Vector is a quantity which has both magnitude and direction. We will use the arrow to designate vectors.

Transcription

1 In this section, we will study the fundamental operations (addition, resolving vectors into components) of force vectors. Vector is a quantity which has both magnitude and direction. We will use the arrow to designate vectors. We will use to designate the magnitude.

2 Assume that we have a particle subjected to a force, Head Line of action, Sense of the force

3 Mathematically, when a vector is multiplied or divided by a scalar, the new vector would have a magnitude of: ( a is a scalar): Now assume that we have two forces acting on our particle.

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6 Vector addition is commutative.

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8 Vector acting in the opposite direction of

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11 Resolution of a vector: Two or more vectors acting on a particle may be replaced by a single force which has the same effect on the particle. Conversely a single force acting on a particle may be replaced with two or more forces which have the same effect on the particle. These forces are called the components of the original force and the process is called the vector resolution. Apparently there are infinite number of possibilities.

12 Two cases are especially interesting: CHAPTER 1: FORCE VECTORS Magnitude and direction of Q can be found graphically or using trigonometry.

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14 Important Math Relations: CHAPTER 1: FORCE VECTORS

15 Example: Beer and Jonhston, 7 th edition, 2.2 A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is 25 KN directed along the axis of the barge, determine The tension in each of the ropes knowing that α=45

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17 Example: CHAPTER 1: FORCE VECTORS Two forces are acting on point A If F 2 =80N, determine a) F 1 if the rod remains vertical b) Magnitude of the resultant force.

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20 x CHAPTER 1: FORCE VECTORS

21 Addition of forces by summing x and y components: When we have three or more forces, it is easier to resolve these forces into their x and y components and then add the respective scalar components.

22 Simply: This also works in 3D.

23 CARTESIAN VECTORS IN 3-D: FORCES IN SPACE We will use a right-handed (Cartesian) rectangular coordinate system z x y y When we curl our right hand from +x to +y the thumb points in +z direction. x Outward from the page. z

24 CONCEPT OF UNIT VECTOR For a vector A with a magnitude of A, an unit vector is defined as U A = A / A. Characteristics of a unit vector: a) Its magnitude is 1. b) It is dimensionless. c) It points in the same direction as the original vector (A). The unit vectors in the Cartesian axis system are i, j, and k. They are unit vectors along the positive x, y, and z axes respectively. CHAPTER 1: FORCE VECTORS

25 3-D CARTESIAN VECTOR TERMINOLOGY Consider a box with sides A X, A Y, and A Z meters long. The vector A can be defined as A = (A X i + A Y j + A Z k) m The projection of the vector A in the x-y plane is A. The magnitude of this projection, A, is found by using the same approach as a 2-D vector: A = (A X 2 + A Y2 ) 1/2. The magnitude of the position vector A can now be obtained as A = ((A ) 2 + A Z2 ) ½ = (A X 2 + A Y 2 + A Z2 ) ½

26 TERMS (continued) The direction or orientation of vector A is defined by the angles,, and. These angles are measured between the vector and the positive X, Y and Z axes, respectively. Their range of values are from 0 to 180 CHAPTER 1: FORCE VECTORS Using trigonometry, direction cosines are found using the formulas These angles are not independent. They must satisfy the following equation. cos ² + cos ² + cos ² = 1 This result can be derived from the definition of a coordinate direction angles and the unit vector. Recall, the formula for finding the unit vector of any position vector: or written another way, u A = cos i + cos j + cos k.

27 ADDITION/SUBTRACTION OF VECTORS Once individual vectors are written in Cartesian form, it is easy to add or subtract them. The process is essentially the same as when 2-D vectors are added. For example, if A = A X i + A Y j + A Z k B = B X i + B Y j + B Z k, and then A + B = (A X + B X ) i + (A Y + B Y ) j + (A Z + B Z ) k or A B = (A X - B X ) i + (A Y - B Y ) j + (A Z - B Z ) k.

28 Example Problem: The screw eye shown in the picture is subjected to two forces. Find the magnitude and the coordinate direction angles of the resultant force.

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30 POSITION VECTOR AND FORCE DIRECTED ALONG A LINEA We start with the definition of a position vector. A position vector locates a point in space relative to another point. The definition of a position vector: In order to reach point B, one must have:

31 More generally if A is not at origin: CHAPTER 1: FORCE VECTORS

32 FORCE VECTOR DIRECTED ALONG A LINEA CHAPTER 1: FORCE VECTORS

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34 Two forces are acting on a pipe as shown in the figure. Find the magnitude and the coordinated direction angles of the resultant force.

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