2. Scalars & Vectors

3. Tug of War

4. Treasure Hunt

5. Scalars Completely described by its magnitude Direction does not apply at all e.g. Mass, Time, Distance, etc.

6. Vectors Characterised by its magnitude & direction Knowledge of direction is necessary e.g. Displacement, Velocity, Acceleration, Force, etc.

7. Vector Quantity 1. By scaled drawing: Draw an arrow of definite length and direction to represent the vector. 2. By a statement: A car is travelling eastward at a velocity of 5 m/s. 5 m/s

8. Vector Quantity For example: A boy travels 10 m along a direction of 20 0 east of north. 20 0 10 m north

9. Adding & Subtracting Scalars Same as in algebra You only have to add algebraically the variables together  i.e. x units + y units = (x + y) units  e.g. Adding Time: 10s + 15s = 25s  e.g. Subtracting volumes: 15cm 3 - 10cm 3 = 5cm 3

10. Adding Vectors If the vectors are acting along the same line: 10 N 8 N12 N Just add them up algebraically!

11. Adding Vectors If the vectors are acting at an angle to each other: Eric leaves the base camp and hikes 11.0 km, north and then hikes 11.0 km east. Determine Eric's resulting displacement. ?

12. Method 1: Graphical Method ¦Graphical Method / Scaled Vector Diagram 1.Decide on a scale (e.g. 1cm : 1 km) 2.Draw the vectors in the desired directions 11 km

13. Graphical Method 1.Complete a parallelogram using the 2 sides given. 2.Draw the diagonal that represents the resultant. 3.Measure the length that represents the magnitude. 4.Use a protractor to measure the angle the resultant makes with a specified reference direction.

14. 11.0 km 15.6 km In this example, Eric’s final displacement is 15.6 km (because the red line is 15.6 cm long) and is at 45 0 East of North. 45 0 Graphical Method

15. ¦ Mathematical Method +We use the Pythagoras’ Theorem 1c = (a 2 + b 2 ) 1where c is the resultant Method 2- Mathematical Method 11 2 + 11 2 = R 2 R = 15.6 m

16. +To find the direction of the resultant, we use the definition of tangent. 1Tan = opposite side / adjacent side 1 = tan -1 (opposite side / adjacent side) ¦ Mathematical Method Mathematical Method = tan -1 (11.0 / 11.0) = 45 o

17. Class Practice Question 1 A barge is pulled at a steady speed through still water by two cables as shown in the plan view below. By means of a vector diagram, determine the magnitude and direction of the resultant force exerted on the barge by the cables. [3]

18. [1] -- for an appropriate scale (take up more than ½ of the space provided) [1] – R = 1.1 x 10 5 N (tolerance of 0.1 x 10 5 N ) [1] – R is 37 o clockwise from F 2 Class Practice Question 1

19. Question? Can we still use Pythagoras's method for mathematical method if the vectors are not perpendicular to each other? ?

20. Solve this problem by Mathematical method. Class Practice Question 1

21. Mathematical Method – when the vectors are not perpendicular N 120 o Hint: Apply cosine rule to this triangle to find magnitude of R Apply sine rule to find direction of R

22. To find magnitude: c 2 = a 2 + b 2 - 2ab cosc = 75 000 2 + 50 000 2 – 2(75 000) (50 000)cos120 o c = 1.09 x 10 5 The magnitude of resultant is 1.09 x 10 5 N. To find direction: 75 000 / sinA = 109 000/ sin120 A = 37 o Cosine Rule sine Rule

23. Question? But can we still use the graphical method is there are more than 2 vectors to be added? 20 m 25 m 15 m

24. Graphical Method – Head-to-tail Method The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position.drawing a vector to scale Where the head of this first vector ends, the tail of the second vector begins (thus, head-to-tail method). The process is repeated for all vectors which are being added. Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish.

25. Graphical Method – for more than two vectors Head-to-tail method

26. Example weight drag Lift Thrust What is the resultant force on the plane?

27. Using Graphical method Head-to-Tail Method (for addition of more than 2 vectors) weight drag Lift Thrust Resultant

28. Question? But can we still use the mathematical method is there are more than 2 vectors to be added? 20 m 25 m 15 m

29. Mathematical Method – for more than two vectors When there are more than two vectors +Simply use any of the above methods and solve this two vectors at a time. A B C D E 1First find the resultant of A and B, and name it D. 1Then find the resultant of D and C, which is E and which is also the resultant of the three vectors. 1It doesn’t matter which two vectors you resolve first, be A & C or B & C, the answer will still be the same.

30. Addition & Subtraction of Vector Quantities A VERY IMPORTANT NOTE +If the vector sum is 0 the object that the vectors are acting on is in equilibrium; it doesn’t move at all. 8N The vector sum is 0. 10N 6N 8N The vector sum is 0.

31. Equilibrium For example, if a box stays in equilibrium,the resultant of F1 and F2 must be equal and opposite to F3. F 1 = 4 N F 2 = 3 N F 3 = 7N

32. Equilibrium For example, if a box stays in equilibrium,the resultant of F1 and F2 must be equal and opposite to F3. F1F1 F2F2 F3F3 R

33. Equilibrium Equilibrium means  the forces acting on that object are balanced  the resultant force is zero  the object does not move

34. Example This system is in equilibrium. Find the weight of the car by graphical method. 736 g 425 g

35. Ans Draw a free-body diagram to show all the forces. T 1 = 4.25 N T 2 = 7.36 N 30 o W

36. Ans From the free-body diagram, it is clear that Resultant of T 1 and T 2 must be equal and opposite to W so that the system remains in equilibrium. Hence, to find W, just find resultant of T 1 and T 2 by graphical method. T 1 = 4.25 N T 2 = 7.36 N Ans: W = 8.5 N