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

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Unformatted text preview: Force Vectors Force Scalars and Vectors LEARNING OBJECTIVES LEARNING Be able to differentiate between Scalar Be and Vector quantities. and Be able to perform vector operations. Be able to resolve forces into their Be respective components. respective PRE-REQUISITE KNOWLEDGE PRE-REQUISITE Units of measurements. Sine and Cosine Rules. Definitions of sine, cosine and tangent. Trigonometry concepts DEFINITIONS DEFINITIONS A Scalar is a quantity that has only Scalar magnitude such as distance, length, mass, volume, age. mass, A vector is a quantity that has both vector 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 same B). QUANTITY DEFINITIONS QUANTITY You want to travel from East Lansing to Grand You Rapids. What pieces of information do you need to do so? to VECTOR OPERATIONS VECTOR Multiplication and Division The product of vector A and scalar a is aA. The A. The magnitude is aA and the sense (direction) and 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 Vector Addition Vectors A and B can be added using the parallelogram Vectors law or triangle construction (head to tail) law A A B B R=A+B R=A+B Triangle construction Parallelogram Law VECTOR OPERATIONS VECTOR Addition of Collinear Vectors R = A+B A B VECTOR OPERATIONS VECTOR Vector Subtraction To subtract vector B from vector A, reverse To the direction of B and add it to A. the A R’ B A -B R’ = A + (-B) = A-B VECTOR OPERATIONS VECTOR Vector Resolution A vector may be resolved into two components vector having known lines of action using the parallelogram law. parallelogram A R B VECTOR OPERATIONS VECTOR Example – Vector Resolution Determine the resultant vector and its direction VECTOR OPERATIONS VECTOR a = 90 – (10 + 15) = 65o Solution R b = d = 180 – 65 = 115o F2 =150 N b 10o a c d 15o F1 = 100N 150 212.6 −1 150 sin (115) = ; c = sin = 39.8o sin ( c ) sin (115) 212.6 R = 150 2 + 100 2 − 2(150) (100 ) cos(115) = 212.6 N Example 2 Determine the magnitudes of FA and FB for θ =15o y VECTOR OPERATIONS VECTOR R = 10kN x VECTOR OPERATIONS VECTOR Solution 2 B β F , F = 10 kN 15o α FA A 30o C FB D α = 15o β = 180o – (30 + 15) = 135o VECTOR OPERATIONS VECTOR Solution 2 B β FR = 10 kN 15o α = 15o β = 180o – (30 + 15) = 135o α FA A 30o C FB D FA/sin(α) = FR/sin(β) = FB/sin(30o) FA = 3.66 kN and FB = 7.07 kN Example 3 Determine θ to minimize the magnitude of FB Determine y VECTOR OPERATIONS VECTOR R = 10kN x VECTOR OPERATIONS VECTOR Solution 3 FA A FB 30o θ 30o 150-θ FR=10 kN Use the sin law as follows: F /sin30 = 10/sin(150-θ ), To minimize FB, differentiate FB relative to θ and set it equal to zero. dFB = 10 sin ( 30o ) cos(150o − θ ) = 0 cos 150o − θ = 0 ⇒ θ = 60 o ⇒ FB = 5 kN ( ) dθ sin 2 (150o − θ ) and FA = 8.66 kN VECTOR OPERATIONS VECTOR Questions 1. Which one of the following is a scalar quantity? a) Force b) Position c) Mass b) 2. Which law do you use in vector addition? b) Arithmetic d) Newton’s Second law Ne d) Velocity a) The triangular law c) Pascal’s law VECTOR OPERATIONS VECTOR What is the magnitude of the resultant force F? 5 = lb 0 40 o F=? 60o F 1 F2 = 80 lb A) F = 132.3 lb B) F = 113.6 lb C) F = 120 lb ...
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