Vectors & Scalars Vectors 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 & scalars Distance VECTORSSCALARS Displacement Mass Volume Force Velocity Speed Acceleration

Vectors & Scalars Vectors 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 & scalars VECTORSSCALARS 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 NE NW SW SE Compass bearings can be used to indicate the direction of a vector. SSE WNW WNW = 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 45 o NW SW SE 90 o 135 o 180 o 225 o 270 o 315 o 360 o /0 Measurements always from North Always measured clockwise.

Direction - Compass EW S N NE NW SW SE 45 o Compass bearings can be used to indicate the direction of a vector. 10 o Direction of purple vector in bearing and NSEW? 67.5 o ENE or…

Direction - Compass EW S N NE NW SW SE 45 o Compass bearings can be used to indicate the direction of a vector. ENE or… 10 o 280 o or 10 o N of W 67.5 o

Bearings Find the bearings for each of the vectors A – D. Two different ways for each vector. EW S N A B C D 30 o Clockwise measurements POSITIVE. Anticlockwise NEGATIVE. 30 o

Bearings A = 30 o or 30 o E of N B = 120 o or 30 o S of E or E 30 o S C = 210 o or 30 o W of S or S 30 o W D = 300 o or 30 o N of W or W 30 o N EW S N 30 o A B C D 120 O 210 O 30 o

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/ 2 circumference A - A displacement = 0! distance = circumference (2  r) 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 + s 1 = 5m START s1s1 s 2 = -1m s tot = …………………………

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 + s 1 = 5m START s1s1 s 2 = -1m s tot = +5 +(-1) = +4m

Resultant Linear Displacements N S 1 = 3km, 90º S 2 = 4km, 90 º 1. Same direction - A person walks 3km east and then 4km further east - find their ……………. displacement. S 1 = ………… S 2 = ………………… 2. Opposite direction - A person walks 3km east and then 4km West. (Take East as ……………….) R =..………………….. Choose one direction (……….) as …………… S 1 = …….. S 2 = ………...: R = ……………………………………………………………. R = ………………………...: R = ………………………..

Resultant Linear Displacements N S 1 = 3km, 90º S 2 = 4km, 90 º 1. Same direction - A person walks 3km east and then 4km further east - find their resultant displacement. S 1 = 3km, 90º S 2 = -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 positive S 1 = +3km S 2 = +4km.: R = S 1 + S 2 = = +7km ie: 7km EAST R = S 1 + S 2 = = -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 WEST 4m 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  (-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 WEST 4m EAST Resultant vector goes from the tail of one vector to the head of the other. (Beginning to END.)

Resultant The 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 = = 25 R 2 = 25 R = 5km 53.1 o E of N sinB = o/h = 3/5 B = sin -1 (0.6) B = 36.9 Bearing = = 53.1 B

Forces as a Vector Same direction Two forces of15 N at 90 o and 40 N at 90 o are applied to a box. Opposite direction Two forces of 100 N at 90 o and 40 N at270 o are applied to a box.

Forces as a Vector Same direction 15 N 90 o and 40 N 90 o R = = 55N 90 o Maximum Resultant Opposite direction 100 N 90 o and 40 N 270 o R = = +60N 90 o minimum resultant 90 o (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 180 o. 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 2NB 8NC 4ND10 N Question 2 If the resultant between the two vectors is 3.5 N which of the following is the most likely angle between them? A 180 O B 0 O C 20 O D 100 O

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 180 o. 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 2NB 8NC 4ND10 N ANS: D Question 2 If the resultant of the two vectors is 3.5 N which of the following is the most likely angle between them? A 180 O B 0 O C 20 O D 100 O ANS:D N The MAXIMUM resultant is experienced when the forces are at 0 o. Resultant (min) Resultant (max)

Components of Vectors Given Vector F F can be expressed as the vector sum of two perpendicular vectors F x & F y F FxFx FyFy y x 

Components of Vectors Given ANY Vector F F can be expressed as the vector sum of two perpendicular vectors F x & F y F FxFx FyFy y x  F x = F cos  F y = F sin  F x is the component of F in the x direction. F y 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)  F v = F sin  F s (m)  F h = 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)  F h = F cos  F s (m)  F h = F cos  Pushing Pulling 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 Plane The system shown is in equilibrium. What is the magnitude and direction of the friction force acting on the block? m  F f = ? F // = -F f = -F g sin  m FgFg   F 90 F || N FfFf Opposite direction

Inclined Plane The system shown is in equilibrium. What is the magnitude and direction of the friction force acting on the block? 250N 30 o

Inclined Plane The system shown is in equilibrium. What is the magnitude and direction of the friction force acting on the block? 250N 30 o F f = F g sin 30 = 250(0.5) = 125N up the slope Sin30 = F f / F g 30 o FgFg

Pulley system The system shown is in equilibrium. What is the magnitude and direction of the friction force acting on the block? W 250N 30 o 10kg

Pulley system The system shown is in equilibrium. What is the magnitude and direction of the friction force acting on the block? W 100kg 250N 30o30o 30 o 250N 100N F f = ? W 90 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