Unit-II ENGINEERING MECHANICS

Transcription

1 ECHNICS in motion. Unit-II ENGINEERING ECHNICS By Prof. V. adhava Rao, SJCE, ysore It s a branch of science, which deals with the action of forces on bodies at rest or ENGINEERING ECHNICS engineering. It deals with the principles of mechanics as applied to the problems in BSIC CONCEPTS 1. atter: nything which has mass and requires space to occupy is called matter. 2. ass: It is a measure of quality of matter contained by the body. SI unit: Kg. 3. Volume: It is a measure of space occupied by the body. Unit: m 3 Note: Liter Unit of volume 1000 liters = 1 m 3 TS Thousand million cubic feet ft ft x ft x 1000 ft 4. State of rest and motion: State of rest and state of motion are relative and depend on the frame of reference. body is said to be in a state of rest w.r.t. a frame of reference if the position of the body w.r.t. that frame of reference is not changing with time. body is laid to be in a state of motion w.r.t. a frame of reference if the position of the body w.r.t. that frame of reference is changing with time. 5. Scalar and Vector Quantities: Quantities which require only magnitude to represent them are called scalar quantities. Eg: ass, Time interval. Quantitites which require both magnitude and direction to represent them are called vector quantities. Eg: orce, Velocity, etc.

2 6. Displacement and distance travelled: The total linear movement made by a body to change its position from one point to another is called distance travelled by the body. It is a scalar quantity. Unit: eter (m) mm illimeter 10-3 m km Kilo eter 10 3 m The total linear movement made by a body to change its position from one B Displacement point to another moving along a particular Distance direction is called displacement. Displacement is a vector quantity. Unit: eter (m). 7. Speed and Velocity: The distance travelled in a unit time is speed. Unit: m/s ms -1 The displacement in unit time is called velocity. Unit: m/s ms Uniform motion and non-uniform motion: If the velocity of the moving body remains constant then the motion is said to be uniform. If the velocity is changing with time, the motion is laid to be non-uniform. 9. cceleration and retardation: The time rate of change of velocity is called acceleration. If the velocity is increasing with time then acceleration is positive. If the velocity is decreasing with time then acceleration is negative. Negative acceleration is called retardation or deceleration. Unit: m/s 2 ms omentum: It is the capacity of a moving body to impart motion to other bodies. omentum of a moving body is given by the product of mass and velocity of the moving body. omentum = ass Velocity Unit: kg m/s or kg ms Newton s I Law of otion: Everybody continues to be in its state of rest or uniform motion unless compelled by an external agency.

3 12. Inertia: It is the inherent property of a body by virtue of which it can retain its state of rest or uniform motion unless compelled by an external agency. 13. orce: It is an external agency, which overcomes or tends to overcome the inertia of a body. It is a vector quantity. 14. Elements of a force: There are four elements: a. agnitude b. Direction c. Line of action d. Point of action or application 15. Newton s II Law of motion: The rate of change of moment of a body is directly proportional to the magnitude of the force applied and takes place in the direction of the force applied. Explanation: Initial momentum = mu inal momentum = mv Change in momentum over a time interval t = mv mu Rate of change of momentum = ccording to Newton s II law, mv mu t mv mu α t v u α m t m u Time interval = t m V α ma = K ma In SI, unit force is defined as that force which acts on a body of unit mass producing unit acceleration. i.e., = 1 when m = 1 and a = 1 then 1 = k k = 1 = ma Unit of force: newton (N) is the unit of force. One newton is that force which acts on a body of mass 1 kg producing an acceleration of 1 m/s 2.

4 kn Kilo newton 10 3 N N ega newton 10 6 N GN Giga newton 10 9 N 16. Newton s III law of motion: or every action there is equal and opposite reaction. 17. Branches of echanics: echanics Solid ech. luid ech. Rigid Body ech. ech. of Deformable Bodies luid Statics luid Kinematics luid Dynamics Statics Dynamics Kinematics Kinetics Statics: Statics deals with the action of forces on bodies at rest or in equilibrium. Dynamics: Dynamics deals with the action of forces on bodies in motion. Kinematics: It deals with the study of geometry of motion without considering the cause of motion. Kinetics: Kinetics deals with a study of motion considering the course of motion. 18. Rigid body: The concept of rigid body is purely theoretical or imaginary. rigid body is said to undergo, no deformation under the action of any external agency such as force and moments. In other words relative positions of the modules of a rigid body are fixed in space. 19. Particle: Concept of particle is purely theoretical or imaginary. particle is said to have mass but requires no space to occupy. In other words, a particle is a point mass.

5 The concept of particle cannot be used if the shape and size of the body is influencing the motion. Eg: i) otion of a swimmer. ii) otion of a body along a curved path. 20. Continuum: The concept of continuum is purely theoretical or imaginary. Continuum is said to be made up of infinite number of molecules packed in such a way that, there is no gap between the molecules so that property functions remain same at all the points. 21. Point force: The concept of point force in purely theoretical or imaginary, here the force is assumed to be acting at a point or over infinity small area. 22. Principle physical independence of forces: 1 a 1 1 a a 1, a a 1, a 2 ction of forces on bodies are independent, in other words the action of forces on a body is not influenced by the action of any other force on the body. 23. Principle of superposition of forces:

6 1 a 1 1 a (a 1 +a 2 ) 1 2 (a 2 +a 1 ) Net effect of forces applied in any sequence on a body is given by the algebraic sum of effect of individual forces on the body. 24. Principle of transmissibility of forces: a = Line of action Rigid body a = Rigid body B Line of action The point of application of a force on a rigid body can be changed along the same line of action maintaining the same magnitude and direction without affecting the effect of the force on the body. Limitation of principle of transmissibility: Principle of transmissibility can be used only for rigid bodies and cannot be used for deformable bodies. 25. ssumptions made in Engineering echanics i) ll bodies are rigid. ii) Particle concept can be used wherever applicable. iii) Principle of physical independence of forces is valid. iv) Principle of superposition of forces is valid. v) Principle of transmissibility of forces is valid.

7 SYSTE O ORCES group or set of forces is called system of forces. Types: 1. Coplanar force system: If the lines of action of forces forming the system lie in the same plane, then the system is said to be coplanar. 2. Non-coplanar forces: If the lines of action of forces forming the system do not lie in the same plane then the system is said to be non-coplanar. Note: Our study is restricted to coplanar forces. 3. Collinear force system: 1 B 2 3 C If the forces forming the system have common line of action then the system is said to be collinear.

8 4. Concurrent force system: x D 4 1 x 0 C 2 B O If the line of action of forces forming the system pass through a common point (point of concurrence) then the system is said to be concurrent. 5. Non-concurrent force system: 1 O If the lines of action of forces forming the system do not pass through a common point, then the system is said to be non-concurrent. 6. Parallel force system: Like Unlike It is a particular case of non-concurrent force system in which the line of action of forces forming the system are parallel. RESOLUTION O ORCE Y

9 Y Displacement Displacement The force is producing, simultaneous x displacement and y-displacement. The part of the force which is producing x displacement is called x component or horizontal component of the force (x). The part of the force which produces y displacement is called y component of the force or vertical component of force (y). The technique of finding a component of a force along any direction is called resolution of force. The component of a force along any direction is called the resolved component. The components of a force determined along two mutually perpendicular direction are called rectangular components. To resolve a force along any direction Y 2 y θ 1 O represents the force both in magnitude and direction θ is the acute angle mode by the force w.r.t. x direction. We have, O Cos θ = O x Cos θ = 1 x = Cos θ ( ) Sin θ = O 1 2 Sin θ = O

10 y Sin θ = y = Sin θ ( ) component of a force is given by the product of magnitude of the force and cosine of acute angle made by the force w.r.t. x-direction. Y component of a force is given by the product of magnitude of the force and sine of acute angle made by the force w.r.t. x-direction. Note: 1. Sign convention for the direction of components θ = 0 x = Cos O = y = Sin O = O The horizontal component or component of a force acting along x direction is the force itself. Whereas, its vertical component or y-component is zero. 3. x = Cos 90 = O y = Sin 90 = x component of a force acting along Y direction is zero. Whereas, its y component is equal to itself.

11 4. If a force is inclined at 45 o w.r.t. x axis or y axis then its x component will be equal to y component ( x = y ). Problems 1. ind and Y components of forces in the following cases. a) 100 kn x = Cos o = kn = kn ( ) y = Sin 30 = kn = kn ( ) b) x = + 20 Cos 70 = kn = kn ( ) 30 o 20 kn y = + 20 Sin 70 = kn = kn ( ) c) ethod-i 3 tan θ = o θ = N x = Cos o = 160 N = 160 N ( )

12 y = Cos o = 120 N = 120 N ( ) ethod-ii 4 Cos θ = = Sin θ = = x = Cos θ = 200 x 0.8 = 160 N = 160 N ( ) y = 200 Sin θ = 200 x 0.6 = 120 N = 120 N ( )

13 RESULTNT ORCE O SYSTE O ORCES 1 α α R The resultant of a system of forces is a single calculated force which is capable of producing the same effect as that of system of forces on the body. It is the vector sum of forces of the system. COPOSITION O ORCES The technique of finding the resultant of forces is called composition of forces. OENT O ORCE It is the capacity of a force to produce rotator motion. In other words moment of a force is its rotating capacity. into oment of about in clockwise Based on the direction of rotation produced moment of a force can be classified a) Clockwise moment oment of about in anticlockwise b) nticlockwise moment / counter clockwise moment.

14 Calculation of oment of a orce about a Point oment of a force about any point is given by the product of magnitude of force and perpendicular distance between the line of action of a force and the point about which moment is considered. = L Unit: Nm Sign Convention for oment of a orce Clockwise moment positive and anticlockwise moment negative. 1) ind moment of force about in the following cases. a) = 10 kn 2 m b) = 10 kn 2 m c) 3 m = 20 kn

15 d) 2 m 50 kn 2) ind moment of the force about and B in the following a) 2 m = 10 kn 4 m B b) B = 20 kn 3 m 4 m