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Ashley Abid Nicole Bogdan Vectors

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Vectors and Scalars A vector quantity is a quantity that is fully described by both magnitude and direction. Scalars are quantities that are fully described by a magnitude (or numerical value) alone. Examples of VectorsExamples of Scalars Displacement (distance in a direction) Distance (m) Velocity (distance over time)Temperature (C or F) Acceleration (velocity over time)Energy (J) Force (newtons * acceleration)Time (s)

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Drawing Vectors All vectors can be represented as arrows. Tail Head

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Magnitude of Vectors in One Dimension Vectors acting in the same direction produce the greatest magnitude force Vectors acting in opposite directions produce the smallest magnitude force At 0 degrees, magnitude is greatest At 180 degrees, magnitude is the smallest

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What is displacement? Displacement is an object's overall change in position. It takes direction into account. If a person walks around the perimeter of the diagram, the total distance traveled would be 4m + 2 m + 4 m + 2 m = 12m However, the total displacement is calculated as 4 m East + 2 m South + 4 m West + 2 m North = 0 m. East and West cancel one another out, just like North and South.

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Vector Fundamentals in Two Dimensions Vectors can be added together to form a resultant vector. The vectors added together are called component vectors. They are represented with compass directions on the x and y axis. Resultant --> <-- components

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Adding Vectors and Calculating Resultants The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together. There are two methods to calculate resultants: Head to Tail Tail to Tail (Parallelogram Method)

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Head to Tail Method 1. Place the two vectors next to each other so that the head of one vector is touching the tail of the other vector. 2. Draw the resultant vector by connecting the remaining head and tail. V1 + V2 = R V1 V2 Resultant Vector (R)

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Head to Tail Method Often, vectors must be rearranged for the head to tail method. The angles must remain the same.

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The Pythagorean Theorem If two vectors are perpendicular to each other, you can solve for their resultant using the

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Parallelogram Method 1. Draw the two components 2. Extend parallel lines to each of with their tails touching. the components so that their lines meet 3. The resultant is the diagonal extended from one corner to the next. V1 V2 R

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Trigonometry Review for Parallelogram Method The hypotenuse represents the resultant force. The adjacent and opposite represent the components.

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Finding Horizontal and Vertical Components to a Vector The vertical and horizontal components make a triangle and so we can use sine and cosine to calculate a missing component. The formulas R x =R cosθ and R y = R sinθ are used. Vertical Component R y Horizontal Component R x θ

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Finding an Equilibrant Equilibrium is any situation where the net force acting on an object is zero. It is called equilibrium because all the forces acting on the object equal out and cancel each other. This third force that would do the cancelling out is called the equilibrant. The equilibrant is a vector that is the exact same size as the resultant would be, but the equilibrant points in exactly the opposite direction. For this reason, an equilibrant touches the other vectors head-to-tail like any other vector being added.

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Drawing Equilibrants The equilibrant is the exact same size as the resultant would be, but the equilibrant points in exactly the opposite direction

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(short animation)

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Review Question #1

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Review Question #1 Solved R y = R sinθ Ry = 300 N * sin(60)= (2) 260 N

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Review Question #2

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Review Question #2 Solved (3) Use the parallelogram method to find the missing component.

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Review Question #3

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Review Question #3 Solved Resultant: (F1^2 + F2^2) = R^2 R = 14 N North East Equilibrant is equal in magnitude but opposite in direction Equilibrant = 14 N South West (1) R = 14 N

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Review Question #4

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Review Question #4 Solved (1) decreases At 0 degrees, both objects work in the same direction. Their magnitude is added. At 90 degrees, the pythagorean theorem is used 6 N ^2 + 8 N ^2 = 10 N ^2 14 N > 10 N. 6 N + 8 N 14 N

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Review Question #5

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Question #5 Solved (1) Law of Triangles The sum of any two sides of a triangle cannot be smaller than the third side 1 N + 3 N < 5 N

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