B_-_Scalar_and_Vector - Force Vectors Scalars and Vectors...

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Unformatted text preview: Force Vectors Scalars and Vectors LEARNING OBJECTIVES Be able to differentiate between Scalar and Vector quantities. Be able to perform vector operations. Be able to resolve forces into their respective components. PRE-REQUISITE KNOWLEDGE Units of measurements. Sine and Cosine Rules. Definitions of sine, cosine and tangent. Trigonometry concepts DEFINITIONS A Scalar is a quantity that has only magnitude such as distance, length, mass, volume, age. A vector is a quantity that has both magnitude and direction such as position, force, moment. The vector is denoted by a capital letter (F or B) while the magnitude is denoted by the same capital letter but in italic (F or B). QUANTITY DEFINITIONS You want to travel from East Lansing to Grand Rapids. What pieces of information do you need to do so? VECTOR OPERATIONS Multiplication and Division The product of vector A and scalar a is aA. The magnitude is aA and the sense (direction) is that of A if a is positive, and it is opposite to A if a is negative. A 2A 0.5 A -2 A VECTOR OPERATIONS Vector Addition Vectors A and B can be added using the parallelogram law or triangle construction (head to tail) A A B B R=A+B R=A+B Triangle construction Parallelogram Law VECTOR OPERATIONS Addition of Collinear Vectors R = A+B A B VECTOR OPERATIONS Vector Subtraction To subtract vector B from vector A, reverse the direction of B and add it to A. A R' B A -B R' = A + (-B) = A-B VECTOR OPERATIONS Vector Resolution A vector may be resolved into two components having known lines of action using the parallelogram law. A R B VECTOR OPERATIONS Example Vector Resolution Determine the resultant vector and its direction VECTOR OPERATIONS a = 90 (10 + 15) = 65o Solution R b = d = 180 65 = 115o F2 =150 N b 10o a c d 15o F1 = 100N R = 150 2 + 100 2 - 2(150)(100 ) cos(115) = 212.6 N 150 212.6 -1 150 sin (115) = ; c = sin = 39.8o sin ( c ) sin (115) 212.6 Example 2 Determine the magnitudes of FA and FB for =15o y VECTOR OPERATIONS R = 10kN x VECTOR OPERATIONS Solution 2 B F , F = 10 kN 15o FA A 30o C FB D = 15o = 180o (30 + 15) = 135o VECTOR OPERATIONS Solution 2 B FR = 10 kN 15o = 15o = 180o (30 + 15) = 135o FA A 30o C FB D FA/sin() = FR/sin() = FB/sin(30o) FA = 3.66 kN and FB = 7.07 kN Example 3 Determine to minimize the magnitude of FB y VECTOR OPERATIONS R = 10kN x VECTOR OPERATIONS Solution 3 FA A FB 30o 30o 150- FR=10 kN Use the sin law as follows: F /sin30 = 10/sin(150- ), To minimize FB, differentiate FB relative to and set it equal to zero. dFB = 10 sin ( 30o ) cos(150o - ) = 0 cos 150o - = 0 = 60 o FB = 5 kN ( ) d sin 2 (150o - ) and FA = 8.66 kN ...
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