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

How to specify a velocity vector?

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 QuantityFor example:

A boy travels 10 m along a direction of 200 east of north.

200

10 m

north

Adding & Subtracting Scalars

• Same as in algebra• You only have to add algebraically the

variables togetheri.e. x units + y units = (x + y) units

e.g. Adding Time: 10s + 15s = 25se.g. Subtracting volumes: 15cm3 - 10cm3 = 5cm3

Adding Vectors

• If the vectors are acting along the same line:

10 N

8 N12 N

the resultant force = (10 + 8 - 12) N

= 6N to the East

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

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

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 450 East of North.

450

Graphical Method

Mathematical MethodWe use the Pythagoras’ Theoremc = (a2 + b2)where c is the resultant

Method 2- Mathematical Method

112 + 112 = R2

R = 15.6 m

To find the direction of the resultant, we use the definition of tangent.Tan = opposite side / adjacent side = tan-1 (opposite side / adjacent side)

Mathematical Method

Mathematical Method

= tan-1 (11.0 / 11.0) = 45o

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 105 N (tolerance of 0.1 x 105 N )

• [1] – R is 37o clockwise from F2

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

120o

Hint:

Apply cosine rule to this triangle to find magnitude of R

Apply sine rule to find direction of R

To find magnitude:

c2 = a2 + b2 - 2ab cosc = 75 0002 + 50 0002 – 2(75 000) (50 000)cos120o

c = 1.09 x 105

The magnitude of resultant is 1.09 x 105 N.

To find direction:75 000 / sinA = 109 000/ sin120A = 37o

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.

• 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 vectorsSimply use any of the above methods and solve this two

vectors at a time.A

B

C

D

E

First find the resultant of A and B, and name it D.

Then find the resultant of D and C, which is E and which is also the resultant of the three vectors.

It 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 NOTEIf the vector sum is 0 the object that the vectors are

acting on is in equilibrium; it doesn’t move at all.

8N 8N

The vector sum is 0.

10N

6N

8NThe 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.

F1 = 4 N

F2 = 3 N

F3 = 7N

Equilibrium

• For example, if a box stays in equilibrium,the resultant of F1 and F2 must be equal and opposite to F3. F1

F2

F3

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.

T1 = 4.25 N

T2 = 7.36 N

30o

W

Ans• From the free-body diagram, it is clear that Resultant of T1

and T2 must be equal and opposite to W so that the system remains in equilibrium. Hence, to find W, just find resultant of T1 and T2 by graphical method.

T1 = 4.25 N

T2 = 7.36 N

Ans: W = 8.5 N