the same level and a smooth ring of weight W, which can slide freely along the string, is pulled by horizontal force P.
Find the value of P and the tension in the string. Let C be the position of the ring. A body of weight 20 N is suspended by two strings 5 m and 12 m long and other ends being fastened to the extremities of a rod of length 13 m. If the rod be so held that the body hangs immediately below its middle point, find out the tensions in the strings. The body is in equilibrium under the action of following forces: Neglect the self-weight of the members. Let P 1 and P 2 be the forces induced in the tie rod and jib respectively.
The tie rod will be under tension and jib will be compression as shown in Fig. Let T be the tension in BC. Weights W and 3 W are tied to the string at the points B and C respectively. At B, the forces in equilibrium are W, T 1 and T 2. A weight W is supported by two strings at the right angle to one another and attached to two points in the same horizontal line. Prove that their tensions are inversely proportional to their lengths.
Find the reactions at the surfaces of contact. Assuming all surfaces smooth, compute the reactions on the spheres at A and B. If the system is in equilibrium the line of action of the weight must pass through O, the point of intersection of the lines of action of the two reactions R A and R B. A uniform wheel 40 cm in diameter rests against a rigid rectangular block 10 cm thick as shown in Fig. Find the least pull through the centre of the wheel to just turn it over the corner of the block.
All surfaces are smooth. Find also the reaction of the block. The wheel weighs N. In this position, the following forces shall act on the wheel: If the pull is to be minimum it must be applied normal to AO. The cylinders shown in Fig.
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Find the reactions at the supports. The unknown in this case can be determined by resolving the forces along O 1 O 2 and in a direction perpendicular to O 1 O 2. Putting the value of R L in eqn. The channel is mm wide at the bottom with one side vertical. Find the reactions. Refer to Figs. R and S being the centres of the cylinders. Except for the reaction R Q , the directions of all other forces are known.
R Q sin R Q cos From eqn. Two forces N and N act at a point as shown in Fig. The resultant of the two forces is N. Direction of the resultant Fig. The force system shown in Fig. The clock shown in Fig. The resultant R of these forces is parallel to the incline. Determine P and R. Does the block move up or down the plane? The block will move up the incline Ans. Two spheres A and B weighing N and 75 N respectively and with the corresponding radii 75 mm and 50 mm are placed in a container as shown in Fig.
Determine the support reactions. They are connected by a bar of negligible weight hinged to each cylinder at their geometric centres by smooth pins.
Find the forces P, as shown that holds the system in the given position. In these figures N A and N B represent the reactions normal on the cylinders A and B respectively and Q represents the force exerted by the connected bar. It may be noted here that the direction of the force Q will be along the line AB, joining the centres of the two cylinders, because the bar AB is a two force member and the two forces acting on it must be equal and opposite to keep it in equilibrium.
Determine the resultant of the forces acting on the board as shown in Fig. Resolving the forces horizontally: Determine the resultant of the force system shown in Fig.
Assume that the co-ordinates are in metres. Ans Example 2. Find the resultant of the system of the forces shown in Fig. Force is something which changes or tends to change the state of rest or of uniform motion of a body in a straight line. The two commonly used units of force are: Gravitational Units are used by engineers for all practical purposes. Characteristics of a force are: Forces may be represented in two way: A force system is a collection of forces acting on a body in one or more planes.
The principle of transmissibility of forces states that when a force acts upon a body its effect is the same whatever point in its line of action is taken as the point of application provided that the point is connected with the rest of the body in the same invariable manner. A resultant force is a single force which can replace two or more forces and produce the same effect on the body as the forces.
The principle of resolved parts states. It states: Triangle law of forces states: Polygon law of forces states: Equilibrium conditions for coplanar concurrent forces: The force polygon must close. Fill in the blanks: State the effects which a force may produce when it acts on the body.
Name the different force systems. Explain briefly the following: State the following law of forces: Discuss graphical and analytical methods for finding resultant of several coplanar concurrent forces.
State equilibrium conditions for coplanar concurrent forces. Unsolved Examples 1. Two forces equal to 2P and P respectively act on a particle ; if the first be doubled, and the second increased by 12 N the direction of the resultant is unaltered, find the value of P.
Determine the resultant in magnitude and direction: The following forces act at a point: Find the magnitude and direction of the resultant force. A particle is acted upon by three forces, equal to 5 N, 10 N and 13 N along three sides of an equilibrium triangle, taken in order. Find graphically or otherwise the magnitude and direction of the resultant force.
Forces of 2, 3, 5, 3 and 2 N respectively act at one of the angular points of regular hexagon towards the other five angular points, taken in order. Forces of 7, 1, 1 and 3 N respectively act at one of the angular points of regular pentagon towards the four other points, taken in order.
Find the magnitude and direction of their resultant force. Find graphically or otherwise the tension in each cord. Find tensions in the two chains. A small ring is situated at the centre of a hexagon, and is supported by six strings drawn tight, all in the same plane and radiating from the centre of the ring, and each fastened to a different angular points of the hexagon.
The tensions in four consecutive strings are 1 N, 3. Find the tensions in the two remaining strings. A body of weight 45 N is suspended by two strings 18 cm and 61 cm long and other ends being fastened to the extremities of rod of length If the rod be so held that the body hangs immediately below its middle point, find out the tensions of the strings.
In a simple jib crane, the vertical post is 2. Find the forces on the jib and tie rod when a weight of 2. A rope AB, 2. A load of N is suspended from a point C on the rope 90 cm from A.
What load connected to a point D on the rope, 60 cm from B, will be necessary to keep the portion CD level? A sphere of radius mm and weight 10 N is suspended against a smooth wall by a string of length mm. The string joins a point in the wall and a point on the surface of the sphere. Find the inclination and the tension of the string and the reaction of the wall.
Two equal heavy spheres of 5 cm radius are in equilibrium with a smooth cup of 15 cm radius. Show that reaction between the cup and one sphere is double than that between the two spheres.
Clockwise and anti-clockwise moments. Principle of moments. Equilibrium conditions for bodies under coplanar non-concurrent forces. Parallel forces. Graphical method for finding the resultant of any number of like or unlike parallel forces. Properties of a couple. Engineering applications of moments—The lever—The balance—The common steel yard—Lever safety valve. This rotational tendency of a force is called moment.
The force multiplied by the perpendicular distance from the point to the line of action of the force is called moment about that point. Unit of moment is equal to the force unit multiplied by the distance unit. It can be in kgfm or Nm etc.
Consider, a finite rigid body capable of rotation about point O as shown in Fig. The diagram shows the section of the body in the plane of the paper. The axis of rotation is the line perpendicular to the paper and passing through the point O. Let us apply a force P directed along the paper and acting on the body at the point A.
The direction PA is the line of action of the force which is perpendicular to OA. Then the moment or torque of the force P about the point O is given by the product of force P and the distance OA, i.
It may be noted that the moment of force varies directly with its distance from the pivot. For example, it is much easier to turn a revolving door by pushing at the outer edge of the door, as in Fig.
If, on the other hand, the force P tends to rotate the body in the anti-clockwise sense, as shown in Fig. Let us consider, four coplanar forces P 1 , P 2 , P 3 and P 4 about P 4 acting simultaneously on a body and keeping the body in equilibrium Fig.
AA is the axis about which the body can rotate l 1 , l 2 , l 3 and l 4 are the respective perpendicular distances from O about which moments are to be taken to the lines of action of these forces. The body, thus can only be in equilibrium if the algebraic sum of all the external forces and their moments about any point in their plane is zero.
Mathematically, the conditions of equilibrium may be expressed as follows: This system will be in equilibrium if it satisfies the conditions of equilibrium viz. When coplanar forces do not meet in a point the system is known as coplanar non- concurrent force system. This system will be in equilibrium if it satisfies all the three conditions of equilibrium viz. Hence in this case, all the three conditions of equilibrium have to be fulfilled.
Case 1. When the two forces meet at a point. Complete the parallelogram ABCD. AC represents the resultant R of P and Q. Join OB and OA. Consider a force P which can be represented in magnitude and direction by the line AB.
Let O be the point, about which the moment of this force is required. Case 2. When the two forces are parallel to each other: Let P and Q be the two parallel forces as shown in Fig. Draw a line AB perpendicular to the forces to meet their lines of action in A and B.
Locate any point O in the plane of the two forces on AB produced. They are said to be like when they act in the same sense, they are said to be unlike when they act in opposite sense. The moment of R about C is zero, so that the algebraic sum of moments of P and Q about C must also be zero. Let P be greater than Q.
The algebraic sum of the moments of P and Q about C must be zero so that these moments must be equal and opposite. The point C is called the centre of parallel forces. It is clear that the position of C is independent of the inclination of the forces to AB.
The resultant force Z which passes through point N is completely given by ae in magnitude and direction. COUPLE A couple is pair of two equal and opposite forces acting on a body in a such a way that the lines of action of the two forces are not in the same straight line. The effect of a couple acting on a rigid body is to rotate it without moving it as a whole. The movement of the whole body is not possible because the resultant force is zero in the case of forces forming a couple. The perpendicular distance between the lines of action of two forces forming the couple is called the arm of couple.
Thus, in Fig. The moment of a couple is known as torque which is equal to one of the forces forming the couple multiplied by arm of the couple. Opening or closing a water tap. The two forces constitute a couple as shown in Fig. Turning the cap of a pen.
Unscrewing the cap of an ink bottle. Twisting a screw driver. Steering a motor-car Fig. Winding a watch or clock with a key.
Water tap Steering wheel Fig. The algebraic sum of the moments of the forces forming a couple about any point in their plane is constant. In all the three cases, we find that the sum of the moments in each case is independent of the position of the point O, and depends only on the constant arm of the couple, so the algebraic sum of moments of the forces forming a couple about any point in their plane is constant. Any two couples of equal moments and sense, in the same plane are equivalent in their effect.
This result is quite useful as it clearly states that moment is the only important thing about a couple. Thus, in a couple we may change the magnitude or direction of the forces or the arm of the couple itself without changing its effect provided that the new couple with changed values has the same moment in the same sense.
Two couples acting in one place upon a rigid body whose moments are equal but opposite in sense, balance each other. A force acting on a rigid body can be replaced by an equal like force acting at any other point and a couple whose moment equals the moment of the force about the point where the equal like force is acting. Any number of coplanar couples are equivalent to a single couple whose moment is equal to the algebraic sum of the moments of the individual couples.
The levers simple curved, bent or cranked and compound levers. The balance. The common steel yard. Lever safety valve. The Lever. It works on the principle of moments i. The principle of lever was first developed by Archimedes. Some common examples of the use of lever are: The energy developed by a force acting through a distance against resistance is known as work. The distance may be along a straight line or along a curved path. Common forms of work include a weight lifted through a height, a pressure pushing a volume of substance, and torque causing a shaft to rotate.
The rate of doing work, or work done per unit time is called power. For example, a certain amount of work is required to raise an elevator to the top of its shaft. A 5 HP motor can raise the elevator, but a 20 HP motor can do the same job four times faster.
It differs from an elastic body in the sense that the latter undergoes deformation under the effect of forces acting on it and returns to its original shape and size on removal of the forces acting on the body. The rigidity of a body depends upon the fact that how far it undergoes deformation under the effect of forces acting on it. In real sense no solid body is perfectly rigid because everybody changes it size and shape under the effect of forces acting on it.
It actual practice, the deformation i. A scalar quantity is one that has magnitude only. Mass, volume, time and density. Vector quantity. A vector quantity is one that has magnitude as well as direction. Force, velocity, acceleration and moment etc.
A vector quantity is represented by a line carrying an arrow head at one end. The length of the line to convenient scale equals the magnitude of the vector. The line, together with its arrow head, defines the direction of the vector. Suppose a force of 60 N is applied to point A in Fig. The vector AB represents this force since its length equals 60 N to scale and its direction is proper. If the vector BA is drawn to same scale Fig.Case a.
Positions of centre of gravity of regular solids. Offer Price: Polygon law of forces. Part I and Part II. A uniform rod 4 m long and of weight N is pivoted at a point 1 m from A.
The authors, using a time-honoured straightforward and flexible approach, present the basic concepts and principles of mechanics in the clearest and simplest form possible to advanced undergraduate engineering students of various disciplines and different educational backgrounds.
Find out the position of the centroid of L section as shown in Fig.