Scalar and vector quantities

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  • Scalar and vector quantities

    1. 1. SCALAR AND VECTOR QUANTITIES RAPHAEL V. PEREZ, CpE
    2. 2. SCALAR AND VECTOR QUANTITIES Define Scalar and Vector
    3. 3. SCALAR QUANTITY has only magnitude. (Only the measure / quantity) VECTOR QUANTITY has both magnitude and direction.
    4. 4. SCALAR VECTOR distance displacement work power acceleration volume pressure velocity speed weight mass force resistance
    5. 5. SCALAR AND VECTOR QUANTITIES WHAT IS RESULTANT VECTOR?
    6. 6. RESULTANT VECTOR is the is the vector that 'results' from adding two or more vectors together. -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0.5 1 1.5 2 2.5 3 3.5 4 y
    7. 7. The goal of this topic is to find the MAGNITUDE OF THE RESULTANT VECTOR (R), and the VECTOR ANGLE (θ)
    8. 8. THERE ARE THREE TECHNIQUES TO FIND THE RESULTANT VECTOR AND THE VECTOR ANGLE: 1. GRAPHICAL METHOD – you need the technical tools like sharp pencil, ruler, protractor and the paper (graphing or bond) to show the vectors graphically. The output is the connection of vectors is like a polygon.
    9. 9. R ɵ
    10. 10. SAMPLE PROBLEM 1. A man walks at40 meters East and 30 meters North. Find the magnitude of resultant displacement and its vector angle. Use Graphical Method.
    11. 11. Solution: Write the given facts Given: A = 40 meters East B = 30 meters North R = ? θ = ?
    12. 12. graph the vectors from the origin (head to tail) head (arrowhead) VECTOR tail
    13. 13. graph the vectors from the origin (head to tail) NOTE: 1 GRID = 10 METERS B=30METERS, NORTH A = 40 METERS, EAST USE RULER TO MEASURE AND TO DRAW A LINE θ = 37 N of E
    14. 14. SAMPLE PROBLEM 2. Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
    15. 15. graph the vectors from the origin (head to tail) NOTE: 1 GRID = 10 km POSSIBLE GRAPH 2. Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
    16. 16. ASSIGNMENT Use graphical Method to find the magnitude of the resultant displacement and the vector angle 1. Given: A= 13cm, 30 N of E B= 20 cm, North R = ? θ = ? 2. Given: M= 5.7 cm, NW N= 2.5 cm, SE O= 1.3 cm, NE R = ? θ = ?
    17. 17. 2. The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90- degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.
    18. 18. SAMPLE PROBLEM 1. A man walks at 40 meters East and 30 meters North. Find the magnitude of resultant displacement and its vector angle. Use Pythagorean Theorem.
    19. 19. B=30METERS,NORTH A = 40 METERS, EAST sketch your problem θ
    20. 20. • 1. ____ is an example of a scalar quantity a) velocity b) force c) volume d) acceleration • 2. ___ is an example of a vector quantity a) mass b) force c) volume d) density
    21. 21. • 3. A scalar quantity: a) always has mass b) is a quantity that is completely specified by its magnitude c) shows direction d) does not have units • 4. A vector quantity a) can be a dimensionless quantity b) specifies only magnitude c) specifies only direction d) specifies both a magnitude and a direction
    22. 22. • 5. A boy pushes against the wall with 50 pounds of force. The wall does not move. The resultant force is: a) -50 pounds b) 100 pounds c) 0 pounds d) -75 pounds • 6. A man walks 3 miles north then turns right and walks 4 miles east. The resultant displacement is: a) 1 mile SW b) 7 miles NE c) 5 miles NE d) 5 miles E
    23. 23. • 7. A man walks at 10 m East, but he returns back at 10 m at west. The resultant displacement is: a) 0 km b) 20 km c) 10 km d) -10 km • 8. The difference between speed and velocity is: a) speed has no units b) speed shows only magnitude, while velocity represents both magnitude (strength) and direction c) they use different units to represent their magnitude d) velocity has a higher magnitude
    24. 24. • 7. A plane flying 500 MI/hr due north has a tail wind of 45 MI/hr the resultant velocity is: a) 545 miles/hour due south. b) 455 miles/hour north. c) 545 miles/hr due north. d) 455 MI/hr due south • 8. The difference between speed and velocity is: a) speed has no units b) speed shows only magnitude, while velocity represents both magnitude (strength) and direction c) they use different units to represent their magnitude d) velocity has a higher magnitude
    25. 25. • 9. The resultant magnitude of two vectors a) Is always positive b) Can never be zero c) Can never be negative d) Is usually zero • 10. Which of the following is not true. a) velocity can be negative b) velocity is a vector b) speed is a scalar d) speed can be negative
    26. 26. 3. ANALYTICAL (COMPONENT) METHOD Each vector has two components : the x-component and the y-component If the vectors are in secondary directions : (NW, NE, SW or SE directions) Ax = A cos θx Ay = A sin θx where: A = the given vector value θx = the given angle from x -axis Ax = the x – component of vector A Ay = y – component of vector A
    27. 27. Component formula for x and y: Ax = A cos θx Ay = A sin θx Sum of x and y Components:
    28. 28. Consider the sign conventions for the Sum of x and y Components Quadrant of Magnitude + + I - + II - - III + - IV 0 Y-axis (North or South) 0 X- axis (West or East)
    29. 29. About the vector angle: Recall it from Trigonometry: if θx is positive: rotation of magnitude is counterclockwise from x-axis if θx is negative: rotation of magnitude is clockwise from x-axis
    30. 30. Recall the SAMPLE PROBLEM Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
    31. 31. Solution: Draw a sketch A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W
    32. 32. Solution: Draw a table Vector and measure X-component Y-component A +5 0 B +6 cos 45 +6 sin 45 C -7 cos30 +7 sin 30 Total (use scientific calculator in degrees mode) Note: for the sum of components: round off the answers into 5 decimal places. Therefore, the direction of the magnitude of resultant vector is in QUADRANT I
    33. 33. Solution: Compute for the magnitude and vector angle WHICH IS NEAR IN OUR PREVIOUS DRAWING IN GRAPHICAL METHOD
    34. 34. Actual happen on vectors (not needed to graph) NOTE: 1 GRID = 10 km POSSIBLE GRAPH A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
    35. 35. graph the vectors from the origin (head to tail)
    36. 36. θ = 67.67 (ƩRx, ƩRy) = (3.18046, 7.74264) Quadrant I
    37. 37. FINAL ANSWER
    38. 38. 4 N Vectors X- component Y- component A B C Total A = B = C =
    39. 39. A = B = C = D = 7 N Vectors X-component Y-component A B C D Total

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