vectors & scalars vectors quantities having both magnitude (size) and direction. for any vector...

Vectors & ScalarsVectors• Quantities having both

MAGNITUDE (size) and DIRECTION.

• For any vector both size and direction must be stated.

Scalars• Quantities having MAGNITUDE

(size) only.

Classify the following as vectors & scalarsDistance VECTORS SCALARSDisplacement

Mass

Volume

Force

Velocity

Speed

Acceleration

Vectors & ScalarsVectors• Quantities having both

MAGNITUDE (size) and DIRECTION.

• For any vector both size and direction must be stated.

Scalars• Quantities having

MAGNITUDE only.

Classify the following as vectors & scalarsVECTORS SCALARS

Displacement Distance Force Mass Velocity Volume Acceleration Speed Weight

Representing Vectors

All vector quantities can be represented by an arrow.

Magnitude = Length of LINE

The arrow is a straight line drawn to a suitable scale and the length of the arrow represents the magnitude of the vector while the arrow head represents the direction.

Scale: eg 1cm = 10m

• The scale that is used must be chosen carefully in order for the whole drawing to fit on one page

• the scale must be shown on the page.

Direction ARROW

Direction - Compass

EW

S

N NENW

SW SE

Compass bearings can be used to indicate the direction of a vector.

SSE

WNWWNW = west of north west

Give the bearing in degrees for each of the directions shown. Always taken from North – clockwise!

Direction - Compass bearings


EW

S

N NE 45o

NW

SW SE

90o

135o

180o

225o

270o

315o

360o/0

Measurements always from NorthAlways measured clockwise.

Direction - Compass

EW

S

N NENW

SW SE

45o

45o


10oDirection of purple vector in bearing

and NSEW?

67.5o

ENE or…

Direction - Compass

EW

S

N NENW

SW SE

45o

45o


ENE or…

10o280o

or10o N of W

67.5o

Bearings

Find the bearings for each of the vectors A – D. Two different ways for each vector.

EW

S

N A

B

C

D

30o

Clockwise measurements POSITIVE.

Anticlockwise NEGATIVE.

30o

30o

30o

Bearings

A = 30o or 30o E of N

B = 120o or 30o S of E or E 30o S

C = 210o or 30o W of S or S 30o W

D = 300o or 30o N of W or W 30o N

EW

S

N

30o

A

B

C

D

120O

210O

30o

Distance vs Displacement

*

A Start

*

B End

Path traveled = DISTANCE

Displacement = Straight line distance from starting point to finishing point.

A B

For rectilinear motion DISTANCE = DISPLACEMENT

Circular motion:• A - B displacement = …………… distance = ……………….

• A - A displacement = …………..• distance = ……………………..

Straight line distance = DISPLACEMENT


*

A Start

*

B End


Displacement = Straight line distance from starting point to finishing point.

A B

For rectilinear motion DISTANCE = DISPLACEMENT

Circular motion:• A - B displacement = diameter distance = 1/2circumference

• A - A displacement = 0!• distance = circumference (2r)

Straight line distance = DISPLACEMENT


*

A Start

*

B End


• Distance cannot be less than displacement.

• A “negative” displacement is a movement in the opposite direction to the one CHOSEN as POSITIVE.

Right as +s1 = 5m

START s1

s2 = -1m

stot = …………………………


*

A Start

*

B End


• Distance cannot be less than displacement.

• A “negative” displacement is a movement in the opposite direction to the one CHOSEN as POSITIVE.

Right as +s1 = 5m

START s1

s2 = -1mstot = +5 +(-1) = +4m

Resultant Linear DisplacementsN

S1 = 3km, 90º S2 = 4km, 90 º

1. Same direction - A person walks 3km east and then 4km further east - find their ……………. displacement.

S1 = ………… S2 = …………………

2. Opposite direction - A person walks 3km east and then 4km West. (Take East as ……………….)

R = ..…………………..

Choose one direction (……….) as ……………S1 = …….. S2 = ……….. .: R = …………………………………………………………….

R = ………………………...: R = ………………………..

Resultant Linear DisplacementsN

S1 = 3km, 90º S2 = 4km, 90 º

1. Same direction - A person walks 3km east and then 4km further east - find their resultant displacement.

S1 = 3km, 90º S2 = -4km, (ie 270º)

2. Opposite direction - A person walks 3km east and then 4km West. (Take East as positive)

R = 7km 90º

R = -1km or 1km 270º

Choose one direction (East) as positiveS1 = +3km S2 = +4km .: R = S1 + S2 = +3 + 4 = +7km ie: 7km EAST

R = S1 + S2 = +3 -4 = -1km.: R = 1km WEST

Example

b) Calculation:

Choose ……………..

West would therefore be

…………………….

• When we add the two vectors together

R = ……………………

1.Vectors in the same or opposite direction

Find the RESULTANT of the following two vectors: 10m East and 6m West.a) Construction:1 cm = 1m

10m EAST

6m WEST4m EAST

N

Resultant vector goes from the tail of one vector to the head of the other. (Beginning to END.)

Example b) Calculation:

Choose East as positive

West would therefore be

negative.

• When we add the two vectors

above together we get

R = +10 + (-6) = +4m

1.Vectors in the same or opposite direction

Find the RESULTANT of the following two vectors: 10m East and 6m West.a) Construction: 5mm = 1m

10m

Motion in a straight line is called rectilinear or linear motion.

10m EAST

6m WEST4m EAST

Resultant vector goes from the tail of one vector to the head of the other. (Beginning to END.)

ResultantThe RESULTANT (R ) of a number of vectors is the ……... ………….. that will have the ……………… as all the original vectors acting together.

It stretches from the ………(tail) of the first vector to the …………… (…………..) of the last vector. (Tail to Head)

Eg. Determine the resultant displacement of a person who walks 4km due east and then 3km north.

Resultant

3km N

4km E

N

The RESULTANT (R ) of a number of vectors is the single vector that will have the same effect as all the original vectors acting together.

It stretches from the beginning of the first vector to the end of the last vector.

R= ?

Eg. Determine the resultant displacement of a person who walks 4km due east and then 3km north.

Resultant

3km N

(3cm)

4km E (4 cm)

N

The resultant (R ) of a number of vectors is the single vector that will have the same effect as all the original vectors acting together. It stretches from the beginning of the first vector to the end of the last vector.

R= 5cm

Using Pythagoras

32 + 42 = 9 + 16 = 25

R2 = 25

R = 5km 53.1o E of N

sinB = o/h = 3/5

B = sin-1(0.6)

B = 36.9

Bearing = 90 - 36.9 = 53.1

B

Forces as a VectorSame direction

Two forces of15 N at 90o and 40 N at 90o are applied to a box.

Opposite direction

Two forces of 100 N at 90o and 40 N at270o are applied to a box.

Forces as a VectorSame direction

15 N 90o and 40 N 90o

R = 15 + 40 = 55N 90o Maximum Resultant

Opposite direction

100 N 90o and 40 N 270o

R = -40 + 100 = +60N 90o minimum resultant

90o (Right)is positive

Resultant forces

• A person in a lift going up.

•

Gravity (…………)

• Exerted by the earth on the person.

……… or Reaction force exerted by the lift on the person (= ………………)

Considering ONLY the forces on the person.

Upward pull of lift.

• If the forces on an object are UNBALANCED the object experiences a NETT or ………………………… FORCE.

Resultant forces

• A person in a lift going up.

•

Gravity (Weight)

• Exerted by the earth on the person.

Normal or Reaction force exerted by the lift on the person (= WEIGHT)

Considering ONLY the forces on the person.

Upward pull of lift.

• If the forces on an object are UNBALANCED the object experiences a NETT or RESULTANT FORCE.

N

As the angle between the forces increases the magnitude (size) of the resultant DECREASES.

The MINIMUM resultant is experienced when the forces are at 180o.

Forces at an angle

Question:A 5N force and a 3 N force act at a point at an angle to each other. Which one of the following resultants is not possible?A 2N B 8N C 4N D10 N

Question 2If the resultant between the two vectors is 3.5 N which of the following is the most likely angle between them?A 180O B 0O C 20O D 100O

N

As the angle between two forces increases the magnitude (size) of the resultant DECREASES.

The MINIMUM resultant is experienced when the forces are at 180o.

Forces at an angle

Question:A 5N force and a 3 N force act at a point at an angle to each other. Which one of the following resultants is not possible?A 2N B 8N C 4N D10 N ANS: D

Question 2If the resultant of the two vectors is 3.5 N which of the following is the most likely angle between them?A 180O B 0O C 20O D 100O ANS:D

N

The MAXIMUM resultant is experienced when the forces are at 0o.

Resultant (min)

Resultant (max)

Components of VectorsGiven Vector F

• F can be expressed as the vector sum of two perpendicular vectors Fx & Fy

F

Fx

Fy

y

x

Components of VectorsGiven ANY Vector F

• F can be expressed as the vector sum of two perpendicular vectors Fx & Fy

F

Fx

Fy

y

x

Fx = F cos

Fy = F sin

• Fx is the component of F in the x direction.

• Fy is the component of F in the y direction. Hwk

Ex 2.2 pg 2-9 nos: 1 and 2

Components of Forces

F

s (m)

…………………F

s (m)

………………

………….

………..

…………………………….

………………………..

Components of Forces

F

s (m)

Fv = F sin F

s (m)

Fh = F cos

Pushing

Pulling

Horizontal Component

Vertical Component

Pushing has a component INTO the ground. This would INCREASE FRICTION and make it more difficult to push.

Components of Forces• Which is easier pushing or pulling a roller??

F

s (m)

Fh = F cos

F

s (m)

Fh = F cos

PushingPulling

Pushing has a component INTO the ground. This would INCREASE FRICTION and make it more difficult to push. Also pulling makes it easier to go over obstacles.

Inclined PlaneThe system shown is in

equilibrium.

What is the magnitude and direction of the friction force acting on the block?

m

Ff = ?

F// = -F f = -Fg sin

m

Fg

F90

F||

NFf

Opposite direction


equilibrium. What is the magnitude and direction of the friction force acting on the block?

250N

30o



250N

30o

Ff = Fgsin 30

= 250(0.5)

= 125N up the slope

Sin30 = Ff / Fg30o

Fg

Pulley systemThe system shown is in


W 250N

30o10kg

Pulley systemThe system shown is in


W

100kg

250N

30o

250N

250N

30o30o

250N

100NFf = ? W90

W||

W||

Changed mass so change answers

Swimmer Problems If the swimmer

attempts to swim directly across the river what is his resultant velocity?

How long would it take to cross?

How far down the bank would he land?

Current 1.5m.s-1

Width 30m

Swimming speed 1.2m.s-1

vectors & scalars vectors quantities having both magnitude (size) and direction. for any vector...

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