Mechanical Science(2)

LESSON PLAN LECTURE 8 INTRODUCTION TO STATICS 1. Statics deals with the conditions of equilibrium of bodies acted upon by forces . 2. A rigid body is defined as a definite quantity of matter , the parts of which are fixed in position relative to one another . The physical bodies are never absolutely rigid but deform slightly under the action of loads, which they have to carry .If the deformation is negligible when compared with the size of body , it is assumed to be rigid. 3. A particle may be defined as an object which has only mass , and no size. Such a body cannot exist theoretically , but when dealing with problems involving distances considerably larger when compared to the size of the body can be neglected without sacrificing accuracy. 4. Force may be defined as any action that tends to change the state o restor motion of a body to which it is applied.The three quantities , which completely define a force, are called its specifications. The specifications of a force are (1) its magnitude , (2) its point of application , and (3) its direction The S.I. unit used by engineers to measure the magnitude of a force is the Newton 5. The point of application of a force acting upon a body is that point in the body at which the force can be assumed to be concentrated . The direction of a force is the direction , along a straight line through its point of application , in which the force tends to move a body to which it is applied. 6. VECTOR AND SCALAR QUANTITIES A quantity is said to be scalar if it is completely defined by its magnitude alone. Examples are length , area and time. A quantity is said ton be a vector , if it is completely defined only when its magnitude and direction are specified . Hence , force is a vector. The other examples of vector are velocity and acceleration. 7. TRANSMISSIBILITY OF A FORCE ( Sliding Vector) When two forces are in equilibrium (equal, opposite, collinear) , their resultant is zero and their combined action on a rigid body is equivalent to that of no fofce at all. 8. LAW OF SUPERPOSITION

The action of a given system of a system of forces on a rigid body will in no way be changed if we add to or subtract from the another system of forces. LECTURE 9 SYSTEM OF FORCES Force System Collinear forces

Characteristics

Line of action of all the forces act along the same line. Coplanar parallel forces All forces are parallel to each other and lie in a single plane. Coplanar like parallel forces All forces are parallel to each other , lie in a single plane and are acting in the same direction. Coplanar concurrent forces Line of action of all forces pass through a single poin αα t and forces lie in the same plane. Coplanar nonconcurrent All forces do not meet at a forces point , but lie in a single plane.

Examples Forces on a rope in a tug of war. System of forces acting on a beam subjected to vertical loads.(including reactions) Weight of a stationary train on a rail when the track is straight. Forces on a rod resting against a wall.

Forces on a ladder resting against a wall when a person stands on a rung which is not at its centre of gravity. Non- coplanar parallel All the forces are parallel to The weight of benches in a forces each other, but not in the class room. same plane. Non –coplanar concurrent All forces do not lie in the A tripod carrying a camera. forces same plane, but their lines of action pass through a single point. Noncoplanar non- All forces do not lie in the Forces acting on a moving concurrent forces same plane and their lines bus. of action do not pass through a single point. LAMI’S THEOREM If three concurrent forces are acting on a body, kept in an equilibrium , then each force is proportional to the sine of the angle between the other two forces and the constant of proportionality is the same.

LECTURE 10 FREE BODY DIAGRAMS A free body is a body not connected with other bodies and which from any given position can be displaced in any direction in space. Free body diagram is a sketch of the isolated body , which shows the external forces on the body and the reactions exerted on it by the removed elements . The general procedure for constructing a free- body diagram is as follows – 1. A sketch of the body is drawn , by removing the supporting surfaces. 2. Indicate on this sketch all the applied forces , which tend to set the body in motion , such as those caused by weight of the body or applied forces. 3. Also indicate on this sketch all the reactive forces, such as those caused by the constraints as supports that tend to prevent motion. 4. All relevant dimensions and angles, reference axes are shown on the sketch. EQUILIBRIUM OF A BODY A body is said to be in equilibrium when it is at rest or continues to be in steady linear motion. According to Newton’s law of motion , it means that the resultant of all the forces acting on a body in equilibrium is zero . In graphical terms , it means that the force polygon closes. The resultant R of a system of concurrent forces is zero only when the following conditions are satisfied: ∑ X =0

∑Y = 0

It may be observed that only one of the above two conditions is not sufficient. TYPES OF FORCES ON A BODY While applying equilibrium conditions to a body it is essential that all forces acting on the body should be considered . The various forces acting on a body may be grouped as (i) Applied forces (ii) Non applied forces Applied forces – Applied forces are the forces applied externally to a body. Each of the forces has got a point of contact with the body.

Non –applied forces – There are two types of non-applied forces (a) Self weight (b) Reactions Self weight – Every object is subjected to gravitational acceleration and hence has got a self weight. W = mg Reactions – These are self adjusting forces developed by the other bodies which come in contact with the body in consideration . The reactions adjust themselves to bring the body in equilibrium. LECTURE 11 THEOREM OF VARIGNON The moment of the resultant of two concurrent forces with respect to a center in their plane is equal to the algebraic sum of the moments of the components with respect to the same center. MOMENT OF A FORCE Moment of a force about a point is the measure of its rotational effect . Moment is defined as the product of the magnitude of the force and the perpendicular distance of the point from the line of action of the force . The ple point about which the moment is considered is called moment center and the perpendicular distance of the point from the line of action of the force is called moment arm. The moment of a force has got direction also. It may be that moment has a clockwise or anticlockwise direction . The S. I. unit of moment is N-m. COUPLE Two parallel forces equal in magnitude and opposite in direction and separated by a definite distance are said to form a couple . The sum of the forces forming a couple in any direction is zero , which means the translatory effect of the couple is zero. Now , we can list the following characteristics of a couple(i) (ii) (iii)

A couple consists of a pair of equal and opposite parallel forces which are separated by a definite distance. The translatory effect of a couple on the body is zero. The rotational effect (moment) of a couple about any point and it is equal to the product of the magnitude of the forces and the perpendicular distance between the two forces.

(iv)

Since the sole effect of a couple is a moment and this moment is the same about any point , the effect of a couple is unchanged if

(a) (b) (c)

The couple is rotated through any angle. The couple is shifted to any other position . The couple is replaced by another pair of forces whose rotational effect is the same.

LECTURE 12 SOME IMPORTANT FACTS 1. Two couples acting in the same plane are equivalent if they have equal moments . A couple is completely defined by its plane of action and the magnitude and sign of its moment. 2. The moment of the resultant couple is equal to the algebraic sum of the moments of the two given couples. 3. We can replace several couples in one plane by a single resultant couple acting in the same plane, the moment of which is equal to the algebraic sum of the moments of the given couples. 4. A couple can be balanced only by another couple which is equal in moment , opposite in sign , and coplanar in action with the given couple. 5. A system of couples acting in one plane is in equilibrium if the algebraic sum of their moments is equal to zero. 6. It is possible to resolve a couple into several component couples by choosing the component couples in such a manner that the algebraic sum of the moments is equal to the moment of the given couple. 7. A given force P applied to a body at any point A can always be replaced by an equal force applied at another point B together with a couple which will be statically equivalent to the original force. 8. Built –in –end (fixed supports ) is considerably more complex constraint than the other types of constraints. It restricts the translatory motion as well as rotation. COMPOSITION OF TWO FORCES The reduction of a given system of forces to the simplest system that will be equivalent is called the problem of composition of forces. RESOLUTION OF A FORCE The replacement of a single force by several components , which will be equivalent in action to the given force , is called the problem of resolution of force. EQUILIBRIUM OF THREE FORCES IN A PLANE

Three non- parallel forces can be in equilibrium only when they lie in one plane , intersect in one point , and their free vectors build a closed triangle.The statement is called the theorem of three forces. LECTURE 13 INTRODUCTION TO VECTOR ALGEBRA A vector quantity is defined as a quantity , which satisfies (i) The qualification of possessing a certain magnitude, including a null vector. (ii)

Having a certain orientation in space , called its direction and (iii)

A sense , which qualifies its nature of action (moving towards or away from a given point)

EQUALITY AND EQUIVALENCE OF VECTORS Two vectors are considered to be equal if and only if they have same magnitude , same direction and same sense. Two vectors are said to be equivalent if they produce the same effect in a certain respect. FREE VECTOR The action of free vector is not confined to a unique line in space . The vector may be moved any where in the space without rotation. Examples: Moment vector BOUND OR FIXED VECTOR Must remain at the same point of application . i.e. , a definite point of application. An example of bound vector is the weight of a body . The weight of a body always passes through the center of gravity of the body and is always directed towards the center of earth irrespective of the orientation of the boby in space. TRIANGLE LAW OF FORCES It can be stated as “ If two forces acting simultaneously on a body are represented by the sides of a triangle taken in order , their resultant is represented by the closing side of the triangle taken in the opposite order. PARALLELOGRAM LAW OF FORCES If two forces, represented by vector AB and AC acting under an angle α , are applied to a body at point A , their action is equivalent to the action of one force , represented by the

vector AD , obtained as the diagonal of the parallelogram constructed on the vectors AB and AC are directed. LECTURE 14 RESOLUTION OF A VECTOR A vector can be resolved along three mutually perpendicular directions by constructing two parallelograms. DIRECTION COSINES OF A VECTOR If the angles made by a vector V with the axes of reference x,y,and z be α , β , χ respectively , the direction cosines l ,m and n of the vectors are defined by UNIT VECTOR A unit vector is defined as a vector having unit magnitude and a sense along the vector.

particular direction and

UNIT CO – ORDINATE VECTOR Unit vectors lying along the axes of reference OX, OY and OZ are called unit co-ordinate vectors. PRODUCT OF TWO VECTORS (i) Scalar product or dot product By definition the scalar product or dot product of two vectors results in a scalar quantity whose magnitude is given by A.B = ABcos α EXAMPLES OF DOT PRODUCT (concept of work) Force F acting on a particle produces a displacement ds , then elementary work done dw = (Fcos θ )ds VECTOR PRODUCT OR CROSS PRODUCT Cross product of two vectors A and B generates another vector C such that (i) The magnitude of C is given by ABSin ϑ (ii) Direction of C is perpendicular to planecontaining A and B Sense is determined by right hand thumb screw rule or by right hand thumb rule. Example of cross product (concept of moment of a force)

IMPORTANT VECTOR QUANTITIES Position Vector : The directed line segment r from the origin of a co-ordinate system to a point P in space is called the position vector. Displacement Vector : Displacement vector is a directed line segment connecting any two points on the path of motion .This represent shortest movement of the particle to get from one position on the path of motion to another. LECTURE 15 PRINCIPLE OF VIRTUAL WORK The principle of virtual work lays emphasis on the fact that a small displacement of the bodies and found out the work done by various forces in the system and said if the body is in equilibrium then the total work done on the system should be zero.This approach is called “ virtual work method “ .The word virtual is used since in reality there is no actual displacement . The term virtual is defined by Webster as “being in essence of effect but not infact.” The work done is virtual if the displacements are virtual or forces acting are virtual .Hence in virtual work approach we may have the following(i) (ii)

Principles of virtual displacements Principles of virtual forces

THE METHOD The method consists of the following steps – 1. Give an elemental virtual displacement to the body in equilibrium in suitable direction. 2. Write down the work done by each force in the system. 3. Sum up total work done by system of forces and equate it to zero to get required equations. Many times such equations are identical to equations of equilibrium .However, the method of getting these equations is completely different in this method. SIGN CONVENTION In this chapter the positive senses of moments and displacements are as given below1. Forces and displacements are positive if they are in positive direction of coordinates. 2. Moments and relations are positive if they are in clockwise direction.

LECTURE 8 INTRODUCTION TO STATICS 9. Statics deals with the conditions of equilibrium of bodies acted upon by forces . 10. A rigid body is defined as a definite quantity of matter , the parts of which are fixed in position relative to one another . The physical bodies are never absolutely rigid but deform slightly under the action of loads, which they have to carry .If the deformation is negligible when compared with the size of body , it is assumed to be rigid. 11. A particle may be defined as an object which has only mass , and no size. Such a body cannot exist theoretically , but when dealing with problems involving distances considerably larger when compared to the size of the body can be neglected without sacrificing accuracy. 12. Force may be defined as any action that tends to change the state o restor motion of a body to which it is applied.The three quantities , which completely define a force, are called its specifications. The specifications of a force are (1) its magnitude , (2) its point of application , and (3) its direction The S.I. unit used by engineers to measure the magnitude of a force is the Newton 13. The point of application of a force acting upon a body is that point in the body at which the force can be assumed to be concentrated . The direction of a force is the direction , along a straight line through its point of application , in which the force tends to move a body to which it is applied. 14. VECTOR AND SCALAR QUANTITIES A quantity is said to be scalar if it is completely defined by its magnitude alone. Examples are length , area and time. A quantity is said ton be a vector , if it is completely defined only when its magnitude and direction are specified . Hence , force is a vector. The other examples of vector are velocity and acceleration. 15. TRANSMISSIBILITY OF A FORCE ( Sliding Vector) When two forces are in equilibrium (equal, opposite, collinear) , their resultant is zero and their combined action on a rigid body is equivalent to that of no fofce at all.

16. LAW OF SUPERPOSITION The action of a given system of a system of forces on a rigid body will in no way be changed if we add to or subtract from the another system of forces. LECTURE 9 SYSTEM OF FORCES Force System Collinear forces

Characteristics

Line of action of all the forces act along the same line. Coplanar parallel forces All forces are parallel to each other and lie in a single plane. Coplanar like parallel forces All forces are parallel to each other , lie in a single plane and are acting in the same direction. Coplanar concurrent forces Line of action of all forces pass through a single poin αα t and forces lie in the same plane. Coplanar nonconcurrent All forces do not meet at a forces point , but lie in a single plane.

Examples Forces on a rope in a tug of war. System of forces acting on a beam subjected to vertical loads.(including reactions) Weight of a stationary train on a rail when the track is straight. Forces on a rod resting against a wall.

Forces on a ladder resting against a wall when a person stands on a rung which is not at its centre of gravity. Non- coplanar parallel All the forces are parallel to The weight of benches in a forces each other, but not in the class room. same plane. Non –coplanar concurrent All forces do not lie in the A tripod carrying a camera. forces same plane, but their lines of action pass through a single point. Noncoplanar non- All forces do not lie in the Forces acting on a moving concurrent forces same plane and their lines bus. of action do not pass through a single point. LAMI’S THEOREM

If three concurrent forces are acting on a body, kept in an equilibrium , then each force is proportional to the sine of the angle between the other two forces and the constant of proportionality is the same. LECTURE 10 FREE BODY DIAGRAMS A free body is a body not connected with other bodies and which from any given position can be displaced in any direction in space. Free body diagram is a sketch of the isolated body , which shows the external forces on the body and the reactions exerted on it by the removed elements . The general procedure for constructing a free- body diagram is as follows – 5. A sketch of the body is drawn , by removing the supporting surfaces. 6. Indicate on this sketch all the applied forces , which tend to set the body in motion , such as those caused by weight of the body or applied forces. 7. Also indicate on this sketch all the reactive forces, such as those caused by the constraints as supports that tend to prevent motion. 8. All relevant dimensions and angles, reference axes are shown on the sketch. EQUILIBRIUM OF A BODY A body is said to be in equilibrium when it is at rest or continues to be in steady linear motion. According to Newton’s law of motion , it means that the resultant of all the forces acting on a body in equilibrium is zero . In graphical terms , it means that the force polygon closes. The resultant R of a system of concurrent forces is zero only when the following conditions are satisfied: ∑ X =0

∑Y = 0

It may be observed that only one of the above two conditions is not sufficient. TYPES OF FORCES ON A BODY While applying equilibrium conditions to a body it is essential that all forces acting on the body should be considered . The various forces acting on a body may be grouped as (iv) Applied forces (v) Non applied forces

Applied forces – Applied forces are the forces applied externally to a body. Each of the forces has got a point of contact with the body. Non –applied forces – There are two types of non-applied forces (a) Self weight (b) Reactions Self weight – Every object is subjected to gravitational acceleration and hence has got a self weight. W = mg Reactions – These are self adjusting forces developed by the other bodies which come in contact with the body in consideration . The reactions adjust themselves to bring the body in equilibrium. LECTURE 11 THEOREM OF VARIGNON The moment of the resultant of two concurrent forces with respect to a center in their plane is equal to the algebraic sum of the moments of the components with respect to the same center. MOMENT OF A FORCE Moment of a force about a point is the measure of its rotational effect . Moment is defined as the product of the magnitude of the force and the perpendicular distance of the point from the line of action of the force . The ple point about which the moment is considered is called moment center and the perpendicular distance of the point from the line of action of the force is called moment arm. The moment of a force has got direction also. It may be that moment has a clockwise or anticlockwise direction . The S. I. unit of moment is N-m. COUPLE Two parallel forces equal in magnitude and opposite in direction and separated by a definite distance are said to form a couple . The sum of the forces forming a couple in any direction is zero , which means the translatory effect of the couple is zero. Now , we can list the following characteristics of a couple(v) (vi)

A couple consists of a pair of equal and opposite parallel forces which are separated by a definite distance. The translatory effect of a couple on the body is zero.

(vii)

The rotational effect (moment) of a couple about any point and it is equal to the product of the magnitude of the forces and the perpendicular distance between the two forces. (viii) Since the sole effect of a couple is a moment and this moment is the same about any point , the effect of a couple is unchanged if (d) (e) (f)

The couple is rotated through any angle. The couple is shifted to any other position . The couple is replaced by another pair of forces whose rotational effect is the same.

LECTURE 12 SOME IMPORTANT FACTS 9. Two couples acting in the same plane are equivalent if they have equal moments . A couple is completely defined by its plane of action and the magnitude and sign of its moment. 10. The moment of the resultant couple is equal to the algebraic sum of the moments of the two given couples. 11. We can replace several couples in one plane by a single resultant couple acting in the same plane, the moment of which is equal to the algebraic sum of the moments of the given couples. 12. A couple can be balanced only by another couple which is equal in moment , opposite in sign , and coplanar in action with the given couple. 13. A system of couples acting in one plane is in equilibrium if the algebraic sum of their moments is equal to zero. 14. It is possible to resolve a couple into several component couples by choosing the component couples in such a manner that the algebraic sum of the moments is equal to the moment of the given couple. 15. A given force P applied to a body at any point A can always be replaced by an equal force applied at another point B together with a couple which will be statically equivalent to the original force. 16. Built –in –end (fixed supports ) is considerably more complex constraint than the other types of constraints. It restricts the translatory motion as well as rotation. COMPOSITION OF TWO FORCES The reduction of a given system of forces to the simplest system that will be equivalent is called the problem of composition of forces. RESOLUTION OF A FORCE The replacement of a single force by several components , which will be equivalent in action to the given force , is called the problem of resolution of force. EQUILIBRIUM OF THREE FORCES IN A PLANE

Three non- parallel forces can be in equilibrium only when they lie in one plane , intersect in one point , and their free vectors build a closed triangle.The statement is called the theorem of three forces. LECTURE 13 INTRODUCTION TO VECTOR ALGEBRA A vector quantity is defined as a quantity , which satisfies (i) The qualification of possessing a certain magnitude, including a null vector. (ii)

Having a certain orientation in space , called its direction and (vi)

A sense , which qualifies its nature of action (moving towards or away from a given point)

EQUALITY AND EQUIVALENCE OF VECTORS Two vectors are considered to be equal if and only if they have same magnitude , same direction and same sense. Two vectors are said to be equivalent if they produce the same effect in a certain respect. FREE VECTOR The action of free vector is not confined to a unique line in space . The vector may be moved any where in the space without rotation. Examples: Moment vector BOUND OR FIXED VECTOR Must remain at the same point of application . i.e. , a definite point of application. An example of bound vector is the weight of a body . The weight of a body always passes through the center of gravity of the body and is always directed towards the center of earth irrespective of the orientation of the boby in space. TRIANGLE LAW OF FORCES It can be stated as “ If two forces acting simultaneously on a body are represented by the sides of a triangle taken in order , their resultant is represented by the closing side of the triangle taken in the opposite order. PARALLELOGRAM LAW OF FORCES

If two forces, represented by vector AB and AC acting under an angle α , are applied to a body at point A , their action is equivalent to the action of one force , represented by the vector AD , obtained as the diagonal of the parallelogram constructed on the vectors AB and AC are directed. LECTURE 14 RESOLUTION OF A VECTOR A vector can be resolved along three mutually perpendicular directions by constructing two parallelograms. DIRECTION COSINES OF A VECTOR If the angles made by a vector V with the axes of reference x,y,and z be α , β , χ respectively , the direction cosines l ,m and n of the vectors are defined by UNIT VECTOR A unit vector is defined as a vector having unit magnitude and a sense along the vector.

particular direction and

UNIT CO – ORDINATE VECTOR Unit vectors lying along the axes of reference OX, OY and OZ are called unit co-ordinate vectors. PRODUCT OF TWO VECTORS (ii) Scalar product or dot product By definition the scalar product or dot product of two vectors results in a scalar quantity whose magnitude is given by A.B = ABcos α EXAMPLES OF DOT PRODUCT (concept of work) Force F acting on a particle produces a displacement ds , then elementary work done dw = (Fcos θ )ds VECTOR PRODUCT OR CROSS PRODUCT Cross product of two vectors A and B generates another vector C such that (iii) (iv)

The magnitude of C is given by ABSin ϑ Direction of C is perpendicular to planecontaining A and B

Sense is determined by right hand thumb screw rule or by right hand thumb rule. Example of cross product (concept of moment of a force) IMPORTANT VECTOR QUANTITIES Position Vector : The directed line segment r from the origin of a co-ordinate system to a point P in space is called the position vector. Displacement Vector : Displacement vector is a directed line segment connecting any two points on the path of motion .This represent shortest movement of the particle to get from one position on the path of motion to another. LECTURE 15 PRINCIPLE OF VIRTUAL WORK The principle of virtual work lays emphasis on the fact that a small displacement of the bodies and found out the work done by various forces in the system and said if the body is in equilibrium then the total work done on the system should be zero.This approach is called “ virtual work method “ .The word virtual is used since in reality there is no actual displacement . The term virtual is defined by Webster as “being in essence of effect but not infact.” The work done is virtual if the displacements are virtual or forces acting are virtual .Hence in virtual work approach we may have the following(iii) (iv)

Principles of virtual displacements Principles of virtual forces

THE METHOD The method consists of the following steps – 4. Give an elemental virtual displacement to the body in equilibrium in suitable direction. 5. Write down the work done by each force in the system. 6. Sum up total work done by system of forces and equate it to zero to get required equations. Many times such equations are identical to equations of equilibrium .However, the method of getting these equations is completely different in this method. SIGN CONVENTION In this chapter the positive senses of moments and displacements are as given below3. Forces and displacements are positive if they are in positive direction of coordinates.

4. Moments and relations are positive if they are in clockwise direction.