ENGINEERING MECHANICS LECTURE NOTES
Unit 1
Mechanics can be defined as the science, which describes and predicts the condition of rest or motion of bodies under the action of forces. Engineering mechanics is the branch of engineering that applies the principles of mechanics to design, which must take into account the effect of forces. Newton is famous for his three laws of motion. On his name a portion of classical mechanics is called Newtonian mechanics. An alternate formulation for mechanics problems was provided by LAGRANGE AND HAMILTON, based on the concept of energy. Classical mechanics fails when a body approaches the speed of light or when the body size approaches a size comparable to that of atom. Relativistic and quantum mechanics are used for those situations. However in this course we limit our discussion to classical mechanics.
Engineering Mechanics Mechanics of
Mechanics
Solids
Fluids
of
Mechanics
Mechanics
1. Ideal Fluid
of
of
Deformable Bodies
Rigid Bodies
2. Viscous Fluids 3.Incompressive Fluids
Statics
Dynamics
Kinematics
Theory
Theory
of
of
Elasticity
Plasticity
Kinetics
Depending upon the nature of problems, mechanics is divided into statics and dynamics. Statics is the study of forces and conditions of equilibrium of material bodies subjected to the action
of forces. Dynamics is the study of motion of rigid bodies and their correlation with the forces causing them. Dynamics is divided into kinematics and kinetics. Kinematics deals with the spacetime relationship of a given motion of body without concerning the forces causing the motion. Kinetics is the study of the relation between the forces and the resulting motion. In other words
kinetics studies the laws of motion of material bodies under the action of forces.
Idealization in mechanics
Mathematical models are idealizations are used in mechanics to simplify the application of theory. Some of the important idealizations 1. Continuum may be defined as a continuous distribution of matter with no voids or empty
spaces. Each body is made up of atoms and molecules. The matter is assumed as continuously distributed since the behaviour of atoms and molecules are too complex deal with. 2. Particle is a point mass or a material point in the abstract sense. A body whose dimensions
can be neglected in studying its motion or equilibrium may be treated as a particle. Examples a. A cricket ball as seen by the audience b. A distant Aeroplane as seen by a ground observer c. A Satellite orbiting the earth viewed by an observer on the earth d. A planet as seen from another or Sun.
point masses. A system of particles particles is constituted 3. System of particles is an idealization of point when two or more bodies represented by particles and are dealt with together. Examples planetary systems, structure of atom i.e., electron-proton-neutron and billiard balls observed by a viewer in the gallery, Sun-Earth-Moon System.
4. Rigid body is the one in which the distance between any two arbitrary points is invariant, it
means that the distance between any points on the rigid body is constant before and after application of external forces. However this is an idealized situatio n since all the bodies undergo deformation under the action of applied forces. In many cases the deformation is negligible compared to the size of the body and the body may be assumed rigid. Examples: a) an Aeroplane observed in roll, pitch and yaw
b) Spinning top c) Wheel of a cart 5. Deformable bodies: Bodies are considered deformable when the changes in the distance
between any two of its points cannot be neglected. Examples a) A beam deflecting under the application of a load b) A shaft twisting under the application of a torque
The branch of mechanics dealing with deformation that are caused by applied loads is named as strength of materials or mechanics of solids or mechanics of deformable bodies. Units and dimensions: Unit is defined as a numerical standard used to measure the qualitative dimensions of a
physical quantity. The three primary dimensions basic to mechanics are length, time and their force or mass. All other quantities are secondary dimensions or derived quantities in terms of these basic quantities. Fundamental units and derived units
The units in which the fundamental quantities are measured are called fundamental or basic units. The three primary units in mechanics are length, mass and time. The derived units are units of derived physical quantities, which are expressed in terms of fundamental units. Examples area, volume, force, velocity, acceleration, pressure etc., SI units The international systems of units, abbreviated SI (Systema international D’ unites) , has
been accepted through the world and is a modern version of metric system. In SI units length in metres (m), mass in kilogram (kg) and time in seconds (s) are selected as the base units and force in Newton’s (N) is derived from Newton’s second law of motion. When the numerical quantity
whether very large or small units defined used to define its size may be modified by using a prefix. Quantity Mass Length Time
Derived unit Newton Joule
SI units Kilogram Metre Second
Symbol Kg m s
Symbol 2 N kg m/s 2 2 J=Nm-kgm /s
Physical quantity Force Energy. Work, Heat
2
Watt Pascal Hertz
2
W = J/s= N m/s kg m /s 2 2 Pa – N/m =kg-m/s -1 Hz = s
Power Pressure, Stress Frequency
RULES FOR USE OF SI UNITS: 1. The symbol is never written with a pl ural “s” 2. Symbols are always written in lower case letters accept the symbols named after an individual example N and J 3. Kilogram is written as kg and not as kgm or kgf. Similarly, second as s not a sec. or sec, no full stops, dots or dashes should be used. For example moment is N m or N.m or N-m etc., 2
4. It is permissible that one be left between any two unit symbols examples kg m /s, m s 5. No space be left after a multiple or sub multiple symbol examples kJ/kgK 6. Always leave a space between the number and the unit symbol example 3 m, 1500 N 7. For numbers less than unity, zero must be out on the left of the decimal Example 0.30 m. for large numbers exceeding 5 figures, one space after every 3 digits counting from right end must be left blank without any commas, Example 1 500 375 is the correct way of writing the number 8. To the exponential power represented for a unit having a prefix refers to both the unit and 2
2
its prefix, Example mm = (mm) = mm . mm 9. Represent the numbers in terms of the base or derived units by converting all prefixes to powers of, while performing calculations, then the final should then the expressed using a single prefix 10. In general avoid the use of a prefix in the denominator of composite unit. Exception for this is the base unit, kilogram. For example do not write N/ m but rather than MN/metre 11. Compound prefixes should not be used, example G N should be expressed as kN since one space GN = 1X10 X10 = 1 (10 )N = 1kN 9
-6
Physical Quantity Acceleration Angular acceleration Angular displacement Angular velocity Angular momentum Area Couple, moment Density Displacement Energy Force Frequency Length Mass Momentum Moment of inertia of mass Plane angle
3
Unit 2 Metre/second 2 Radian/second Radian Radian/second 2 Kilogram metre /second Square metre Newton metre 3 Kilogram/metre Metre Joule Newton Per second Metre Kilogram Kilogram metre/second 2 Kilogram metre Radian
Symbol 2 m/s 2 rad/s rad rad/s 2 Kg m /s 2 M Nm 3 Kg/m m J N Hz m Kg kg m/s (=N s ) 2 Kg m rad
Power Pressure Speed Time
Watt Pascal Metre/second Second
Multiplication factor 12 10 9 10 106 3 10 -3 10 -6 10 -9 10 -12 10 -15 10 -18 10
W Pa m/s s
Prefix Tera Giga Mega Kilo Milli Micro Nano Pico Femto Atto
Symbol T G M K m μ N p f a
Dimensional Formula:
Dimensional analysis deals with dimensional of quantities Basic unit Mass Length Time Temperature Electric current Laminous Intensity Amount of substance
S.No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Physical quantity Displacement (S) Velocity (V) Acceleration (a) Force (F) Momentum Impulse Work or Energy Power Pressure Frequency Angular Velocity
Expression Distance Distance/Time Velocity/Time Mass X Acceleration Mass X Velocity Force X Time Force X Displacement Work/Time Force/Area No. of Vibrations/Time Angle/time
Dimension M L T K A Cd mol
Dimensional Formula L -1 LT -2 LT -2 MLT -1 MLT -1 MLT 2 2 ML T 2 -3 ML T -1 1 ML T -1 T -1 T
Basic concepts
For the investigation of problems of engineering mechanics, we must introduce the concept of space, mass, time and the force. Space refers of the geometric region occupied by the bodies. The positions of the bodies are
described by linear and angular measurements relative to a coordinate system. The concept of point, direction, length and displacement are required for the measurement and location in space. Example, a point is just an exact indication of location in space. Length is concept for describing the size of a body quantitatively by comparing it with a second body of same size and denoted by L. the unit of measure in SI system is metre (m). The international metre may be defined as 1650763.73 wavelengths of a certain radiation of the krypton-86 atom at 15C and barometric pressure of 76cm of mercury. For 2d problems two independent coordinates are needed and likewise 3 independent coordinates are required for 3d problems. Mass is the quantity of matter in a body which is made up of atoms and molecules. Mass can
also be regarded as that invariant property which measures its resistance to change of motion. Denoted by M. the SI unit of mass is kilogram denoted by kg. The international kilogram may be defined as the mass of platinum iridium cylinder kept at the international breau of weights and measures at Sevres near Paris. Time is the measure of sequence of events. It is related to the concepts of before, after and
simultaneous occurrence of two or more events. Time is the basic quantity in dynamics and is not directly involved in the analysis of statics problems. Denoted by “t” earlier denoted by T. subsequently T is used for absolute temperature and time is denoted by “t” . The SI unit for time
is seconds. The second is defined as the duration of 9192631770 periods of the radiation of a certain state of the Cesium-133 atom. SI units are absolute system of units which are independent of location where the measurements are made. FORCE is defined as any action that tends to change the state of rest of a body to which it is
applied. Its external effect upon a body is manifested by a change in, or a tendency to change, the state of motion of a body upon which acts. The internal effect of a force is to produce stress and deformation in the body. INTERNAL AND EXTERNAL FORCES
Internal forces are those which hold together the material content of the body or the parts of the system under consideration. Internal forces resistant or tend to resistant the external forces. For example if a car is pulling a trailer by a rope, coupling the two as shown in the figure, then the force in the rope is an internal force for the car and trailer system. The tractive force developed by a vehicle is transmitted through a series of components between the engines and wheels; these forces are internal as far as whole vehicle is concerned. (fig 1.5 of kumar)
External forces are those which on act on a body or a system from outside. Infact these are the forces exerted on a body from outside that govern its state of motion. For example for the trailer pulled by a car by a rope the effect of the car is experienced by the force in the rope; hence, for the trailer, the pull by the car i.e., the tension in the rope is an external force as an figure. Similarly the tension in the rope is an external for the car as in figure. EFFECTS OF A FORCE:
1. It may change the motion of a body i.e., if a body is at rest the force may set it in motion. And if the body is already in uniform motion the force may accelerate it. 2. It may retard the motion of a body. The effect of a force on a body in motion on which forces are already acting may be bring to rest or keep it in equilibrium 3. It may give raise to internal stresses in the body on which it acts. The characteristics of a force are :
i.
Its magnitude
ii.
The direction of the line, along which the force acts. It is also known as line of action of the force.
iii.
The direction (or sense) in which the force acts along its line of action- Whether force is push or pull. This denoted by spacing an arrow head on the line of action of the force.
iv.
The point at which the force acts on the body. Force is denoted by “F”. The unit of force in SI system is “N” which is defined as the for ce 2
required to accelerate one kilogram of mass by 1 m/s . Engineers prefer force instead of mass as the basic dimension because most of the data used in the design of components involve direct measurement of force. Mass is sued as fundamental dimension by physicist. Force and mass are not independent but are related by Newton’s second law of motion. Axioms in mechanics
The principles of mechanics are postulated upon several more or less self-evident facts which cannot be proved mathematically but can only be demonstrated to be true. We shall these facts the axioms of mechanics as given below: 1. Parallelogram law: the resultant of two forces is the diagonal formed on the vectors of these two forces.
2. Two forces are in equilibrium only when equal in magnitude, opposite in direction, and collinear in action. 3. A set of forces in equilibrium may be added to any of system of forces without changing the effect of the original system. 4. Action and reaction forces are equal but oppositely directed.
SCALAR AND VECTOR QUANTITIES: Scalar quantities: quantities which possess magnitude only and can be combined
arithmetically are defined as scalars. Common examples are area, energy, mass, time and temperature etc., Vector quantities: are defined as quantities having both magnitude and direction which
combines geometrically according to the parallelogram law. The result must be commutativemeaning independent of the order of geometric addition. Examples are velocity, force and torque etc., A vector of a quantity is represented graphically by drawing a line acting in the direction of the quantity, the length of this line representing to some scale the magnitude of the quantity. An arrow head is spaced at the end of this line, to denote the sense of the direction of the quantity. A Null or Zero vector is defined as a vector whose magnitude is zero. A roll of a zero vector in vector operations is equivalent to the roll of zero value in scalar operations. P
P’
P’= -P
The above system forms a null or a zero vector. Two vectors are said to be equal vectors if their magnitudes, directions and sense are the same. Two vectors are said to be equivalent or equipollent vectors if, in a certain sense they produce the same effect. However, the equality of vectors does not necessarily mean their equivalence of effect. A vector is said to be negative of another vector, if they have the same magnitude and direction but are opposite in sense P1
P1 =P2
P2
P1 P’2
P’2 = -P1
SYSTEM OF FORCES:
When several forces of various magnitudes and directions act upon a body they are set to constitute, a system of forces. The system of forces are classified according to the orientation of the lines of action (LOA) of the forces as follows: Force systems in plane: system of forces consisting of a set of forces with their lines of action lying in the same plane, which are also known as coplanar forces. Force system in space: system of forces consisting of a set of forces with their lines of action lying in the space. Both force systems in plane and force system in space can be further be classified into i)
Concurrent force system
ii)
Parallel force system
iii)
Non concurrent or general force system
The classification is shown in the following figure.
System of Forces
Coplanar
Non-Coplanar
(Plane force) or 2D System
(Spatial force) or 3D system
Concurrent
Concurrent
Non-Concurrent
Parallel
General
Non-Concurrent
Parallel
General
Concurrent Force System in a Plane: In this system, the lines of action of all forces pass through a
single point and forces lie in the same plane (Fig. 1.2)
Parallel Force System in a Plane : in this system, the lines of action of all forces lie in the same plane
and are parallel to each other (Fig. 1.3)
General Force System in a Plane: the lines of action of these forces lie in the same plane but they
are neither parallel nor concurrent (Fig. 1.4)
Concurrent Force System in Space: the lines of action of all forces pass through a single point but
not lie in the same plane. Tripod carrying a camera shown in Fig. 1.5 i s an example.
Parallel Force System in Space: The lines of action of all forces are parallel to each other, but not lie
in the same plane (Fig. 1.6).
General Force System in Space: The lines of action of these forces do not lie in the same plane and
they are neither parallel nor concurrent (Fig. 1.7).
Concurrent force systems can act on a particle or a rigid body. Parallel and general force systems can act only on a system of particles, a rigid body or a system of rigid bodies. RESULTANT:
The effect a system of forces on a body is usually expressed in terms of its resultant, since the value of the resultant determines the motion of the body. If the resultant is zero the body will be in equilibrium and will not change the original state of motion (either at rest or of uniform motion). This is the domain of statics. If the resultant a force is not zero the body will have a varying state of motion there by creating a problem in dynamics. PARALLELOGRAM LAW OF FORCES:
The method of vector addition is based on the parallelogram law which is one of the fundamental axioms of mechanics. Stevinus (in 1586) was the first to demonstrate indirectly this law and finally Varignon and Newton formulated this law in 1687, that forces could be combined by representing them by arrows to some suitable scale, and then forming a parallelogram in which the diagonal represents the sum of the two forces. The law states that “the resultant of two forces is the diagonal of the parallelogram whose
initial sides are the vectors of these forces. The diagonal to be used is that which emanates from the intersection of the initial sides.”
GRAPHICAL METHOD:
The parallelogram formed by two vectors P and Q is divided by the resultant R into two congruent triangles as shown in the figure. If the triangle ABC were alone as shown the vector joining the tail of P to the tip of Q would have the same magnitude and direction as the resultant R defined by the parallelogram law. In this instance force Q has been represented by free vector BC. A free vector is defined as one which may be freely moved in space as distinguished from a localized vector which is fixed or bound to a specific point of application. ANALYTICAL METHOD
If two given forces P and Q acting under the angle α are applied to a body at A, we will now find analytically the formulae for calculating the magnitude of resultant R and the angles β and which its line of action makes with those of the given forces. Fig 2.61 a page 21 of J V Rao shows the parallelogram of forces constructed in the usual manner while fig b shows the triangle of forces
obtained by the geometric addition of their free vectors. From the triangle of forces we find R = √ . The magnitude of the resultant R being known from the above equation, we may determine the angles β and by using the following equations Sin β =
Sin =
It is sometimes convenient to use these formulae to determine the resultant instead of making an accurate construction to scale of the triangle law of forces. For the special i) when ii) When
R = P+Q
, R=√ = √ = √ =(P-Q)
therefore R= P-Q iv)
When , then R= √ and Tan = angle between resultant R and the force Q
Triangle Law:
Triangle law is a corollary of a parallelogram law. If two forces are represented by their free vectors placed tip to tail, their resultant is the vector directed from the tail of the first vector to the tip of the second vector. Equilibrium of Collinear forces
From the principle of parallelogram law of forces the forces applied at one point can always be replaced by their resultant which is equivalent to them. Thus we conclude that two concurrent forces can be in equilibrium, only if their resultant is zero. This means that if we have two forces of equal magnitude acting in opposite directions along the same line then their resultant is zero and the two concurrent forces are in equilibrium. The above when generalized is the second principle of statics. Equilibrium law: Two forces can be in equilibrium only if they are equal in magnitude,
opposite in direction and collinear in action. In engineering problems very often we deal with the equilibrium of a body in the form of a bar on the ends of which two forces are acting as shown in the figure 2.9 page 24 of J.V. Rao.
Neglecting their own weights, it follows from the above equilibrium law that their bar can be in equilibrium only when the forces are equal in magnitude, opposite in direction and collinear in action, which means they must act along the line joining their points of application. If these points of application are assumed to be on the symmetrical axis of the bar, the force act must along this axis. When such central forces are directed as shown in fig. 2.9a, we say that the bar is in tension. When they act as shown in fig. 2.9 b, t he bar is said to be in compression. Considering the equilibrium of the bar a portion of the bar in figure a to the left of section mn, we conclude that to balance the external forces at A the portion to the right must exert on the portion to the left an equal, opposite, and collinear force S as shown in figure c. the magnitude of this internal axial force which one part of a bar in tension exerts on another part is called the tensile force in the bar or simply the force in the bar. In general it may be either a tensile force or a compressive force. Such internal is actually distributed over the cross sectional area of the bar and its intensity i.e., the force per unit cross-sectional area is called the stress in the bar. INTERNAL FORCES are the forces which hold together the particles of a body. For e.g., if we
try to pull a body by applying two equal, opposite and collinear forces, an internal force comes into play
Therefore, the resultant of all these internal forces is zero and does not affect the external
motion of the body or its state of equilibrium. EXTERNAL FORCES or applied forces are the forces that act on the body due to contact with
other bodies or attraction forces from other, separated bodies. These forces may be surface forces (contact forces) or body forces (such as gravitational attraction). At times, we have to deal with the equilibrium of a prismatic bar on each end of which two forces are acting as shown in fig. 2.10(a), instead of a single force at each end as shown in fig. 2.9(a). Then the forces at A and B are replaced with their respective resultants R a and Rb, as shown in the
fig. 2.10(b), which is reduced to the previous case, where the bar is subjected to two equal, opposite and collinear forces. Other examples of two force members held in equilibrium are shown in the fig.2.11.
Next, we consider two forces acting on a body at an angle α in between them as shown in fig. 2.4(a). From the equilibrium law, we conclude that we can hold these two forces in equilibrium, by applying at point A, a force equal and opposite to their resultant. This force is called the equilibrant of the two given forces. A force, which is equal, opposite and collinear to the resultant of the two given forces, is known as the equilibrant of the two given forces.
SUPERPOSITION AND TRANSMISSIBILITY:
When two are in equilibrium (equal, opposite and collinear), their resultant is zero and their combined action on a rigid body is equivalent to that of no force at all. A generalization of this observation gives us the third principle of statics, sometimes called the law of superposition Law of superposition: The action of a given system of forces on a rigid body will in no way be
changed if we add to or subtract from them another system of forces in equilibrium.
Let us consider a rigid body AB under the action of a force P applied at A (fig 2.12 page 26 of JV rao) and acting along DA as shown in the figure from the principle of superposition stated above, we conclude that the application at point B of two oppositely directed forces, each equal to and collinear with P, will in no way alter the action of the given force P. i.e., the action of the body on three forces in figure b is identical to the action of single force P in figure a. Repeating the same reasoning, we remove from the system as shown in figure b the equal, opposite, and collinear forces P and P’’ as a system in equilibrium. Thus we obtain the condition
shown in figure c wher e, instead of the original P applied at A we have equal force P’ applied at B. this proves that the point of application of a force may be transmitted along its line of action without changing the effect of the force on any rigid body to which it may be applied. This statement is called the principle of transmissibility of a force .
Principle of Transmissibility of force states that the external effect of a force on a rigid body is the
same for all points of application along its line of action i.e., it is independent of the point of application. (fig. 1-3.1 a) However its internal effect definitely is associated with the point of application of the force. In fig.a the motion of the block will be the same whether it is pushed at a or pulled at b. But the local internal effects at a and b will be quite different. Note that the principle of transmissibility applies only to the external effects of a force on the same rigid body . Example: a rigid under the action of two equal and opposite forces as shown in the figure. The principle of transmissibility would state that the forces in case1 and case2 are equivalent and each case the net external force is zero. (fig 1.8 I from kumar) this statement is true only from point view of external behavior of the body. Let us look at the development of the internal forces to keep the body and its parts in equilibrium. The resistive forces are developed at a 1 and a2 as shown by dotted arrows in the two cases. For case1 the bar is in tension and for case2 it is in compression. These are entirely different effects.
ACTION AND REACTION:
Very often we are required to study the conditions of equilibrium of bodies which are not free to move. Restriction to the free motion of a body in any direction is called constraint (fig 2.15 a b c Jv Rao) in figure a we have a ball resting on a horizontal plane such that it is free to move along the plane but cannot move vertically downward.
Similarly the ball in fig 2a (2.2a page 18) although it can swing as a pendulum, is constrained against moving vertically downward by the string AB. In figure 2.16 a (page 28), a ball of weight W is supported by a string BC and resting against a smooth vertical wall at A. with such constraints all motions of the ball in the plane of the figure are prevented. There are other kinds of constraints than those mentioned above.
A body that is not entirely free to move and is acted upon some applied forces will, in general, exert pressures against its supports. For example the ball in fig 2.2a exerts a downward pull on the end of the supporting string as shown in fig. 2.2b. Similarly the ball in fig. 2.15a exerts a vertical push against the surface of the supporting plane at the point of contact A as shown in figure 2.15b. For the case shown in figure 2.16a, the ball not only pulls downward on the string BC but also pushes to the left against the wall at A as shown in figure 2.16 b. Now in every case the action of a constrained body against its supports induces reactions from the supports on the body, and as the fourth principle of statics we take the following statement.
Law of action and Reaction: Any pressure on a support causes an equal and opposite pressure from
the support so that action and reaction are two equal and opposite forces. It is seen that this principle of statics is nothing but Newton’s third of motion stated in a form suitable for the
discussion of problems of statics. 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:
One of the most important concepts in mechanics is that of the free body diagrams. To investigate the equilibrium of a constrained body, we shall always imagine that we review the supports and replace them by the reactions which they exert on the body. Thus the case of the ball min figure 2.2a we remove the supporting string and replace it by the reaction R a that it exerts on the ball.
We know that the point of application of this force must be the point of contact B, and
from the law of equilibrium of two forces, we conclude that it must be along the string i.e., vertical and equal to the weight W; thus it iis completely determined. The sketch in figure 2.2c in which the ball is completely isolated from its supports and in which all forces acting on it are shown by vectors is called a free body diagram.
Free body diagram is a sketch of an isolated body, which shows the external forces on the body and the reactions exerted on it by the removed elements.
Procedure for constructing a Free-Body diagram: 1. A sketch of the body is drawn, by removing the supporting surfaces. 2. Indicate on this sketch all the applied or active forces, which tend to set the body in motion,
such as those caused by the weight of the body or applied forces etc., 3. Also indicate on this sketch all the reactive forces, such as those caused by the constraints or
supports that tend to prevent motion. The sense of unknown reaction should be assumed. The correct sense will be determined by the solution of the problem. A +ve result indicates that the assumed direction is correct. A –ve result indicates that the correct sense is opposite to the assumed one. 4. All relevant dimensions and angles, reference axes are shown on the sketch.
The technique of constructing a correct free body diagram co nsists of applying the preceding steps in conjunction with the equations of equilibrium equations. Similarly, in the case of the ball in fig. 2.15(a), we remove the supporting surface and replace it by the reaction R a that it exerts on the ball. We know that the point of application of this force must be the point of contact A, and from the law of equilibrium of two forces, we conclude that it must be vertical and equal to the weight W; thus it is completely determined. The free-body diagram of the ball in fig. 2.15(a) is shown in fig. 2.15(c). In the case of the ball if Fig. 2.16(a), we again remove the supports and isolate the ball as a free body [Fig. 2.16(c)]. Then besides the weight W acting at C, we have two reactive forces to apply, one replacing the string BC and another replacing the wall AB. Since the string is attached to the ball at C and since a string can pull only along its length, we have the reactive force S applied at C and parallel to BC. Its magnitude remains unknown. Regarding the reaction Ra, we have for its point of application the point of contact A. Furthermore, we assume that the surface of the wall is perfectly smooth so that it can withstand only a normal pressure from the ball. Then, accordingly, the reaction R a will be horizontal and its line of action will pass through C as shown. Again only the magnitude remains unknown and the free-body diagram is completed. The question of finding the magnitudes of S and R a, will not be discussed here, although it is only necessary to so proportion these vectors that their resultant is equal and opposite to the vertical gravity force W. From the above discussion, we come across two types of supports namely string support and a smooth surface or support. A flexible weightless and in-extensible string is a constraint prevents a body moving away, from the point of suspension of the string, in the direction of the string. The reaction of the string is directed along the string towards the point of suspension. So, string or cable can support only a tension and this force always acts in the direction of the string. The tension force developed in a continuous string, which passes over a frictionless pulley, must have a constant magnitude to keep the string in equilibrium (Fig. 2.17). Hence, the string or cord, for any angle , is subjected to a constant tension S throughout its length.
A small surface is one whose friction can be neglected. Smooth surface prevents the displacement of a body normal to both contacting surfaces at their point of contact. The reaction of a smooth surface or support is directed normal to both contacting surfaces at their point of contact and is applied at that point (Fig. 2.18). If one of the contacting surfaces is a point, then the reaction is directed perpendicular or normal to the other surface (Fig. 2.15). if two of the contacting surfaces are points, then the reaction is directing perpendicular or normal to the tangent of contacting surfaces [fig. 2.19 (a) and 2.20 (a)].
The free body diagrams of the bodies are shown in figs 2.15 (b), 2.18 (b), 2.19 (b), 2.20(b), respectively. Another type of support is linear elastic spring (fig. 2.21). The magnitude of force developed by a linear elastic spring which has a stiffness k, and is deformed a distance x measured from its unloaded position, is S= kx Note: x is determined from the difference in the spring’s deformed length and its initial length. If
x is positive, S ‘pulls’ on the spring; whereas if x is negative, S must ‘push’ on it. In the case of the body in fig. 2.21 (a), we remove the supporting spring and replace it by the spring force S that it exerts on the body. We know that the point of application of this force must be the point of contact, and from the law of equilibrium of two forces, we conclude that it must be along the spring, i.e., vertical and equal to the weight W; thus it is completely determined. The free body diagram of the body in fig. 2.21 (a) is shown in fig. 2.21 (b).
One more example of free-body diagram is considered here. The lawn roller, of weight W, being pushed up the inclined smooth plane as shown in fig. 2.22(a). In the case of the lawn roller in fig. 2.22(a), we again remove the support and isolate the body as a free body [fig. 2.22 (b)]. Then beside the weight W and push P acting at centre O, we have one reactive force R a to apply, replacing the inclined plane. The reactive force that the surface of the inclined plane is perfectly smooth so that it can withstand only a normal pressure from the roller. Then the reaction R a will be normal to the inclined surface and its line of action will pass through O as shown here the magnitude remains unknown and the free body diagram is completed the question of finding the magnitudes of P and R a will not be discussed here. Proceeding as above with constrained bodies, we shall always obtain two kinds of forces acting on the body: the given forces, usually called active forces, such as the gravity force W in fig. 2.16 (c), and reactive forces, replacing the supports such as the forces S and R a in fig. 2.16 (c). To have equilibrium of the body, it is necessary that the active forces and reactive forces together represent a system of forces in equilibrium. Thus it is by means of the free-body diagram that we determine the system of forces with which we must deal in our investigation of the conditions of equilibrium of any constrained body. The construction of this diagram should be first step in the analysis of every problem of statics, and it must be evident that any errors or omissions here will themselves on all subsequent work. The essential problem of statics may now be briefly recapitulated as follows: we have a body either partially or completely constrained which remains at rest under the action of applied forces. We isolate the body from its supports and show all forces acting on it by vectors, both active and reactive. We then consider what conditions this system of forces must satisfy in order to be in equilibrium, i.e., in order that they will have no resultant. LAWS OF MECHANICS:
1. Newton’s laws: a) First law: “Everybody continues in its state of rest or of unif orm motion, in a straight line, unless it is acted upon by some external force”. b) Second law: the rate of change of momentum is directly proportional to impressed force
and takes place in the same direction, in which the force acts. c)
Third law: to every action there is always an equal and opposite reaction.
Observations of the above laws
1. The word “body” is undefined. It either refers to a particle only or to the centre of mass of a rigid body.
2. The term motion in a Straight line appears in the first and second laws but no attempt has been made to govern the rotational and general motion of the bodies of finite size. 3. Only the “forces” have been considered; the actio n of the moment is not included ./ 4. The second law which relates acceleration to the forces impressed assumes constant C of the mass of the body. (F=d/dt(mv)=m(dv/dt)=ma) Law of Gravitation- weight of Bodies
Any two particles will be attracted each other along a line connecting their centers with a mutual force whose magnitude is directly proportional to the product of their masses and inversely proportional to the distance between them. (fig 1.12 kumar)
The law of gravitation requires that the force of attraction between two particles of masses m1 and m2 separated by a distance r as shown in the figure is given by Where G is the universal constant of gravitation; its value been 6.67X10
-11
2
2
3
2
Nm /kg or m /kg. s
F = Mutual force of attraction between particles or bodies r= distance between centres of the particles or bodies -11
Quantitatively an attractive force of 6.67X10 N is exerted by a body of mass 1kg on another body of mass 1 kg at 1 m distance from it. Obviously the attractive force of reaction by the other body on it must also be equal to the same value. Gravitational attraction of the earth
Gravitational forces exist between every pair of bodies. On the surface of the earth the only gravitational force of appreciable magnitude is the force due to attraction of the earth. For example consider two iron spheres of 100 mm diameter are attracted to the earth with a gravitational force of 37.1N each, which is their weight. On the other hand the force of -8
mutual attraction between the spheres if they are just touching is 9.51X10 N. this force is clearly negligible compared to the earth attraction of 37.1N. Consequently the gravitational attraction of the earth is the only gravitational force we need to be considered for most engineering applications on the surface of the earth. EFFECT OF ALTITUDE:
The force of gravitational attraction of the earth on a body depends on the position of the body relative to the earth. If the earth were a perfect homogenous sphere, a body with a mass of exactly one kg would be attracted to the earth by a force of 9.825N on the surface of the earth, 9.822N at an altitude of 1km, 9.523N at an altitude of 100km, 7.340N at an altitude of 1000km, and 2.456N at an altitude equal to the mean radius of the earth, 6371km. thus the
variation in gravitational attraction of high altitude rockets and space craft’s becomes a major consideration. Every object which falls in vacuum at a height near the surface of the earth wi ll have the same acceleration g, regardless of its mass. By combining equations 1 and 2 i.e., F = ma and and cancelling the term representing the mass of the falling object. This combination gives
me = mass of the earth R = radius of the earth The mass m e and the mean radius R of the earth have been found through experimental 24
6
measurements to be 5.976X10 kg and 6.371X10 m, respectively. These values, together with the value of G already cited, when substituted into the expression for g, give a mean value of g = 2
9.825 m/s . The variation of g with altitude is easily determined from the gravitational law.
Where g0 is the absolute acceleration due to gravity at sea level R is the radius of the earth. APPARENT WEIGHT
The gravitational attraction of the earth on a body of mass m may be calculated from the results of a simple gravitational experiment. The body is allowed to fall freely in a vacuum, and its absolute acceleration is measured. If the gravitational force of attraction or true weight of the body is W then, because the body falls with an acceleration of g, the equation F=ma gives W=mg. The weight W will be in Newton ’s (N) when the mass m is in kilograms(kg) and the 2
2
2
acceleration due to gravity is in meters/seconds (m/s ). The standard value for g is 9.81m/s in M.K.S. and S.I. units will be sufficiently accurate for the calculations in statics. The true weight (gravitational attraction) and the apparent weight (a s measured by a spring balance) calibrated to read a correct force and attached to the surface of the earth, will be slightly less than its true weight. The difference which is due to the rotation of the Earth is quite small and will be neglected. FORCES AND COMPONENTS COMPOSITION:
The reduction a system of forces to the simplest system that will be its equivalent is called the problem of composition of forces. If several forces F1, F2,F3,….., applied to a body at one point, all act in the same plane, they represent a system of forces that can be reduced to a
single resultant force. It then becomes possible to find this resultant by successive applications of the parallelogram law. Let us consider, for example, the four forces F 1, F2, F3,F4 acting on a
̅ of body at point A [fig. 2.31 (a)]. To find their resultant, we begin by obtaining the resultant
̅ the two forces F1 and F2. Combining this resultant with the force F 3, we obtain the resultant ̅ and F4, we obtain the which must be equivalent to F 1, F2 and F3. Finally, combining the forces resultant R of the given forces acting at one point in a plane.
It is evident, in the above case that exactly the same resultant R will be obtained by successively geometric addition of the free vectors r epresenting the given force F 1. From the end B of this vector we construct the vector ̅ representing the force F 2, and afterward, the vector
̅ and ̅, representing the forces F 3
and F4 . The polygon ABCDE obtained in this way is the
same as the polygon ABCDE in fig. 2.13 (a), and the vector
̅ to the end E of the Vector vector
̅ , from the beginning A of the
̅ , gives the resultant R which, of course, must be applied
at point A in the fig. 2.13 (a). The polygon ABCDE in fig. 2.31 (b) is called polygon of forces and the resultant is given by the closing side of the polygon. It is always directed from the beginning of the first vector to the end of the last vector. Thus, we may say that the resultant of any system of concurrent forces in a plane is obtained as the geometric sum of the given forces. The construction of the polygon of forces, for determining the resultant, is much more direct for a large number of forces than successive applications of the parallelogram law and is preferable in the solution of problems. It is evident that the resultant R will not depend upon the order in which the free vectors representing the given forces are geometrically added. For instance, in the above example, we can begin with the force F 1, add to it the force F4 and afterward the forces F 2 and F3. Proceeding in this way the polygon of forces shown in fig 2.31(c) will be obtained. The closing
̅ of the polygon gives the same resultant R as before. side In the particular case where the given forces are all acting along one line, the sides of
̅ line and the geometric summation will be replaced the polygon of forces will all lie along one, by an algebraic summation. The resultant, in this case, is the algebraic sum of its components. If the end of the last vector coincides with the beginning of the f irst, the resultant R is equal to zero and the given system of forces is in equilibrium.
Alternatively, polygon law which is equivalent to the repeated application if parallelogram law can be applied to determine the resultant of a number of concurrent coplanar forces. The Law of Polygon of Forces: it may be stated as “if a number of coplanar forces are acting at a
point such that they can be represented in magnitude and direction by the sides of a polygon taken in an order, their resultant is represented in both magnitude and direction by the closing side of the polygon taken in the opposite order. RESOLUTION OF FORCES:
The parallelogram law shows how to combine to two forces into a resultant force. Of the equal importance is the inverse operation, called resolution, in which a given force is replaced by two components which are equivalent to the given force. The method is demonstrated in the following figure in which we are to replace the force F by components directed along the lines OA and OB radiating from tail O of force F. we need nearly to draw lines from the tip of F parallel to the specified directions to form the parallelogram as shown. The initial sides P and Q of this parallelogram are the desired components. Obviously for the parallelogram law to apply, the components P and Q must intersect on F. (fig2-3.1 page19 of singer-oblique components). The graphical construction can also be made by the triangle law as shown in part b using the singular relation (Lami s theorem) . Although the triangle law is convenient for analytical solution it is localized components shown in part which completely replaces F. ’
Depending on the directions specified, there are infinite number of pairs of oblique components of F that may be formed. Such non-orthogonal components, however, of limited use. Analytically, it is much more convenient to resolve a force into a pair of perpendicular components such rectangular components are then readily combined with similarly oriented rectangular components of other forces by adding these components algebraically. RECTANGULAR COMPONENTS: (FIG3-22 PAGE 20)
Consider the above the fig. in which the force F acts on the given body. The effect of the force is to move the body right ward and upward. Choosing these directions as the positive directions of perpendicular X and Y reference axis, we project the force F upon them to obtain the perpendicular components F x and FY. More precisely, we should draw parallels to the X and Y axis to obtain the basic parallelogram, but when the reference axes are perpendicular, the projected length of the force yields the same components. The relations between these components and F is determined by the basic definitions of Sine and Cosine of the angle between F and the X-axis i.e., Sin and Cos = which are usually rewritten in the following form Fx = F Cos and Fy = F Sin . The components and F x and Fy are considered positive if they act in the directions of X and Y axis and negative if they directed oppositely. Usually X and Y axis are horizontal and vertical, and their positive directions are those as the common Cartesian coordinate axes.
However the orientation of X and Y axis is arbitrary; their directions are adapted to the situation. The relations given above are independent of the orientation of the X axis. It is obvious that when the rectangular components of a force are known, they completely specify the magnitude, inclination, and the direction of the force.
