Couple

A pair of two equal and unlike parallel forces (i.e. forces equal in magnitude, with lines of action parallel to each other and acting in opposite directions) is known as a couple.

A couple is unable to produce any translatory motion (i.e., motion in a straight line). But it produces a motion of rotation in the body on which it acts. The simplest example of a couple is the forces applied to the key of a lock, while locking or unlocking it.

Arm of a Couple

The perpendicular distance (a), between the lines of action of the two equal and opposite parallel forces is known as arm of the couple as shown in Fig.1.

Fig. 1

Moment of a Couple

The moment of a couple is the product of the force (i.e., one of the forces of the two equal and opposite parallel forces) and the arm of the couple.

Mathematically,

                  Moment of a couple = P × a

Where,

     P = Magnitude of the force

     a = Arm of the couple

Classification of Couple

The couple may be classified into the following two categories, depending upon their direction, in which the couple tends to rotate the body, on which it acts.

1) Clockwise Couple

A couple, whose tendency is to rotate the body, on which it acts, in a clockwise direction, is known as a clockwise couple as shown in Fig. 2 (a). Such a couple is also called positive couple.

Fig. 2

2) Anticlockwise Couple

A couple, whose tendency is to rotate the body, on which it acts, in an anticlockwise direction, is known as an anticlockwise couple as shown in Fig. 2 (b). Such a couple is also called a negative couple.

Characteristics of a Couple

A couple (whether clockwise or anticlockwise) has the following characteristics.

1. The algebraic sum of the forces constituting the couple is zero.
2. The algebraic sum of the moments of the forces constituting the couple about any point is the same and equal to the moment of the couple itself.
3. A couple cannot be balanced by a single force. But it can be balanced only by a couple of opposite sense.
4. Any no. of coplanar couples can be reduced to a single couple, whose magnitude will be equal to the algebraic sum of the moments of all the couples.

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Varignon’s Principle of Moments (Law of Moments)

Varignon’s theorem states that “If a number of coplanar forces are acting simultaneously on a particle, the algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant force about the same point.”

Proof

Case (i) When the forces are concurrent

Let ‘P’ and ‘Q’ be any two forces acting at a point O along lines OX and OY respectively and let D be any point in their plane as shown in Fig. 1. Line DC is drawn parallel to OX to meet OY at B. The line OB represent the force Q in magnitude and direction and OA represent the force P in magnitude and direction.

Fig. 1

With OA and OB as the adjacent sides, parallelogram OACB is completed and OC is joined. Let ‘R’ be the resultant of forces P and Q. 

Then, according to the “Theorem of parallelogram of forces”, R is represented in magnitude and direction by the diagonal OC of the parallelogram OACB.

The point D is joined with points O and A. The moments of P, Q and R about D are given by 2 x area of ΔAOD, 2 x area of ΔOBD and 2 x area of ΔOCD respectively.

With reference to Fig1. (a), the point D is outside the <AOB and the moments of P, Q and R about D are all anti-clockwise and hence these moments are treated as positive.

Now, the algebraic sum of the moments of P and Q about

      D = 2ΔAOD + 2ΔOBD

          = 2 (ΔAOD + ΔOBD)

          = 2 (ΔAOC + ΔOBD)

          = 2 (ΔOBC + ΔOBD)

          = 2ΔOCD

          = Moment of R about D

[As AOC and AOD are on the same base and have the same altitude. ΔAOD = ΔOBC. As AOC and OBC have equal bases and equal altitudes. ΔAOC = ΔOBC]

With reference to Fig 1. (b), the point D is within the <AOB and the moments of P, Q and R about D are respectively anti-clockwise, clockwise and anti-clockwise.

Now, the algebraic sum of the forces P and Q about

      D = 2ΔAOD - 2 ΔOBD

          = 2 (ΔAOD-ΔOBD)

          = 2 (ΔAOC- ΔOBD)

          = 2(ΔOBC - ΔOBD)

          = 2ΔOCD

          = Moment of R about D

Case (ii) When the forces are parallel

Let P and Q be any two like parallel forces (i.e. the parallel forces whose lines of action are parallel and which act in the same sense) and O be any point in their plane.

Let R be the resultant of P and Q.

Then,

       R = P + Q

From O, line OACB is drawn perpendicular to the lines of action of forces P, Q and R intersecting them at A, B and C respectively as shown in Fig 2.

Fig 2

Now, algebraic sum of the moments of P and Q about O

     = P x OA + Q x OB

     = P x (OC - AC) + Q x (OC + BC)

     = P x OC – P x AC + Q x OC + Q x BC

  But P x AC = Q x BC

Algebraic sum of the moments of P and Q about O

     = P x OC + Q x OC

     = (P + Q) x OC

     = R x OC

     = Moment of R about O

In case of unlike parallel forces also it can be proved that the algebraic sum of the moments of two unlike parallel forces (i.e. the forces whose lines of action are parallel but which act in reverse senses) about any point in their plane is equal to the moment of their resultant about the same point.

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Moment of a Force

Moment is the turning effect produced by a force, on the body, on which it acts. The moment of a force is equal to the product of the force and the perpendicular distance of the point, about which the moment is required and the line of action of the force.

Fig. 1

Mathematically, moment,

                        M = P × d

  Where,

           P = Force acting on the body

           d = Perpendicular distance between the point, about which the moment is required and the

                  line of action of the force.

Fig. 2

Let a force ‘P’ act on a body which is hinged at O. Then, moment of P about the point O in the body is

                                       Moment = F x ON

Where 

           ON = perpendicular distance of O from the line of action of the force F.

Graphical Representation of Moment

Consider a force P represented in magnitude and direction, by the line AB. Let O be a point, about which the moment of this force is required to be found out, as shown in Fig. 3. From O, draw OC perpendicular to AB. Join OA and OB.

Fig. 3 Representation of Moment

Now moment of the force P about O

                       = P × OC

                       = AB × OC

But AB × OC is equal to twice the area of triangle ABO.

Thus the moment of a force about any point is equal to twice the area of the triangle, whose base is the line to some scale representing the force and whose vertex is the point about which the moment is taken.

Units of Moment

Since the moment of a force is the product of force and distance, therefore the units of the moment will depend upon the units of force and distance. Thus, if the force is in Newton and the distance is in meters, then the units of moment will be Newton-meter (N-m). Similarly, the units of moment may be kN-m (i.e. kN × m), N-mm (i.e. N × mm) etc.

Types of Moment

The moments are of two types.

1) Clockwise Moment

It is the moment of a force, whose effect is to turn or rotate the body, about the point in the same direction in which hands of a clock move as shown in Fig. 4 (a).

2) Anticlockwise Moment

It is the moment of a force, whose effect is to turn or rotate the body, about the point in the opposite direction in which the hands of a clock move as shown in Fig. 4 (b).

The general convention is to take clockwise moment as positive and anticlockwise moment as negative. But there is no hard and fast rule regarding sign convention of moments.

Fig. 4 Clockwise and Anticlockwise Moments

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Analytical Method for Finding Resultant Force - Parallelogram Law of Forces

The resultant force of a given system of forces may be found out analytically by Parallelogram law of forces.

Parallelogram Law of Forces

It states, “If two forces acting simultaneously on a particle, be represented in magnitude and direction by the two adjacent sides of a parallelogram then their resultant may be represented in magnitude and direction by the diagonal of the parallelogram, which passes through their point of intersection.”

Let forces ‘P’ and ‘Q’ acting at a point O be represented in magnitude and direction by OA and OB respectively as shown in Fig.1. Then, according to the theorem of parallelogram of forces, the diagonal OC drawn through O represents the resultant of P and Q in magnitude and direction.

Fig.1

Determination of the Resultant of Two Concurrent Forces with the Help of Law of Parallelogram of Forces

Consider two forces P and Q acting at and away from point A as shown in Fig. 2. Let the forces P and Q are represented by the two adjacent sides of a parallelogram AD and AB respectively as shown in Fig. 2. Let ‘θ’ be the angle between the force P and Q and ‘α’ be the angle between R and P. Extend line AB and drop perpendicular from point C on the extended line AB to meet at point E.

Fig. 2

Consider Right angle triangle ACE,

           AC² = AE² + CE2

                   = (AB + BE)² + CE2

                   = AB² + BE² + 2.AB.BE + CE2

                   = AB² + BE² + CE² + 2.AB.BE …………………….. (1)

Consider right angle triangle BCE,

            BC² = BE² + CE²    and     BE = BC. Cos θ

Putting BC² = BE² + CE² in equation (1), we get

            AC² = AB² + BC² + 2.AB.BE      ……………………….. (2)

Putting BE = BC. Cos θ in equation (2)

            AC² = AB² + BC² + 2.AB. BC. Cos θ

But,   AB = P, BC = Q and AC = R

               R = √ (P2 + Q2 + 2PQ Cos θ)

In triangle ACE

      

Now let us consider two forces F1 and F2 are represented by the two adjacent sides of a parallelogram

i.e. F1 and F2 = Forces whose resultant is required to be found out,

θ = Angle between the forces F1 and F2

α = Angle which the resultant force makes with one of the forces (say F1).

Then resultant force

            R= √ (F1² + F2² + 2F1 F2 Cos θ)

                  

If ‘α’ is the angle which the resultant force makes with the other force F2, then

                 

Cases

1) If θ = 0⁰ i.e., when the forces act along the same line, then

                                R_max = F1 + F2

2) If θ = 90⁰ i.e., when the forces act at right angle, then

                              R= √ (F1² + F2²)

3) If θ = 180⁰ i.e., when the forces act along the same straight line but in opposite directions, then

                               R_min= F1 – F2

In this case, the resultant force will act in the direction of the greater force.

4) If the two forces are equal i.e., when F1 = F2 = F then

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Resolution of a Given Force into Two Components in Two Assigned Direction

Let ‘P’ be the given force represented in magnitude and direction by OB as shown in Fig. 1. Also let OX and OY be two given directions along which the components of P are to be found out.

Fig. 1

Let < BOX = α and < BOY = β

From B, lines BA and BC are drawn parallel to OY and OX respectively. Then the required components of the given force P along OX and OY are represented in magnitude and direction by OA and OC respectively.

Since AB is parallel to OC, 

       <BAX = <AOC = α + β

      < AOB = 180⁰ – (α + β)

Now, in Δ OAB 

Determination of Resolved Parts of a Force

Resolved parts of a force mean components of the force along two mutually perpendicular directions. Let a force F represented in magnitude and direction by OC make an angle θ with OX. Line OY is drawn through O at right angles to OX as shown in Fig.2.

Fig. 2

Through C, lines CA and CB are drawn parallel to OY and OX respectively. Then the resolved parts of the force F along OX and OY are represented in magnitude and direction by OA and OB respectively.

Now in the right angled Δ AOC,

  

       i.e. OA = F cos θ

Since OA is parallel to BC, 

        <OCB = <AOC = θ

In the right angled Δ OBC,

 

i.e., OB = F sin θ

Thus, the resolved parts of F along OX and OY are respectively F cos θ and F sin θ.

Significance of the Resolved Parts of A Force

Fig. 3

Let 50 KN force is required to be applied to a body along a horizontal direction CD in order to move the body along the plane AB. Then it can be said that to move the body along the same plane AB, a force of 50kN is to be applied at an angle of 60° with the horizontal as CD = 50 cos60° = 25kN.

Similarly, if a force of 43.3kN is required to be applied to the body to lift it vertically upward, then the body will be lifted vertically upward if a force of 50kN is applied to the body at an angle of 60° with the horizontal, as the resolved part of 50kN along the vertical CE = 50 sin60° = 43.3kN.

Thus, the resolved part of a force in any direction represents the whole effect of the force in that direction.

Equilibriant

Equilibrant of a system of forces is a single force which will keep the given forces in equilibrium. Evidently, equilibrant is equal and opposite to the resultant of the given forces.

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Composition and Resolution of Vectors

The process of finding a single vector which will have the same effect as a set of vectors acting on a body is known as composition of vectors. The resolution of vectors is exactly the opposite process of composition i.e., it is the process of finding two or more vectors which will have the same effect as that of a single vector acting on the body.

Parallelogram Law of Vectors

The parallelogram law of vectors enables us to determine the single vector called resultant vector which can replace the two vectors acting at a point with the same effect as that of the two vectors. This law was formulated based on experimental results on a body subjected to two forces. This law can be applied not only to the forces but to any two vectors like velocities, acceleration, momentum etc.

This law states that ‘if two forces (vectors) acting simultaneously on a body at a point are represented in magnitude and directions by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes thorough the point of intersection of the two sides representing the forces (vectors)’.

Fig.1 Representation of Parallelogram Law of Vectors

In the Fig. 1, the force F1 = 4 units and the force F2 = 3 units are acting on a body at a point A. To get the resultant of these forces, according to this law, construct the parallelogram ABCD such that AB is equal to 4 units to the linear scale and AC is equal to 3 units. Then according to this law, the diagonal AD represents the resultant in magnitude and direction. Thus the resultant of the forces F1 and F2 is equal to the units corresponding to AD in the direction α to F1.

Triangle Law of Vectors

Referring to Fig. 1. (b), it can be observed that the resultant AD may be obtained by constructing the triangle ABD. Line AB is drawn to represent F1 and BD to represent F2. Then AD should represent the resultant of F1 and F2.

Thus we have derived the triangle law of forces from the fundamental law of parallelogram. The Triangle Law of Forces (vectors) may be stated as ‘if two forces (vectors) acting on a body are represented one after another by the sides of a triangle, their resultant is represented by the closing side of the triangle taken from the first point to the last point.’

Polygon Law of Forces (Vectors)

If more than two forces (vectors) are acting on a body, two forces (vectors) at a line can be combined by the triangle law and finally resultant of all forces (vectors) acting on the body may be obtained.

Fig. 2 Representation of Polygon Law of Vectors

A system of four concurrent forces acting on a body are shown in Fig. 2. AB represents F1 and BC represent F2. Hence according to triangle law of forces AC represents the resultant of F1and F2, say R1. If CD is drawn to represent F3, then from the triangle law of forces AD represents the resultant of R1 and F3. In other words, AD represents the resultant of F1, F2 and F3. Let it be called as R2.

Similarly, the logic can be extended to conclude that AE represents the resultant of F1, F2, F3 and F4. The resultant R is represented by the closing line of the polygon ABCDE in the direction from A to E. Thus we have derived the polygon law of the forces (vectors) and it may be stated as ‘if a number of concurrent forces (vectors) acting simultaneously on a body are represented in magnitude and direction by the sides of a polygon, taken in an order, then the resultant is represented in magnitude and direction by the closing side of the polygon, taken from the first point to the last point’.

Graphical (Vector) Method for the Resultant Force

It is another name for finding out the magnitude and direction of the resultant force by the polygon law of forces. It is done as discussed below.

1) Construction of space diagram (position diagram)

It means the construction of a diagram showing the various forces (or loads) along with their magnitude and lines of action.

2) Use of Bow’s notations

All the forces in the space diagram are named by using the Bow’s notations. It is a convenient method in which every force (or load) is named by two capital letters, placed on its either side in the space diagram.

Fig. 3 Denoting a Force by Bow’s Notation

3) Construction of vector diagram (force diagram)

It means the construction of a diagram starting from a convenient point and then go on adding all the forces by vector addition one by one (keeping in view the directions of the forces) to some suitable scale. Now the closing side of the polygon, taken in opposite order, will give the magnitude of the resultant force (to the scale) and its direction.

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Basic Laws of Engineering Mechanics

The structure of engineering mechanics rests on relatively few basic laws. They are given below.

1) Newton’s Laws of Motion

2) Newton’s Law of Gravitation

3) Principle of Transmissibility of Forces

4) Parallelogram Law of Forces

5) Principles of Physical Independence of Forces

6) Principles of Superposition

1) Newton’s Laws of Motion

i) Newton’s First Law of Motion

Newton’s first law states that ‘everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled by an external agency acting on it’. This leads to the definition of force as ‘force is an external agency which changes or tends to change the state of rest or uniform linear motion of the body’.

ii) Newton’s Second Law of Motion

Magnitude of force is defined by Newton’s second law. It states that ‘the rate of change of momentum of a body is directly proportional to the impressed force and it takes place in the direction of the force acting on it’. As the rate of change of velocity is acceleration and the product of mass and velocity is momentum we can derive expression for the force as given below.

From Newton’s second law of motion,

                                            Force ∝ rate of change of momentum

                                            Force ∝ rate of change of (mass × velocity)

Since mass do not change,

                                           Force ∝ mass × rate of change of velocity

                                                      ∝ mass × acceleration

                                                   F ∝ m × a

                                                   F = k × m × a

where 'F' is the force, 'm' is the mass, 'a' is the acceleration and 'k' is the constant of proportionality.

In all the systems, unit of force is so selected that the constant of the proportionality becomes unity. For example, in S.I. system, unit of force is Newton, which is defined as the force that is required to move one kilogram (kg) mass at an acceleration of 1 m/sec².

                                 ∴ One newton = 1 kg mass × 1 m/sec²

   Thus k = 1

                                                     F = m × a

ii) Newton’s Third Law of Motion

Newton’s first law gave definition of the force and second law gave basis for quantifying the force. Newton’s third law states that ‘for every action there is an equal and opposite reaction’.

Consider the two bodies in contact with each other. Let one body apply a force F on another. According to this law the second body develops a reactive force R which is equal in magnitude to force F and acts in the line same as F but in the opposite direction. Fig.1 shows the action of a ball on the floor and the reaction of floor to this action. In Fig. 2 the action of a ladder on the wall and the floor and the reactions from the wall and the floor are shown.

Fig. 1 Action of a Ball on the Floor and the Reaction of Floor

Fig. 2 Action of a Ladder on the Wall and the Floor and the Reactions from the Wall and the Floor

2) Newton’s Law of Gravitation

It states that everybody attracts the other body. ‘The force of attraction between any two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them’. Thus the force of attraction between the bodies of mass m1 and mass m2 at distance ‘d’ between them as shown in Fig. 3 is

where G is the constant of proportionality and is known as constant of gravitation.

Fig. 3 Force of Attraction between Two Bodies

From above equation,

It has been proved by experiments that the value of G = 6.673 × 10^–11 Nm²/kg². Thus if two bodies one of mass 10 kg and the other of 5 kg are at a distance of 1 m, they exert a force 

on each other.

Similarly, 1 kg-mass on earth surface experiences a force of

Since, mass of earth = 5.96504 × 10²⁴ kg and radius of earth = 6371 × 10³ m. This force of attraction is always directed towards the centre of earth. In common usage the force exerted by earth on a body is known as weight of the body. Thus weight of 1 kg-mass on/near earth surface is 9.80665 N, which is approximated as 9.81 N for all practical problems. Compared to this force the force exerted by two bodies on each other is negligible. Thus in statics

• Weight of a body W = mg
• Its direction is towards the centre of the earth, in other words, vertically downward. 
• The force of attraction between the other two objects on the earth is negligible.

3) Principle of Transmissibility of Forces

According to this law ‘the state of rest or motion of the rigid body is unaltered, if a force acting on the body is replaced by another force of the same magnitude and direction but acting anywhere on the body along the line of action of the replaced force’.

Let F be the force acting on a rigid body at point A as shown in Fig. 4. According to this law, this force has the same effect on the state of body as the force F applied at point B, where AB is in the line of force F.

Fig. 4 Representation of Principle of Transmissibility of Forces

In using law of transmissibility it should be carefully noted that it is applicable only if the body can be treated as rigid. Hence if we are interested in the study of internal forces developed in a body, the deformation of body is to be considered and hence this law cannot be applied in such studies.

3) Parallelogram Law of Forces

The parallelogram law of forces enables us to determine the single force called resultant force which can replace the two forces acting at a point with the same effect as that of the two forces. This law was formulated based on experimental results on a body subjected to two forces. This law can be applied not only to the forces but to any two vectors like velocities, acceleration, momentum etc. 

This law states that ‘if two forces (vectors) acting simultaneously on a body at a point are represented in magnitude and directions by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes thorough the point of intersection of the two sides representing the forces (vectors)’.

Fig. 5 Representation of Parallelogram Law of Forces

In the Fig. 5, the force F1 = 4 units and the force F2 = 3 units are acting on a body at a point A. To get the resultant of these forces, according to this law, construct the parallelogram ABCD such that AB is equal to 4 units to the linear scale and AC is equal to 3 units. Then according to this law, the diagonal AD represents the resultant in magnitude and direction. Thus the resultant of the forces F1 and F2 is equal to the units corresponding to AD in the direction α to F1.

4) Principles of Physical Independence of Forces

It states that the action of a force on a body is not affected by the action of any other force on the body.

5) Principles of Superposition of Forces

It states that ‘the net effect of a system of forces on a body is same as the combined of individual forces acting on the body’. Since a system of forces in equilibrium do not have any effect on a rigid body this principle is stated in the following form also: ‘The effect of a given system of forces on a rigid body is not changed by adding or subtracting another system of forces in equilibrium.’

Fig. 6 Representation of Principle of Superposition of Forces

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Scalar and Vector Quantities

Scalar Quantities

The scalar quantities (or sometimes known as scalars) are those quantities which have magnitude only such as length, mass, time, distance, volume, density, temperature, speed, energy work etc.

Vector Quantities

The vector quantities (or sometimes known as vectors) are those quantities which have both magnitude and direction such as force, displacement, velocity, acceleration, momentum etc. Following are the important features of vector quantities.

Representation of a Vector

A vector is represented by a directed line as shown in Fig. 1. It may be noted that the length OA represents the magnitude of the vector OA. The direction of the vector OA is from O (i.e., starting point) to A (i.e., end point). It is also known as vector P.

Fig. 1 Vector OA

Unit Vector

A vector, whose magnitude is unity, is known as unit vector.

Equal Vectors

The vectors, which are parallel to each other and have same direction (i.e., same sense) and equal magnitude are known as equal vectors.

Like Vectors

The vectors, which are parallel to each other and have same sense but unequal magnitude, are known as like vectors.

Addition of Vectors

Consider two vectors PQ and RS, which are required to be added as shown in Fig. 2 (a). Take a point A and draw line AB parallel and equal in magnitude to the vector PQ to some convenient scale. Through B, draw BC parallel and equal to vector RS to the same scale. Join AC which will give the required sum of vectors PQ and RS as shown in Fig. 2 (b).

Fig. 2 Method of Addition of Vectors

This method of adding the two vectors is called the Triangle Law of Addition of Vectors. Similarly, if more than two vectors are to be added, the same may be done first by adding the two vectors, and then by adding the third vector to the resultant of the first two and so on. This method of adding more than two vectors is called Polygon Law of Addition of Vectors.

Subtraction of Vectors

Consider two vectors PQ and RS in which the vector RS is required to be subtracted as shown in Fig. 3 (a). Take a point A, and draw line AB parallel and equal in magnitude to the vector PQ to some convenient scale. Through B, draw BC parallel and equal to the vector RS, but in opposite direction, to that of the vector RS to the same scale. Join AC, which will give the resultant when the vector PQ is subtracted from vector RS as shown in Fig. 3 (b).

Fig. 3 Method of Subtraction of Vectors

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Basic Terminologies in Mechanics

1) Mass (m)

The quantity of the matter possessed by a body is called mass. The mass of a body will not change unless the body is damaged and part of it is physically separated. If the body is taken out in a space craft, the mass will not change but its weight may change due to the change in gravitational force. The body may even become weightless when gravitational force vanishes but the mass remain the same.

2) Weight (w)

Weight of a body is the force with which the body is attracted towards the centre of the earth. The weight of the body is equal to the product of mass and the acceleration due to gravity. This quantity of a body varies from place to place on the surface of the earth.

Mathematically,

w=mg

Where ‘w’ is the weight of the body, ‘m’ is the mass of the body and ‘g’ is the acceleration due to gravity.

Table 1 Difference between Mass and Weight

Mass	Weight
Mass is the total quantity of matter contained in a body.	Weight of a body is the force with which the body is attracted towards the centre of the earth.
Mass is a scalar quantity, because it has only magnitude and no direction.	Weight is a vector quantity, because it has both magnitude and direction.
Mass of a body remains the same at all places. Mass of a body will be the same whether the body is taken to the centre of the earth or to the moon.	Weight of body varies from place to place due to variation of ‘g’ (i.e., acceleration due to gravity.
Mass resists motion in a body.	Weight produces motion in a body.
Mass of a body can never be zero.	Weight of a body can be zero.
Using an ordinary balance (beam balance), the mass can be determined.	Using a spring balance, the weight of the body can be measured.
The SI unit of the mass is the kilogram (kg).	The SI unit of the weight is Newton (N).

3) Time

The time is the measure of succession of events. The successive event selected is the rotation of earth about its own axis and this is called a day. To have convenient units for various activities, a day is divided into 24 hours, an hour into 60 minutes and a minute into 60 seconds. Clocks are the instruments developed to measure time. To overcome difficulties due to irregularities in the earth’s rotation, the unit of time is taken as second, which is defined as the duration of 9192631770 period of radiation of the cesium-133 atom.

4) Space

The geometric region in which study of body is involved is called space. A point in the space may be referred with respect to a predetermined point by a set of linear and angular measurements. The reference point is called the origin and the set of measurements as coordinates. If the coordinates involved are only in mutually perpendicular directions, they are known as cartesian coordination. If the coordinates involve angles as well as the distances, it is termed as Polar Coordinate System.

5) Length

It is a concept to measure linear distances. Meter is the unit of length. However depending upon the sizes involved micro, milli or kilo meter units are used for measurements. A meter is defined as length of the standard bar of platinum-iridium kept at the International Bureau of weights and measures. To overcome the difficulties of accessibility and reproduction now meter is defined as 1690763.73 wavelength of krypton-86 atom.

5) Continuum

A body consists of several matters. It is a well known fact that each particle can be subdivided into molecules, atoms and electrons. It is not possible to solve any engineering problem by treating a body as conglomeration of such discrete particles. The body is assumed to be a continuous distribution of matter. In other words the body is treated as continuum.

6) Particle

A particle may be defined as an object which has only mass and no size. Theoretically speaking, such a body cannot exist. However in dealing with problems involving distances considerably larger compared to the size of the body, the body may be treated as a particle, without sacrificing accuracy.

For example:

• A bomber aeroplane is a particle for a gunner operating from the ground.
• A ship in mid sea is a particle in the study of its relative motion from a control tower.
• In the study of movement of the earth in celestial sphere, earth is treated as a particle.

7) Rigid Body

A body is said to be rigid, if the relative positions of any two particles do not change under the action of the forces acting on it i.e., the distances between different points of the body remain constant. No body is perfectly rigid. Rigid body is ideal body.

Fig. 1 Rigid Body due to the action force F

8) Deformable Body

When a body deforms due to a force or a torque it is said deformable body. Material generates stresses against deformation. All bodies are more or less elastic.

Fig. 2 Deformable Body due to the action force F

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Force and System of Forces

Force is that which changes or tends to change the state of rest of uniform motion of a body along a straight line. It may also deform a body by changing its dimensions. The force may be broadly defined as an agent which produces or tends to produce, destroys or tends to destroy motion. It has a magnitude and direction.

Mathematically,

                                                            Force = Mass× Acceleration

                                                                   F = m a

Where, F - Force

            m - Mass

             a - Acceleration

Characteristics of Force

1) Magnitude: Magnitude of force indicates the amount of force (expressed as N or kN) that will be exerted on another body

2) Direction: The direction in which it acts

3) Nature: The nature of force may be tensile or compressive

4) Point of Application: The point at which the force acts on the body is called point of application

Units of Force

1) In C.GS. System

In this system, there are two units of force: (i) Dyne and (ii) Gram force (gmf). Dyne is the absolute unit of force in the C.G.S. system. One dyne is that force which acting on a mass of one gram produces in it an acceleration of one centimeter per second².

2) In M.K.S. System

In this system, unit of force is kilogram force (kgf). One kilogram force is that force which acting on a mass of one kilogram produces in it an acceleration of 9.81 m/ sec².

3) In S.I. Unit

In this system, unit of force is Newton (N). One Newton is that force which acting on a mass of one kilogram produces in it an acceleration of one m /sec².

                                                                1 Newton = 10⁵ Dyne

Effect of Force

A force may produce the following effects in a body, on which it acts.

1. It may change the motion of a body. i.e. if a body is at rest, the force may start its motion and if the body is already in motion, the force may accelerate or decelerate it.
2. It may retard the forces, already acting on a body, thus bringing it to rest or in equilibrium.
3. It may give rise to the internal stresses in the body, on which it acts.
4. A force can change the direction of a moving object.
5. A force can change the shape and size of an object

Principle of Physical Independence of Forces

It states, “If a number of forces are simultaneously acting on a particle, then the resultant of these forces will have the same effect as produced by all the forces.”

System of Forces

When two or more forces act on a body, they are called to form a system of forces. Force system is basically classified into the following types.

1) Coplanar Forces

The forces, whose lines of action lie on the same plane, are known as coplanar forces.

Fig. 1 Coplanar Forces

2) Collinear Forces

The forces, whose lines of action lie on the same line, are known as collinear forces.

Fig. 2 Collinear Forces

3) Concurrent Forces

The forces, which meet at one point, are known as concurrent forces. The concurrent forces may or may not be collinear.

Fig. 3 Concurrent Forces

4) Coplanar Concurrent Forces

The forces, which meet at one point and their line of action also lay on the same plane, are known as coplanar concurrent forces.

Fig. 4 Coplanar Concurrent Forces

5) Coplanar Non-Concurrent Forces

The forces, which do not meet at one point, but their lines of action lie on the same plane, are known as coplanar non-concurrent forces.

Fig. 5 Coplanar Non-Concurrent  Forces

6) Non-Coplanar Concurrent Forces

The forces, which meet at one point, but their lines of action do not lie on the same plane, are known as non-coplanar concurrent forces.

Fig. 6 Non-Coplanar Concurrent  Forces

7) Non-Coplanar Non-Concurrent Forces

The forces, which do not meet at one point and their lines of action do not lie on the same plane, are called non-coplanar non-concurrent forces.

Fig. 7 Non-Coplanar Non-Concurrent  Forces

8) Parallel Forces

The forces, whose lines of action are parallel to each other, are known as parallel forces.

Fig. 8 Parallel Forces

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