vectors & scalars vectors quantities having both magnitude (size) and direction. for any vector...
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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
Compass bearings can be used to indicate the direction of a vector.
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
Compass bearings can be used to indicate the direction of a vector.
10oDirection of purple vector in bearing
and NSEW?
67.5o
ENE or…
Direction - Compass
EW
S
N NENW
SW SE
45o
45o
Compass bearings can be used to indicate the direction of a vector.
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
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 = diameter distance = 1/2circumference
• A - A displacement = 0!• distance = circumference (2r)
Straight line distance = DISPLACEMENT
Distance vs Displacement
*
A Start
*
B End
Path traveled = DISTANCE
• 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 = …………………………
Distance vs Displacement
*
A Start
*
B End
Path traveled = DISTANCE
• 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)
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
Inclined PlaneThe system shown is in
equilibrium. What is the magnitude and direction of the friction force acting on the block?
250N
30o
Inclined PlaneThe system shown is in
equilibrium. What is the magnitude and direction of the friction force acting on the block?
250N
30o
Ff = Fgsin 30
= 250(0.5)
= 125N up the slope
Sin30 = Ff / Fg30o
Fg
Pulley systemThe system shown is in
equilibrium. What is the magnitude and direction of the friction force acting on the block?
W 250N
30o10kg
Pulley systemThe system shown is in
equilibrium. What is the magnitude and direction of the friction force acting on the block?
W
100kg
250N
30o
250N
250N
30o30o
250N
100NFf = ? W90
W||
W||
Changed mass so change answers