Force Vectors and Vector Operations in Engineering Mechanics Statics, Lecture notes of Mechanics

Download Force Vectors and Vector Operations in Engineering Mechanics Statics and more Lecture notes Mechanics in PDF only on Docsity! CHAPTER-2 Force Vectors Book: "Engineering Mechanics Statics, R. C. Hibbeler, 12 Ed FORCE VECTORS Scalars & Vectors Vector Operation Vector Addition of Forces Addition of a System of Coplanar Forces Cartesian Vectors Addition of Cartesian Vectors Position Vectors Force Vector Directed along a Line Dot Product VECTOR OPERATIONS * Vector Addition (Parallelogram Law): — All vector quantities obey the parallelogram law of addition — The two “component” vectors A and B are added to form a “resultant” vector R = A + B using the parallelogram law as follows: ¢ First join the tails of the components at a point so that it makes them concurrent * From the head of B, draw a line parallel to A. Draw another line from the head of A that is parallel to B. These two lines intersect at point P to form the adjacent sides of a parallelogram * The diagonal of this parallelogram that extends to P forms R, which then represents the resultant vectorR=A+B B L B R=A+B Parallelogram law VECTOR OPERATIONS * Vector Addition (Triangle Rule): Triangle rule is a special case of the parallelogram law Vector B is added to vector A in a “head-to-tail” fashion, i.e., by connecting the head of A to the tail of B. The resultant R extends from the tail of A to the head of B. In a similar manner, R can also be obtained by adding A to B Vector addition is commutative, i.e., the vectors can be added in either order, R= A+B=B +A If the two vectors A and B are collinear, i.e., both have the same line of action, the parallelogram law reduces to an algebraic or scalar addition R= A+B ZO 2O™ R MN, R=A+B Triangle rule B A R=B+A Triangle rule R=A+B Addition of collinear vectors VECTOR OPERATIONS * Vector Subtraction: — The resultant of the difference between two vectors A and B of the same type may be expressed as: R’ =A-B=A+(-B) — Subtraction is therefore defined as a special case of addition, so the rules of vector addition also apply to vector subtraction. Parallelogram law Triangle construction Vector subtraction VECTOR ADDITION OF FORCES ¢ Addition of Several Forces: — For more than two forces to be added, successive applications of the parallelogram law can be carried out in order to obtain the resultant force — If three forces F,, F2, F3 act at a point O, the resultant of any two of the forces is found, say, F, + F, — Then this resultant is added to the third force, yielding the resultant of all three forces; i.e., Fa = (F, + F2)+F3 VECTOR ADDITION OF FORCES Examples: 2.1, 2.2, 2.3, 2.4 Fundamental Problems: F2-1, F2-5, F2-6 Practice Problems: 2-1, 2-13, 2-16, 2-20, 2-28, 2-31 EXAMPLE 2-1 The screw eye in figure is subjected to two forces, F, and F,. Determine the magnitude and direction of the resultant force. 10° Fy) = 150N PROBLEM 2-31 Three cables pull on the pipe such that they 600 Ib create a resultant force having a magnitude of 900 lb. If two of the cables are subjected to known forces, as shown in the figure, determine the angle 6 of the third cable so that the magnitude of force F in this cable is a minimum. All forces lie in the x—y plane. What is the magnitude of F? ADDITION OF A SYSTEM OF COPLANAR FORCES * When a force is resolved into two components along the x and y axes, the components are then called rectangular components * For analytical work we can represent these components in one of two ways, using either Scalar Notation or Cartesian Vector Notation ADDITION OF A SYSTEM OF COPLANAR FORCES Scalar Notation: Rectangular components of force F are found using the Parallelogram Law so that F=F, + F, Because these components form a right triangle, their magnitudes can be determined from: F, = F Cos 8 and F, = F Sin 8 Instead of using the angle 0, however, the direction of F can also be defined using a small “slope” triangle Since this triangle and the larger shaded triangle are similar, the proportional length of the sides gives: a 2 F, b Foe Fre ee) = -e(?) The y component is a negative scalar since F, is directed along the negative y axis ADDITION OF A SYSTEM OF COPLANAR FORCES * Coplanar Force Resultants: — Components of the resultant force of any number of Coplanar Forces can be represented symbolically by the algebraic sum of the x and y components of all the forces y Fp, = DF, Fry = LFy — Once these components are determined, they may be sketched along the x and y axes with their proper sense of direction, and the resultant force can be determined from vector addition Fr= VFR, + Fry Fry Fes 6 = tan! ADDITION OF A SYSTEM OF COPLANAR FORCES Examples: 2.5, 2.6, 2.7 Fundamental Problems: F2-7, F2-9, F2-11 Practice Problems: 2-33, 2-37, 2-43, 2-49, 2-51 EXAMPLE 2-7 The end of the boom O in figured is subjected to three concurrent and coplanar forces. Determine the magnitude and direction of the resultant force. CARTESIAN VECTORS * Operations of vector algebra in Three Dimensions are greatly simplified if the vectors are first represented in Cartesian Vector Form * Right-handed Coordinate System: — A rectangular coordinate system is said to be Right- handed if: * Thumb of the Right Hand points in the direction of the positive z-axis and * when the Right-hand Fingers are curled about this = axis (i.e. z) and directed from the positive x towards the positive y-axis CARTESIAN VECTORS : * Rectangular Components of a Vector: — A vector A may have 1, 2, or 3 rectangular components along the xX, y, Z coordinate axes, depending on it’s orientation in the space — When A is directed within an octant of the x, y, z frame then by two successive applications of the parallelogram law: A’ =A, +A, A=A’+A, A=A,+A,+A, * Cartesian Unit Vectors: — In 3-dimensions, the set of Cartesian Unit Vectors, i, j, k, is used to designate the directions of the x, y, z axes respectively CARTESIAN VECTORS * Cartesian Vector Representation: — Three components of A act in the positive i, j, and k directions [A= 4i+ 4,i+ Ax] * Magnitude of a Cartesian Vector: A' = VA + Aj A= VA"? + A?. A=VA +A + A ADDITION OF CARTESIAN VECTORS Addition (or Subtraction) of two or more vectors are greatly simplified if the vectors are expressed in terms of their Cartesian components eg. A=A,ji+Ajt+ Ak B=B,i+ B,j + Bk Then the resultant vector, R, has components which are the scalar sums of the i, j, k components of A and B, i-e., R=A+B=(A,+B,)i+ (Ay + By)j + (A, + Bk If this is generalized and applied to a system of several concurrent forces, then the force resultant is the vector sum of all the forces in the system and can be written as: F, = LF = XFit+ LF, j + DEK Here XF,, XF,, and XF, represent the algebraic sums of the respective x, y, z ori, j, k components of each force in the system CARTESIAN VECTORS Examples: 2.8, 2.9, 2.10, 2.11 Fundamental Problems: F2-13, F2-17 Practice Problems: 2-59, 2-63, 2-69, 2-73, 2-80, 2-83 EXAMPLE 2-9 Determine the magnitude and the coordinate direction angles of the resultant force acting on the ring. F, = {501 — 100j + 100k} Ib | F, = (60j + 80k} Ib PROBLEM 2-83 Three forces act on the ring. If the resultant force Fp has a magnitude and direction as shown, determine the magnitude and the coordinate direction angles of force F3. & POSITION VECTORS * Position Vector “r” is a fixed vector which locates a point in space relative to another point * Ifr extends from the origin of coordinates, O, to point P(x, y, z), r can be expressed in Cartesian vector form as: r=xit yj+zk * Starting at the origin O, one “travels” x in the +i direction, then y in the +j direction, and finally z in the +k direction to arrive at point P(x, y, z) or POSITION VECTORS For a more general case, position vector r may be directed from point A to point B in space — rcan also be designated as rap, to indicate from and to the point where it is directed By the Head-to-tail vector addition, using the triangle rule: rmnt+r=rz Solving for r and expressing r, and rg in Cartesian vector form yields: r=rg- Vy = (Xgit+ yp jt Ze k) - (Xaity,jt+z,.k) A(X 4, ¥ 4:74) Z B(xp, Ya, Zp) EXAMPLE 2-12 An elastic rubber band is attached to points A and B as shown in figure. Determine its length and its direction measured from A toward B. EXAMPLE 2-13 The man shown pulls on the cord with force of 70 lb. Represent this force acting on the support A as a Cartesian vector and determine its direction. PROBLEM 2-91 Determine the magnitude and coordinate direction angles of the resultant force acting at A. DOT PRODUCT * Introduction: — In statics, one has to find the angle between two lines or the components of a force parallel and perpendicular to a line — In two dimensions, these problems can readily be solved by trigonometry since the geometry is easy to visualize — In three dimensions, however, this is often difficult, and consequently vector methods should be employed for the solution — The dot product, defines a particular method for “multiplying” two vectors, and can be used to solve the above-mentioned problems DOT PRODUCT The dot product of vectors A and B, written A . B, and read “A dot B” is defined as the product of the magnitudes of A and B and the cosine of the angle 0 between their tails. Expressed in equation form, A:'B = ABcosé The dot product is often referred to as the scalar product of vectors since the result is a scalar and not a vector Laws of Operation: — Commutative Law: A.B=B.A — Multiplication by a scalar: a(A.B) = (aA).B = A.(aB) — Distributive Law: A.(B+D) = (A.B) + (A.D) DOT PRODUCT * Cartesian Vector Formulation — If we want to find the dot product of two general vectors A and B that are expressed in Cartesian vector form, then we have: A-B = (A,i + A,j + A,k)- (Bi + Byj + Bk) = A,B .i-i) + A,By(i *j) + A,BAi *k) + AyBAj “i) + (AyB(j-j) + AyBAj *k) + A,B,(k+i) + A,B,(k-j) + A,B(k:k) — Carrying out the dot-product operations, the final result becomes: A+B = A,B, + A\B, + A;B; — Thus, to determine the dot product of two Cartesian vectors, multiply their corresponding x, y, z components and sum these products algebraically — The result will be either a positive or negative scalar DOT PRODUCT - Applications * The components of a vector parallel and perpendicular to a line: — The component of A that is perpendicular to line aa can also be obtained — Since A=A,+A.,then Ai =A-A, — There are two possible ways of obtaining A * One way would be to determine 8 from the dot product, 8 = Cos-1 (A.u,/A) * Then A.=A Sin 8 * Alternatively, if A, is known, then by Pythagorean’s theorem we can also write A:=v(A2-A,2) A, =A cos 6 U, DOT PRODUCT Examples: 2.16, 2.17, 2.18 Fundamental Problems: F2-26, F2-29 Practice Problems: 2-112, 2-115, 2-119, 2-123, 2-128, 2-132 EXAMPLE 2-17 The frame is subjected to a horizontal force F = {300j}. Determine the magnitude of the components of this force parallel and perpendicular to member AB. (a) (b) PROBLEM 2-115 Determine the magnitudes of the components of F = 600N acting along and perpendicular to segment DE of the pipe assembly. PROBLEM 2-132 Determine the magnitudes of the projected components of the force F = 300 N acting along line OA.