Engineering Mechanics Part I: Statics PDF

Engineering Mechanics Part I: Statics L5 Dr. El-Sayed Awad Sayed Ahmed [email protected] Force Analysis  Introduction The force has been defined as : action of one body on another body. It is a vector quantity and must be added accordin...

Engineering Mechanics Part I: Statics L5 Dr. El-Sayed Awad Sayed Ahmed [email protected] Force Analysis  Introduction The force has been defined as : action of one body on another body. It is a vector quantity and must be added according to the definition of vector addition. The complete specification of the action of a force must include magnitude, direction, and point of application, and in this case , the force is treated as a fixed vector. When only the resultant external effects of a force system are to be investigated, the force may be treated as a sliding vector. 1-2  The direction of a force The direction of a force is defined by the line of action and the sense of the force The line of action of a force Is the infinite straight line along which the force acts. It is characterized by the angle it forms with some fixed axis. 1-3 Principle of Transmissibility It states: “If a force acts at any point on a rigid body, it may also be considered to act at any other point on its line of action, provided this point is rigidly connected with the body.” 1-4  Effect of a Force on a body A force may produce (or tend to produce) the following effects in a body on which it acts: It may change the motion of a body. It may also tend to rotate the body about an axis. (chapter 5 ) It may retard the motion of a body. It may retard the forces, already acting on a body, thus bringing it to rest or in equilibrium. ( chapters 4 and 6 ). It may give rise to the internal stresses in the body on which it acts. 1-5 Systems of Forces System of Force consists of two or more forces act on a body and It may be: Coplanar forces: The forces, whose lines of action lie on the same plane. Collinear forces: The forces, whose lines of action lie on the same line. Concurrent forces: The forces, which meet at one point. Coplanar concurrent forces: The forces, which meet at one point and their lines of action also lie on the same plane. 1-6 Systems of Forces (Cont.) Coplanar non-concurrent forces: The forces, which do not meet at one point, but their lines of action lie on the same plane. Non-coplanar concurrent forces: The forces, which meet at one point, but their lines of action do not lie on the same plane. 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 1-7 External force When the force is exerted by a different object. Internal force: When one part of a given object is subjected to a force by another part of the same object. Distributed Force Any contact force that be applied over a large area is called a distributed force 1-8 Concentrated Force Any contact force that be applied over a very small area compared with the other dimensions of the body, is called a Concentrated Force 1-9  Types of Forces Gravitational Force w=mg In SI units : g = 9.81 m/s2 at sea level. In U.S customary units: g = 32.2 ft/s2 at sea level. The gravitational force (weight) of an object is represented by a vector from the center of mass of the object and its direction is vertically downward pointing to earth center. 1 - 10 Body Force The force is called a body force when acting on the volume of the object e.g. (gravitational force on an object) Surface Force The force is called a surface force when acting on the surface of the object. Surface force is generated through direct physical contact between two bodies 1 - 11  Contact Forces Are the forces that result from the contact between deferent objects.For example, you exert a contact force when you push on a wall. The surface of your hand exerts a force on the surface of the wall that can be represented by a vector F. The wall exerts an equal and opposite force - F on your hand.(action-reaction) Contact forces exerted on objects may : By the surface contact of other objects. By cables and ropes. By pulleys By springs. 1 - 12  1- Surface contact force and Friction Force Two surfaces in contact exert forces on each other that are equal in magnitude and opposite in direction. Each force F can be resolved into the normal force N and the tangential force friction force f. If the friction force is negligible in comparison to the normal force, the surfaces are said to be smooth,. Otherwise, they are rough, Fig. (c). 1 - 13 If the contacting surface is curved the normal force is perpendicular and the friction force is parallel to the plane tangent to the surface at their point of contact, respectively. 1 - 14 2-Ropes and cables If a rope (or a cable) is attached to a crane, it exerts a force –T on it, whose magnitude is equal to the tension T, in the cable, and the line of action of T is collinear with the cable. 1 - 15  3-Pulleys A pulley, is a wheel with a grooved rim.It can be used to change the direction of a rope or a cable. When a pulley turns freely and the rope or cable either is stationary or turns the pulley at a constant rate, the tension is approximately the same on both sides of the pulley. 1 - 16 4-Springs The springs are used to exert contact forces in mechanical devices. When the Un-stretched length of the spring is L0 is stretched to a length L < L0 ( it pulls on the object with a force F ,(Fig. c). The object exerts an equal and opposite force -F on the spring. F =s k where : s = L − L0 and k is called Stiffness costant 1 - 17 Composition and Resolution of Forces Forces must be treated as vector quantities Composition (Resultant) of Forces or Resolution of a Force into two or three components require applications of the vector operations. 1 - 18 Resultant Force of a Coplanar Force System. 1. Graphical method Parallelogram law of forces (two forces) Triangle law of forces. (two forces) Polygon law of forces. (more than two forces) 1 - 19  Composition of two Forces Graphical method R = F1 + F2 Analytical method R = F + F + 2F1F2cosθ 1 2 2 2 The angle α which the resultant force makes with one of the two forces (say F1 ). F2 sinθ  = tan ( −1 ) F1 + F2 cosθ 1 - 20  NB: The magnitude R and direction of the resultant is computed from the trigonometry of the Triangle of forces. 1 - 21 Resultant of more than two coplanar forces graphically Polygon law of forces 1 - 22  Resolution of a force into two given directions We can use the properties of the parallelogram to resolve a given force R into two components F1 and F2 in the two given directions OA &OB which make angles of measure θ1 , θ2 respectively, as shown in the figure. From the properties of parallelogram and applying sine rule on the force triangle OAC we get: 𝑭1 𝑭2 𝑹 = = 𝒔𝒊𝒏𝜽2 𝒔𝒊𝒏𝜽1 𝒔𝒊𝒏(𝜽1 + 𝜽 2 ) 1 - 23  Resolution method for the resultant force of a system of coplanar forces 1. Scalar Notation Resolve all the forces horizontally (x- direction) and find the algebraic sum of all the horizontal components, i.e., Rx = ∑Fx Resolve all the forces vertically (y-direction) and find the algebraic sum of all the vertical components i.e., Ry = ∑ Fy. The resultant force R= R +R 2 x 2 y The resultant R will be inclined at an angle α, with the horizontal, such that : Ry tan  = Rx 1 - 24 2. Vector Notation (I) 2-Dimensional Force System FR = F1 + F2 + F3 then, FR = (F1x + F2x + F3x) i + ( F1y + F2y + F3y) j FR = (∑Fx ) i + (∑Fy) j FR = FRx i + FRy j FRy FR = F + F and  = tan ( 2 Rx 2 Ry −1 ) FRx 1 - 25  (II) 3-Dimensional Force System FR = F1 + F2 + F3 FR = (∑Fx ) i + (∑Fy) j+ (∑Fz) k FR = FRx i + FRy j + FRz k Magnitude FR = F + F + Fx 2 y 2 z 2 Coordinate direction angles FRx F F  = cos −1  = cos −1 Ry  = cos −1 Rz FR FR FR 1 - 26 Cartesian Force Vector F by knowing uF and F 1 - 27  ( x 2 − x1 )i + ( y 2 − y 1 ) j + ( z 2 − z1 )k F = Fu F = F ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2 + ( z 2 − z1 ) 2 1 - 28  Solved Example 1. Determine the magnitude and the coordinate direction angles of the resultant force. 1 - 29 Solution 1 - 30 1 - 31 2. Determine the magnitude and coordinate direction angles of the resultant force. (problem 3.13) 1 - 32 Fy =-80 cos 30Osin40O 1 - 33 Solution 1 - 34  3. Determine the magnitude and coordinate direction angles of the resultant force. (problem 3.14) 1 - 36 1 - 37  4. Determine the magnitude and coordinate direction angles of the resultant force and sketch this vector on the coordinate system (problem 3.16) 1 - 38 1 - 39