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Vector Addition

Graphical Analytical Component Method. Vector Addition. What does an ordered pair mean in math? Ex:(2,3). Math sTUFF. Quantities having both magnitude and direction Magnitude: How much (think of it as the length of the line) Direction: Which way is it pointing?

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Vector Addition

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  1. Graphical Analytical Component Method Vector Addition

  2. What does an ordered pair mean in math? Ex:(2,3) Math sTUFF

  3. Quantities having both magnitude and direction • Magnitude: How much (think of it as the length of the line) • Direction: Which way is it pointing? • Can be represented by an arrow-tipped line segment • Examples: • Velocity • Acceleration • Displacement • Force Vectors

  4. Compare the two vectors. What makes them different Question:

  5. Direction The magnitude or length is exactly the same Answer

  6. Two or more vectors acting on the same point are said to be concurrent vectors. • The sum of 2 or more vectors is called the resultant (R). • A single vector that can replace concurrent vectors • Any vector can be described as having both x and y components in a coordinate system. • The process of breaking a single vector into its x and y components is called vector resolution. Vector Terminology

  7. Vectors are said to be in equilibrium if their sum is equal to zero. • A single vector that can be added to others to produce equilibrium is call the equilibrant (E). • Equal to the resultant in magnitude but opposite in direction. More Vector Terminology E + R = 0 E = - R E = 5 N R = 5 N at 180 ° at 0°

  8. E= 10 N at 0 degrees R = 20 N at 0 degrees What is the resultant of the following vectors?

  9. 30 N at 0 degrees Answer

  10. 20 N at 45 degrees 10 N at 225 degrees Do not get freaked out by the angles, Think about it for a second. Question

  11. 10 N at 45 degrees Answer

  12. Vectors are drawn to scale and the resultant is determined using a ruler and protractor. • Vectors are added by drawing the tail of the second vector at the head of the first (tip to tail method). • The order of addition does not matter. • The resultant is always drawn from the tail of the first to the head of the last vector. Using the Graphical Method of Vector Addition:

  13. Example Problem A 50 N force at 0° acts concurrently with a 20 N force at 90°. R   R and  are equal on each diagram.

  14. Adding vectors!

  15. Question: Add these two vectors together a b

  16. Answer b a R= a+b

  17. Perpendicular vectors act independently of one another. In problems requesting information about motion in a certain direction, choose the vector with the same direction. Motion Applications

  18. A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s. • What is the resultant velocity of the boat? Example Problem:Motion in 2 Dimensions

  19. A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s. 5.00 m/s N 8.00 m/s E River width

  20. What is the resultant velocity of the boat? Draw to scale and measure. 5.00 m/s N 8.00 m/s E R = 9.43 m/s at 32°

  21. Advantages and Disadvantages of the Graphical Method • Can add any number of vectors at once • Uses simple tools • No mathematical equations needed • Must be correctly draw to scale and at appropriate angles • Subject to human error • Time consuming

  22. A rough sketch of the vectors is drawn. • The resultant is determined using: • Algebra • Trigonometry • Geometry Solving Vectors Using the Analytical Method

  23. Quick Review Right Triangle c is the hypotenuse B c2 = a2 + b2 c sin = o/h cos = a/h tan = o/a a A + B + C = 180° B = 180° – (A + 90°) C A b tan A = a/b tan B = b/a

  24. These Laws Work for Any Triangle. A + B + C = 180° C Law of sines: a = b = c sin A sin B sin C b a B A c Law of cosines: c2 = a2 + b2 –2abCos C

  25. Use the Law of: • Sines when you know: • 2 angles and an opposite side • 2 sides and an opposite angle • Cosines when you know: • 2 sides and the angle between them

  26. Draw a tip to tail sketch first. • To determine the magnitude of the resultant • Use the Pythagorean theorem. • To determine the direction • Use the tangent function. For right triangles:

  27. Find the resultant for the first two vectors. Add the resultant to vector 3 and find the new resultant. Repeat as necessary. To add more than two vectors:

  28. Advantages and Disadvantagesof the Analytical Method • Does not require drawing to scale. • More precise answers are calculated. • Works for any type of triangle if appropriate laws are used. • Can only add 2 vectors at a time. • Must know many mathematical formulas. • Can be quite time consuming.

  29. Each vector is replaced by 2 perpendicular vectors called components. Add the x-components and the y-components to find the x- and y-components of the resultant. Use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant. Solving Vector Problems using the Component Method

  30. Vector Resolution y = h sin  x = h cos  h y  x • + ++ • - +-

  31. Components of Force: y x

  32. What are the components of the following force 25N @ 12 degrees North of West Question

  33. West is 180 degrees to 12 degrees north of west is 168 degrees The X component is -24.45N The Y component is 5.20N You can confirm you answer –X and +Y would be found in the second quadrant on a graph so this answer makes sense Answer

  34. Example: 6 N at 135° 5 N at 30° R = (0.09)2 + (6.74)2 = 6.74 N  = arctan 6.74/0.09 = 89.2°

  35. The tangent function has 2 places that it is not defined (you get an error on your calculator) • 90 degrees and 270 degrees • The x and y components tell you the angle range Tangent Function

  36. My X component was negative and my y component was negative as well. My calculator told me that my answer was 22 degrees. What is my true angle? Question: Critical THinking

  37. My evidence: • Negative X • Negative Y • We are in quadrant three(between 180 degrees and 270) • I got 22 degrees, so I must take 180+22 to get 202degree as my angle! Using my tangent rules Answer

  38. Solve the following problem using the component method. 10 km at 30 6 km at 120

  39. Adding Vectors To find the magnitude: pythagoreantheorum To find the direction: 1. Take into account if either X or y is + or – 2. Use any trig function SOH CAH TOA to find angle

  40. I get a positive x and a negative y component when I add them together. What degree range is my angle in? Critical Thinking Question 2

  41. X is positive so that can only mean either quadrant 1 or 4 Y is negative so that means you have to get quadrant 4 as your answer 270 to 360 degrees Answer

  42. Make sure that all angles are measured from the x axis (0 degrees) Report both the magnitude and the direction otherwise the vector is wrong! Keep track of signs, They give you a clue to where the angle of the vector actually is. Notes

  43. Adding two vectors Find the resultant magnitude: Find the resultant direction:

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