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VECTORS

VECTORS. Do you remember the difference between a scalar and a vector?. Scalars are quantities which are fully described by a magnitude alone. Vectors are quantities which are fully described by both a magnitude and a direction. Vector Representation.

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VECTORS

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  1. VECTORS

  2. Do you remember the difference between a scalar and a vector? Scalars are quantities which are fully described by a magnitude alone. Vectors are quantities which are fully described by both a magnitude and a direction.

  3. Vector Representation 1. The length of the line represents the magnitude and the arrow indicates the direction. 2. The magnitude and direction of the vector is clearly labeled.

  4. Scaling!!! The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn precisely to length in accordance with a chosen scale.

  5. Compass Coordinate System Direction!!! Sometimes vectors will be directed due East or due North. However we will encounter vectors in all sorts of directions and be forced to find the angle!

  6. Compass Coordinate System Use the same scale for all vector magnitudes • Δx = 30 m @ 20º E of N • V = 20 m/s @ 30º W of N • a = 10 m/s2@ 40º W of S • F = 50 N @ 10º S of E N E W To Draw direction: Ex. 20º E of N: Start w/ North and go 20° East S Navigational System?

  7. All these planes have the same reading on their speedometer. (plane speed not speed with respect to the ground (actual speed) What factor is affecting their velocity?

  8. A B C

  9. Vector Addition

  10. Easy Adding…

  11. Resultant Vectors The resultant is the vector sum of two or more vectors.

  12. Vector Addition Graphical Method

  13. Vector Addition: Head to Tail Method or (tip to tail) • Select an appropriate scale (e.g., 1 cm = 5 km) • Draw and label 1st vector to scale. • *The tail of each consecutive vector begins at the head of the most recent vector* • Draw and label 2nd vector to scale starting at the head of the 1st vector. • Draw the resultant vector (the summative result of the addition of the given vectors) by connecting the tail of the 1st vector to the head of the 2nd vector. (initial to final pt.) • Determine the magnitude and direction of the resultant vector by using a protractor, ruler, and the indicated scale; then label the resultant vector.

  14. Velocity Vectors 120 km/h 20 km/h 80 km/h 100 km/h = = 100 km/h 20km/h B. Headwind (against the wind) • Tailwind • (with the wind)

  15. Velocity Vectors C. 90º crosswind Using a ruler and your scale, you can determine the magnitude of the resultant vector. Or you could use the Pythagorean Theorem. Then using a protractor, you can measure the direction of the resultant vector. Or you could use trigonometry to solve for the angle. Resultant 80 km/h 100 km/h 60 km/h

  16. The Parallelogram Method • Find the resultant force vector of the two forces below. • 25 N due East, 45 N due South 25 N, East Decide on a scale!!! 51 N 59º S of E 51 N 31º E of S 45 N, South

  17. An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N 180o 0o 270o Scale: 50 mph = 1 inch

  18. An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N W E S Scale: 50 mph = 1 inch

  19. An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N 50 mph 200 mph W E VR = 165 mph @ 40° N of E S Scale: 50 mph = 1 inch

  20. 2. Find the resultant velocity vector of the two velocity vectors below. 700 m/s @35 degrees E of N; 1000 m/s @ 30 degrees N of W V2 Vr V1

  21. Intro to Vectors Warm-up A bear walks one mile south, then one mile west, and finally walks one mile north. After his brisk walk, the bear ends back where he started. What color is the bear???

  22. Vector Addition Component Method

  23. In what direction is the leash pulling on the dog?

  24. Breaking Down Vectors into Components

  25. What would happen to the upward and rightward Forces if the Force on the chain were smaller?

  26. Breaking Down Vectors into Components

  27. Breaking Down Vectors into Components

  28. Breaking Down Vectors into Components

  29. Breaking Down Vectors into Components

  30. Adding Component Vectors

  31. Adding Component Vectors

  32. Adding Component Vectors

  33. VR=? V2 V1 Warm Up sohcahtoa 1) Find the resultant Magnitude:__________ of the two vectors Direction:___________ Vector #1 = 20.5 N West Vector #2 = 14.3 N North Vector Diagram 24.99 N 34.90º N of W

  34. V2 V1 Warm Up sohcahtoa 2) Find the component of the resultant = 255m 27º South of East Vector # 1 _______ Direction__________ Vector # 2________ Direction__________ Vector Diagram: 115.8 m South (-) East (+) 227.2 m Conventions: + Vr - + -

  35. Vector Addition: The Method of Components Practice: Find FR =Fnet =? 200 N due South, 100 N at 40º N of W Skip • Draw vector diagram. (Draw axis) • Resolve vectors into components using trig: Vadj = V cos θ Vopp = V sin θ 3. Sum x and y components: ΣVxiΣVyi • Redraw!! Determine resultant vector using Pythagorean’s Theorem and trig: Magnitude= √(Σ Vxi)² + (Σ Vyi)² Direction Action: θ = tan-1(opp/adj) Answer: Fnet = N @ ˚ W of S

  36. An airplane flies at an engine speed of 100 m/s at 50º W of S into a wind of 30 m/s at 200 E of N. What is the airplane’s resultant velocity? Solve using the components method!! How far has the plane traveled after 1 hr?? Answer: 75.52 m/s @ 28.54˚ S of W 168.89 miles per 1 hour

  37. You Try!!! A motor boat traveling 4.0 m/s, East encounters a current traveling 3.0 m/s, North. a. What is the resultant velocity of the motor boat? b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore? c. What distance downstream does the boat reach the opposite shore?

  38. Balanced & Unbalanced Forces Equilibrium

  39. Balanced? or Unbalanced?

  40. Statics Forces are in Equilibrium

  41. Are these Objects are in Equilibrium? 151 N Ceiling 5 kg 25 N 50 N 10 N 15 N 102 N

  42. Solving Static Problems • The First Condition of Equilibrium: The upward forces must balance the downward forces, and the leftward forces must balance the rightward forces. • SFx = 0 SFy = 0 at equilibrium • Solution of Problems in Static's: • Isolate a body. What point or object are you going to talk about? • Draw the forces acting on the body you have isolated, and label them. (If their value is not know, give them a symbol such as F1, FP, T1, etc.) Remember…this is a free-body diag.

  43. Solving Static Problems Cont. • Split each of the forces into its x and y components, and label the components in terms of the symbols given in rule 2 and the proper sines and cosines. • 4) Write down your summation equations for the first condition of equilibrium. • 5) Solve the equations for the unknowns.

  44. Equilibrium problem Solve for Fx & Fy? 300 N 20º 10 kg Fx Fy Ans: Fx = 102.61 N Fy = 183.91N

  45. Equilibrium problem 200 N Solve for F? 10 kg F 30º ө 50 N Answer: F = 88.34 N @ 60.65º S of W

  46. 45º 37º W Tension Warm-up • Find the tension in each rope if the weight (W) is 50 N. • Be sure to pick an appropriate point to draw your free-body-diagram. Then sum your x forces and then your y-forces. Answers: T1= N, T2= N

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