Engineering Mechanics PDF Notes

1 ENGINEERING MECHANICS Mechanics: Mechanics may be defined as the science, which describes and predicts the condition of rest or motion of the bodies under the action of forces. Mechanics is divided into statics and Dynamics. Statics: Statics is the study of forces a...

1 ENGINEERING MECHANICS Mechanics: Mechanics may be defined as the science, which describes and predicts the condition of rest or motion of the bodies under the action of forces. Mechanics is divided into statics and Dynamics. Statics: Statics is the study of forces and condition of equilibrium of bodies at rest or moving with constant velocity (zero acceleration) under the action of forces. Dynamics: Dynamics is the study of motion of rigid bodies and their correlations with the forces causing the motion. Dynamics is divided into kinematics and kinetics. Kinematics: It is the study of motion of rigid bodies without considering the forces causing the motion. It deals the relationship between displacement, velocity, and acceleration, and their variation with time. Kinetics: It is the study of the relationship between the forces and the resulting motion. Idealization in Mechanics Idealizations are used in mechanics to simplify the application of theory. Here, some important idealizations are being discussed which are as follows: 1. Continuum: It may be defined as the continuous distribution of matter with no holes, voids or empty spaces. 2. Particle: A body whose dimensions can be neglected in studying its motion or condition of equilibrium may be treated as a particle. Example: While studying the planetary motion. 3. Rigid body: A body is assumed to be rigid, if the deformation is negligible compared to the size of the body. Example: A lever supporting weights at its ends. Fundamental Principles of Mechanics 1. Newton’s First law (law of Inertia): Everybody continues in a state of rest or of uniform motion in a straight line unless it is compelled to change that state by a force imposed on the body. First law helps us to define a force. 2. Newton’s Second Law: The acceleration of a given particle is proportional to the impressed force and takes place in the direction of the straight line in which the force is impressed. F = ma This law helps us to measure a force quantitatively. 3. Newton’s Third Law: To every action there is equal and opposite reaction FORCE: Force may be defined as any action that tends to change the condition of rest or uniform motion of a body to which it applied. Force is a vector quantity. Unit of force  2 SI unit: N (Newton) MKS unit: kgf (kilogram-force) CGS unit: Dyne 1 Kgf= wt of 1 kg mass at sea level=1 kg = 9.81 N -5 1 Dyne = 10 N CHARACTERISTIC or SPECIFICATIONS OF A FORCE To completely define a force there is a need of the following quantities, which are as follows: 1. MAGNITUDE 2. DIRECTION: The angle made by the magnitude or line of action of force with the horizontal or vertical reference line. 3. POINT OF APPLICATION: The point at which or from which a force may be assumed to be concentrated. 4. LINE OF ACTION: When a force acts on a body which tends to move in a straight line in the direction of applied force, if this straight line passes through the point of application of the applied force then this straight line is called line of action of force. The LOA goes up to the infinite while line of magnitude has limited value. LOA indicated by dotted line while line of magnitude indicated by thick line. 5. SENSE: it is indicated by arrowhead. Line of action Magnitude Sense θ O Point of application Figure 1 Principle of Transmissibility of forces: It states that the condition of equilibrium or of motion of rigid body will remain unchanged if the point of application of force acting on the rigid body is transferred to act at any other point along its line of action while the magnitude and direction of the force is the same. Figure 2  3 The principle of transmissibility can be used to discuss the condition of equilibrium of a rigid body to determine the external forces acting on the rigid body. The principle of transmissibility cannot be used to determine the internal forces in the body. SYSTEM OF FORCES When more than one force acts on a body at a particular instant, they are said to constitute a system of forces. System of forces Coplanar forces or plane forces Non-coplanar or space or spatial forces Concurrent forces Non-Concurrent forces Non-Concurrent Concurrent forces forces Collinear Parallel General forces forces forces Collinear Parallel General forces forces forces Coplanar Forces: Forces which lie in the same plane are known as coplanar forces. Non-Coplanar Forces: Forces which do not lie in the same plane are known as non-coplanar forces. Forces F1 and F2 are coplanar forces lie in X-Z plane Forces F3 and F4 are coplanar forces lie in X-Y plane Forces F1 and F3 are non-coplanar forces lie in X- Z and X-Y forces. Forces F3 and F4 are non-coplanar forces lie in X- Y and Y-Z plane. Figure 3 Concurrent forces: The forces, whose lines of action intersect at a common point, are called concurrent forces.  4 Non- Concurrent Forces: The forces, whose lines of action do not intersect at a common point, are called concurrent forces. Combined system of Forces F1 and F2 Combined system of Forces F1 and F3 Combined system of Forces F1 and F4 Combined system of Forces F2 and F3 Combined system of Forces F2 and F4 Combined system of Forces F3 and F4 Combined system of Forces F1, F2 and F3 All the above force systems individually are concurrent force systems. Figure 4 But combined system of forces F1, F2, F3 and F4 is non-concurrent force system. Collinear Forces The forces, whose lines of action lie on the same line, are known as collinear force. Figure 4 Parallel Forces The forces, whose lines of action are parallel to each other, are called parallel forces. The parallel forces can be classified into two types: 1. Like parallel forces 2. Unlike parallel forces Like Parallel Forces: Forces whose lines of action are parallel to each other and all of them act in same direction, are called like parallel forces. Unlike Parallel Forces: Forces whose lines of action are parallel to each other but all of them do not act in the same direction, are called unlike parallel forces.  5 COPLANAR CONCURRENT FORCES Forces whose lines of action intersect at a common point and lie in the same plane are known as coplanar concurrent forces. Example: Forces acting on a rod which is hinged at point c in the wall and supported by a string AB The lines of action of forces W, T and reaction force R intersect at a common point O and all C these forces lie in the same plane. COPLANAR NON-CONCURRENT FORCES Forces whose lines of action do not intersect at a common point and lie in the same plane are known as coplanar non-concurrent forces. Example: A system of forces acting on a ladder resting against a wall and floor. B RB G Forces W, RB, RA lie in the same plane but do not RA intersect at a common point. W A COPLANAR PARALLEL FORCES The Forces; whose lines of action parallel to each other and lie in the same plane are known as coplanar parallel forces. Example: A system of vertical forces acting on a beam.  6 COPLANAR COLLINEAR FORCES Forces whose lines of action lie in the same line and the same plane are known as coplanar collinear forces. Example: The forces acting on the rope in the tug of war. NON-COPLANAR CONCURRENT FORCES Forces whose lines of action do not lie on the same plane, but they intersect at a common point. Example: A tripod carrying a camera as shown in figure. NON-COPLANAR NON-CONCURRENT FORCES Forces whose line of action do not lie on the same plane and they do not intersect at a common point. Example: The forces acting on a moving bus NON-COPLANAR PARALLEL FORCES Forces whose line of action do not lie on the same plane and they are parallel to each other. Example: The weight of benches in a class.  7 NON-COPLANAR COLLINEAR FORCES Not possible Different force system will have different effects on the rigid bodies. For an instance, concurrent forces in the plane and in space tend to move or translate the body as a whole. As there is no rotational motion involved and, in such cases, body can be idealized as a particle i.e., a body without any extent. However, non-concurrent forces in plane and in space tend to rotate the body in addition to translating the body. In such cases the body cannot be idealized as a particle while treated as a rigid body itself. Parallelogram Law of forces It states that if two forces acting at a point of a rigid body are represented in magnitude and direction by the two adjacent sides of a parallelogram then their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point. The Sum or resultant of P and Q is independent of the order in which they are added. R=P+Q = Q+P Analytical method of parallelogram: B C Q R θ α θ O P A D If the angle between the forces P and Q is θ, then R = P 2 + Q 2 + 2 PQ cos  8 The direction of resultant force with the force P is α, then  Q sin   = tan −1    P + Q cos   Special Cases: (i) When two forces are equal and θ is the angle between them: R= P 2 + Q 2 + 2 PQCOS = P 2 + P 2 + 2 P  PCOS   R = 2 P 2 (1 + COS ) = 2 P 2  2COS 2 = 2 PCOS 2 2  QCOS  −1  P sin   and  = tan −1   = tan    P + QCOS   P + P cos       2sin cos   sin   2 2 = tan −1  −1  = tan    1 + cos   2  2 cos   2    = tan −1  tan   2  = 2 i.e., the resultant bisects the angle between the forces (ii) When the two forces act at right angles, i.e., θ=90° R= P 2 + Q 2 + 2 PQ cos 90 R = P2 + Q2 (iii) When two forces act in the same line and the same sense, i.e., θ=0°. In this case, the value of resultant will be maximum. R= P 2 + Q 2 + 2 PQ cos 0 = P 2 + Q 2 + 2 PQ cos 0 Rmax = P + Q (iv) When two forces have the same line of action but opposite senses, i.e., θ= 180°. In this case, the value of resultant will be minimum. R= P 2 + Q 2 + 2 PQ cos180 = P 2 + Q 2 − 2 PQ R min = P − Q Law of Triangle of Forces: This can be stated as “If two forces acting at a point of a rigid body are represented in magnitude and direction by two sides of a triangle taken in an order, then their resultant is represented in magnitude and direction by the third side taken in the opposite order”.  9 Sine Rule: P Q R = = sin  sin  sin  Cosine Rule: a 2 + c2 − b2 cos  = 2ac a + b2 − c2 2 cos  = 2ab b + c2 − a2 2 cos  = 2bc Resolution of Force When a single force F acting on a body is replaced by two or more forces which together have the same effect on the body as the force F, these forces are called the components of the original force F and the process is called resolution of force into its components. Resolving a Force into Rectangular Components: The components of force which are perpendicular to each other, are called rectangular components. 10  11 Resultant of several coplanar concurrent forces (Method of Projection): This is the analytical method of polygon law. The forces R and R are perpendicular forces. Now we see the sign convention of forces: X Y Now as per the sign convention, we take R X =  FX = F1 cos 1 + F4 cos  4 − F2 cos  2 − F3 sin 3 R Y =  FY = F1 sin 1 + F2 sin  2 − F3 cos 3 − F4 sin  4 Now, here are the conditions as the sign of R and R X Y 1. When R and R both have +ive sign, then X Y Resultant R = RX 2 + RY 2 RY tan  = RX  RY   = tan −1   Direction of resultant  RX  Resultant lies in first quadrant. 2. When R is -ive and R is +ive , then X Y  12 Resultant R = RX 2 + RY 2 RY tan  = RX Direction of resultant  RY   = tan −1    RX  Resultant lies in second quadrant. 3. When R and R both have -ive sign, then X Y Resultant R = RX 2 + RY 2 RY tan  = RX  RY  Direction of resultant  = tan −1    RX  Resultant lies in third quadrant. 4. When R is +ive and R is -ive , then X Y Resultant R = RX 2 + RY 2 RY tan  = RX  RY  Direction of resultant  = tan −1    RX  Resultant lies in fourth quadrant.  13 ASSIGNMENT-1 (Resultant of concurrent force system) Q1. The forces 20 N, 30 N, 40 N, 50 N and 60 N are acting at one of the angular points of a regular hexagon, towards the other five angular points, taken in order. Find the magnitude and direction of the resultant force. Ans: R= 155.8 N, α= 76.6° Q2. Find the resultant of given force system. Q3. Determine the resultant of given force systems as shown in both figures.