Scalars & Vectors

Tug of War

Treasure Hunt

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

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

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

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

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 cm 3 = 5cm 3

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

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. ?

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

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.

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 Graphical Method

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

+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

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]

[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

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

Solve this problem by Mathematical method. Class Practice Question 1

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

To find magnitude: c 2 = a 2 + b 2 - 2ab cosc = – 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: / sinA = / sin120 A = 37 o Cosine Rule sine Rule

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

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.

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

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

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

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

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.

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.

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

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

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

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

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

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