Physics M1 Vectors

OBJECTIVESOBJECTIVES

Representing a vector quantity

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Adding VectorsAdding Vectors

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INTRODUCTIONINTRODUCTION

XXBefore you begin, see diagram below. Look for the Security Warning below the ribbon and click the Options button. In the Security Alert box, select “Enable this Content”. Click OK.

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Physics Module 1 Unit 1 Lessons 16-17



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XX

The diagram shows Tom about to drive out of a parking lot in his community.

1. What is Tom’s direction from the house?

2. What Tom’s distance from the house?

3. What is Tom’s distance from the school?

INTRODUCTION

1. South East

2. 250 m

3. 180 m

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100 m

150 m 80 m

Tom’s car

house school

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Tom in his community cont’d

4. What directions would Tom take in order to get the house by road?

5. What direction could Tom take to go in a straight line to the house?

6. Is the distance to go straight to the house shorter or longer than by road?

INTRODUCTION

4. North then West

5. Northwest

6. Shorter

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100 m

150 m 80 m

Tom’s car

house school

CHECKCHECKCHECKCHECK



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XXINTRODUCTION

http://www.mrfizix.com/home/vectors.htm

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XXOBJECTIVES

When you complete this slide presentation, you will be able to:•Represent vectors graphically;•State three ways in which the directions of vectors are written•Draw vectors based on their definitions•Explain what is meant by the negative of a vector.•Draw the negatives of given vectors•Determine the magnitude and direction of the resultant parallel and anti-parallel vectors•Use the Pythagoras' theorem to calculate resultant vectors which perpendicular•Determine the resultant of vectors that are neither parallel nor anti-parallel using scale drawings.•Use the parallelogram method to determine the resultant of concurrent vectors

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XXHow do you represent a vector quantity

graphically?

A vector quantity that has both magnitude and a specified direction.

A vector quantity is represented graphically by a straight line with an arrowhead.

The arrowhead points in the direction of the vector and the length of the line is proportional to the magnitude of the vector

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Head of vector

Tail of vector



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XXWhat are the ways of writing the

direction of a vector quantity?

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W

S

N

E

S

N

1300

Boat travels with velocity of 15 m/s North East

Helicopter travels 200 km on a bearing of 130o

COMPASS POINT BEARINGS



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XXWhat are the ways of writing the

direction of a vector quantity?

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300

Rocket flies with acceleration of 12 m/s2 vertically upwards

Aircraft flies with velocity of 35 m/s at 30o above the horizontal

RELATIVE TO THE VERTICAL AND HORIZONTAL



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XX

Click on the number to reveal the step.

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2

1

How can these vectors be represented? The Steps

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Choose a suitable scale that would give a reasonable length line. For example: If a scale of 1 cm: 100 km was chosen for the helicopter vector, the line representing vector would have length 200/ 100 = 2 cm which is too short. If it was 1cm : 5 km the line representing the vector would have length 200/5 = 40 cm which is too long.

Choose a suitable scale that would give a reasonable length line. For example: If a scale of 1 cm: 100 km was chosen for the helicopter vector, the line representing vector would have length 200/ 100 = 2 cm which is too short. If it was 1cm : 5 km the line representing the vector would have length 200/5 = 40 cm which is too long.

Draw the line of length 10 cm at a bearing of 130o using a protractor to measure the angle.Draw the line of length 10 cm at a bearing of 130o using a protractor to measure the angle.

Put an arrowhead on the line to show the direction and label the line with the magnitude of vector and NOT the length of the line.

Put an arrowhead on the line to show the direction and label the line with the magnitude of vector and NOT the length of the line.



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XXHow are the vectors represented?

Here is the way the vectors mentioned can be represented:

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S

N

1300

200 km

300

35 m/s

Length of vector: 10 cmDirection: 130o

Scale 1cm: 20 km

Length of vector: 7 cmDirection: 30o above the horizontal

Scale 1cm: 5 m/s



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XXHow do you represent a vector quantity

graphically?

Draw scale diagrams of the following vectors:1. A ship travels 20 km at 40o West of North

2. A canoe travels with velocity 2.0 m/s on a bearing of 130o

3. A boy pushing a trolley, with a force of 70 N at an angle of 30o below the horizontal.

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Click here to check your answer





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XXNegative of a Vector

The negative of a vector is another vector of same magnitude as the vector but of opposite direction.

The vector – p is the negative of the vector p.

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p

– p



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State the magnitude and direction of the negative of the vector 2.0m/s on a bearing of 130o .

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S

N

310o

2.0 m/s

2.0 m/s

Negative of the vector is 2.0 m/s on a bearing of 310o



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State the magnitude and direction of the negative of the vector 20 km at 40o West of North

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W

S

N

E

40020 km

20 km

500

Negative of the vector is 20 km at 50o South of East



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XXCan vectors be added?

When two vectors are added, the sum vector is called the resultant. There are two methods of adding vectors

1. Head to tail method. Used with free vectors.

2. Parallelogram method. Used with concurrent vectors

Only vectors representing quantities with the same units can be added.The magnitude and direction of a resultant vector can either be found by scale drawing or calculation.

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XX Adding Vectors

Parallel and Anti-parallel Vectors

1. Adding two parallel vectors

2. Adding two anti-parallel vectors

When adding parallel vectors, add the magnitudes of the vectors. The direction of the resultant is the same as the vectors. When adding anti-parallel vectors, subtract the magnitudes. The direction of the resultant is in the direction of the vector of greater magnitude.20

2 m/s 5 m/s

7 m/s

2 m/s 5 m/s

3 m/s



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XXAdding Vectors: Vectors that are neither

parallel nor anti-parallel

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100 m

150 m80 m

Tom’s car

house school

Recall in the introductory activity that Tom could get to his house two ways. One way is to drive 100 m North together with 150 m West. The other way (which is not possible) is to drive directly from the parking lot to his gate



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parallel nor anti-parallel

The single resultant vector that would take Tom to his house has the same effect as the two vectors put head to tail.

When adding vectors that are neither parallel nor anti-parallel, place the vectors head to tail and the vector that is drawn from the tail of the first vector to the head of the second vector is the resultant of the two vectors.

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Problem: What is the distance and direction of Tom’s house from the parking lot? To answer the question, the magnitude of the vector R and the angle θ must be found.

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Adding Vectors: Vectors that are neither parallel nor anti-parallel

100 m

150 m

R

Mathematical solutionUsing Pythagoras’ equationR2 = 1002 + 1502

= 10000 + 22500 = 32500R = √32500 = 180 m

Tan θ = 150/100 = 1.5 θ = tan-1 1.5 = 56.3o

θ

AnswerTom’s house is 180 m from the parking lot in a direction of 56.3o West of North.



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parallel nor anti-parallel Fill in the blank spaces in the instructions for a scale drawing to find the distance and direction of the house1.Choose a suitable scale of 1 cm : 10 m2.Draw a line of length ____ cm with arrowhead pointing in the ________ direction to represent the 100 m vector, and a line of length ______ cm with arrowhead pointing in the ________ direction to represent the 150 m vector.3.The tail of the 150 m vector must be joined to the head of the 100 m vector.4.Draw a line from the tail of the 100 m vector to the head of the 150 m vector. 5.Measure the length of the line and multiply the result by _____ for the length of the vector in m.6.Measure the angle _____ with a protractor.7.Complete a drawing for yourself.

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Adding Vectors: Vectors that are neither parallel nor anti-parallel

Fill in the blank spaces in the instructions for a scale drawing to find the distance and direction of the house1.Choose a suitable scale of 1 cm : 10 m2.Draw a line of length 10 cm with arrowhead pointing in the North direction to represent the 100 m vector, and a line of length 15 cm with arrowhead pointing in the West direction to represent the 150 m vector.3.The tail of the 150 m vector must be joined to the head of the 100 m vector.4.Draw a line from the tail of the 100 m vector to the head of the 150 m vector. 5.Measure the length of the line and multiply the result by 10 for the length of the vector in m.6.Measure the angle θ with a protractor.7.Complete a drawing for yourself.



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XXAdding Vectors: Concurrent Vectors

Concurrent vectors are vectors that originate at the same point or their lines of action intersect.

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Forces on the boat are concurrent because they are applied at the same point

Forces on the boat are concurrent because their lines of action intersect



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XXAdding Vectors: Concurrent Vectors

Completing the Parallelogram Method

1. Open the compass to the length of one vector.

2. Place the compass point at the head of the other vector and draw an arc.

3. Repeat with the other vector and draw an intersecting arc.

4. Draw the resultant vector from the origin of the two vectors to the point of intersection of the arcs.

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Click to see method



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Adding VectorsActivity 2.2

In the illustration the helicopter heads in the direction shown with velocity of 50 m/s and a cross wind blows with a velocity of 20 m/s. Use a scale drawing to determine the magnitude and direction of the resulting velocity of the helicopter.

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300

N

wind

W

Click for answer



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XXAdding Vectors

Feedback to Activity 2.21. Choose a suitable scale2. Draw vector of length 10 cm

pointing 30o to North (helicopter vector)

3. Draw vector of length 4 cm at the head of the helicopter vector pointing west (wind vector).

4. Draw a vector from tail of the helicopter vector to the head of the wind vector. This is the resultant vector

5. Measure the length of the resultant vector and multiply the length by 5 to get the magnitude of the resultant in m/s

6. Measure the angle from North of the resultant 29

50 m/s

20 m/s

Scale 1 cm : 5 m/s

300

The resultant velocity of the The resultant velocity of the helicopter is 62.5 m/s at 46helicopter is 62.5 m/s at 46oo East of NorthEast of North

12.5 cm x 5 N/cm = 62.5 N46o

62.5 m/s



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Adding VectorsActivity 2.3

In the illustration the forces of two ropes tied to the boat act on the boat. Use scale drawing to determine the magnitude of the resultant force on the boat. State the direction in which the boat will move relative to the 160 N force.

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160 N 240 N

Click to check your answer.

110o



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Adding Vectors

1. Choose a suitable scale.2. Draw a force vector of length

8 cm to represent the 160 N force

3. Use the protractor to measure an angle of 110o and draw a force vector of length 10 cm to represent the 200 N force.

4. Use compasses to draw arcs.5. Draw the resultant vector6. Measure the length of the

vector and calculate its magnitude in N by multiplying the length by 20

7. Measure the angle between the resultant vector and the 160 N force

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110o

160 N 200 N

Scale 1 cm : 20 N

The magnitude of the resultant force on the boat is 210 N at 63.5o to the 160 N force.

210 N

63.5o



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XXQUIZ

1. Are you ready to test yourself? 2. There are seven multiple choice items that

you are required to do. 3. For each item, click on the letter of the

correct response and then click the button ‘NEXT QUESTION’ to continue.

4. Click the BEGIN QUIZ button to start the quiz.

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BEGIN QUIZ!BEGIN QUIZ!



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XXQUIZ: Question 1

Which one of the following is the best example of a vector quantity?

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A. A current of 5 A flowing from positive to negative

A. A current of 5 A flowing from positive to negative

B. A distance of 200 km from Kingston to Montego Bay

B. A distance of 200 km from Kingston to Montego Bay

C. A speed of 12 m/s around a bendC. A speed of 12 m/s around a bend

D. A force of 50 N to the rightD. A force of 50 N to the right

FEEDBACKFEEDBACKFEEDBACKFEEDBACK



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XXQuiz: Feedback to Question 1

Which one of the following is the best example of a vector quantity?

34NEXT QUESTIONNEXT QUESTIONNEXT QUESTIONNEXT QUESTION

A is incorrect: Positive to negative does not state a specific directionB is incorrect: Kingston to Montego Bay does not state a specific directionC is incorrect: Direction changes continuously when going around a bend. It is not specific.D is correct: To the right is a specific direction that can be given relative to a known direction.



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Two forces are applied to a body, as shown. What is the magnitude of the resultant force acting on the body?

QUIZ: Question 2

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A. 19N A. 19N

D. 10.9N D. 10.9N

C. 7N C. 7N

B. 13N B. 13N




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Two forces are applied to a body, as shown. What is the magnitude of the resultant force acting on the body?


The forces are perpendicular vectors R2 = F1

2 + F22 R2 = 52 +122

R = 144 25 169 13N



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XXQUIZ: Question 3

Which correctly states the magnitude and direction of the resultant force on the train carriage in the diagram?

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A. 160 N to the leftA. 160 N to the left

D. 1160 N to the rightD. 1160 N to the right

B.920 N to the right B.920 N to the right

C. 840 N to the rightC. 840 N to the right


1000 N

80 N80 N



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Which correctly states the magnitude and direction of the resultant force on the train carriage in the diagram?


A is incorrect: It states the resultant force to the left onlyB is incorrect: It states the resultant force between the 1000 N and one of the 80 N forcesC is correct: It states the resultant force between the 1000 N and the two 80 N forces. D is incorrect: The forces in opposite directions are added and not subtracted.

1000 N

80 N80 N



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XXQuiz: Question 4

A vector is defined as 2.0 m/s heading North West. Which best states the negative of this vector?

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A. 2.0 m/s heading South WestA. 2.0 m/s heading South West

D. 2.0 m/s heading North D. 2.0 m/s heading North

C. 2.0 m/s heading North EastC. 2.0 m/s heading North East

B. 2.0 m/s heading South East B. 2.0 m/s heading South East




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A vector is defined as 2.0 m/s heading North West. Which best states the negative of this vector?


The diagram shows the negative of the vector. A. C and D are incorrect. The negative of the vector is in the South East direction.

W

S

N

E

2.0 m/s

2.0 m/s



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XXQUIZ: Question 5

Which is the correct way to subtract vector A from vector B that are neither parallel nor anti-parallel?

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B. Add vector A to vector B and multiply the magnitude of the resultant by – 1.

B. Add vector A to vector B and multiply the magnitude of the resultant by – 1.

D. Subtract the magnitude of vector A from the magnitude of vector B.

D. Subtract the magnitude of vector A from the magnitude of vector B.

C. Add vector A to vector B and find the negative of the resultant vector

C. Add vector A to vector B and find the negative of the resultant vector

A. Add the negative of vector A to vector BA. Add the negative of vector A to vector B




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Which is the correct way to subtract vector A from vector B that are neither parallel nor anti-parallel?

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A is correct: Subtraction of vectors is the same as adding the negative of one vector to the otherB is incorrect: This would only be making the magnitude of the sum of the vectors negative.C is incorrect: This would give the negative of the sum of the vectors and not the difference between them.D is incorrect: This is only applicable when vectors are parallel

END OF QUIZ END OF QUIZ END OF QUIZ END OF QUIZ



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XXEnd of Quiz

You have reached the end of the quiz. If you wish to review this lesson, go to the first

slide.If you have finished viewing the lesson, click the

Close (X) button.

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XXReference

www.phy.syr.edu/courses/.../PHY_101_Chapter_2_Lectures.ppt

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Education

Physics M1 Vectors