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## Engineering Mechanics: Statics

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**Engineering Mechanics: Statics**Chapter 2: Force Systems**ForceSystems**Part A: Two Dimensional Force Systems**Force**An action of one body on another Vector quantity External and Internal forces Mechanics of Rigid bodies: Principle of Transmissibility Specify magnitude, direction, line of action No need to specify point of application Concurrent forces Lines of action intersect at a point**Vector Components**• A vector can be resolved into several vector components • Vector sum of the components must equal the original vector • Do not confused vector components with perpendicular projections**Rectangular Components**• 2D force systems • Most common 2D resolution of a force vector • Express in terms of unit vectors , y q x Scalar components – can be positive and negative**Rectangular components are convenient for finding the sum or**resultant of two (or more) forces which are concurrent Actual problems do not come with reference axes. Choose the most convenient one! 2D Force Systems**Example 2.1**• The link is subjected to two forces F1 and F2. Determine the magnitude and direction of the resultant force. Solution**Example 2/1 (p. 29)**Determine the x and y scalar components of each of the three forces**y**x Rectangular components • Unit vectors • = Unit vector in direction of qy qx**Problem 2/4**• The line of action of the 34-kN force runs through the points A and B as shown in the figure. • (a) Determine the x and y scalar component of F. • (b) Write F in vector form.**Moment**• In addition to tendency to move a body in the direction of its application, a force tends to rotate a body about an axis. • The axis is any line which neither intersects nor is parallel to the line of action • This rotational tendency is known as the momentM of the force • Proportional to force F and the perpendicular distance from the axis to the line of action of the force d • The magnitude of Mis M = Fd**Moment**• The moment is a vector M perpendicular to the plane of the body. • Sense of M is determined by the right-hand rule • Direction of the thumb = arrowhead • Fingers curled in the direction of the rotational tendency • In a given plane (2D),we may speak of moment about a point which means moment with respect to an axis normal to the plane and passing through the point. • +, - signs are used for moment directions – must be consistent throughout the problem!**Moment**• A vector approach for moment calculations is proper for 3D problems. • Moment of F about point A maybe represented by the cross-product where r = a position vector from point A to any point on the line of action of F M = r x F M = Fr sin a = Fd**Example 2/5 (p. 40)**Calculate the magnitude of the moment about the base point O of the 600-N force by using both scalar and vector approaches.**Problem 2/43**(a) Calculate the moment of the 90-N force about point O for the condition q = 15º. (b) Determine the value of q for which the moment about O is (b.1) zero (b.2) a maximum**Couple**• Moment produced by two equal, opposite, and noncollinear forces = couple • Moment of a couple has the same value for all moment center • Vector approach • Couple M is a free vector M = F(a+d) – Fa = Fd M = rA x F + rB x (-F) = (rA-rB) x F = r x F**Couple**• Equivalent couples • Change of values F and d • Force in different directions but parallel plane • Product Fd remains the same**Force-Couple Systems**• Replacement of a force by a force and a couple • Force F is replaced by a parallel forceFand a counterclockwise couple Fd • Example Replace the force by an equivalent system at point O • Also, reverse the problem by the replacement of a force and a couple by a single force**Problem 2/67**The wrench is subjected to the 200-N force and the force P as shown. If the equivalent of the two forces is a for R at O and a couple expressed as the vector M = 20 kN.m, determine the vector expressions for P and R**Resultants**• The simplest force combination which can replace the original forces without changing the external effect on the rigid body • Resultant = a force-couple system**Resultants**• Choose a reference point (point O) and move all forces to that point • Add all forces at O to form the resultant force R and add all moment to form the resultant couple MO • Find the line of action of R by requiring R to have a moment of MO**Problem 2/79**Replace the three forces acting on the bent pipe by a single equivalent force R. Specify the distance x from point O to the point on the x-axis through which the line of action of R passes.**ForceSystems**Part B: Three Dimensional Force Systems**Three-Dimensional Force System**• Rectangular components in 3D • Express in terms of unit vectors , , • cosqx, cosqy , cosqz are the direction cosines • cosqx = l, cosqy = m, cosqz= n**Three-Dimensional Force System**• Rectangular components in 3D • If the coordinates of points A and B on the line of action are known, • If two angles q and f which orient the line of action of the force are known,**Problem 2/98**• The cable exerts a tension of 2 kN on the fixed bracket at A. Write the vector expression for the tension T.**Three-Dimensional Force System**• Dot product • Orthogonal projection of Fcosa of F in the direction of Q • Orthogonal projection of Qcosa of Q in the direction of F • We can express Fx = Fcosqx of the force F as Fx = • If the projection of F in the n-direction is**Example**• Find the projection of T along the line OA**Moment and Couple**• Moment of force F about the axis through point O is • r runs from O to any point on the line of action of F • Point O and force F establish a plane A • The vector Mo is normal to the plane in the direction established by the right-hand rule • Evaluating the cross product MO = r x F**Moment and Couple**• Moment about an arbitrary axis known astriple scalar product (see appendix C/7) • The triple scalar product may be represented by the determinant where l, m, n are the direction cosines of the unit vector n**Sample Problem 2/10**A tension T of magniture 10 kN is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment Mz of T about the z-axis passing through the base O.**Resultants**• A force system can be reduced to a resultant force and a resultant couple**Wrench Resultants**• Any general force systems can be represented by a wrench**Problem 2/143**• Replace the two forces and single couple by an equivalent force-couple system at point A • Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts**Resultants**• Special cases • Concurrent forces – no moments about point of concurrency • Coplanar forces – 2D • Parallel forces (not in the same plane) – magnitude of resultant = algebraic sum of the forces • Wrench resultant – resultant couple M is parallel to the resultant force R • Example of positive wrench = screw driver**Problem 2/142**• Replace the resultant of the force system acting on the pipe assembly by a single force R at A and a couple M • Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts