Displacement and Force in

Two Dimensions

Lesson 1: Vectors

OBJECTIVE: 1. To graphically add vectors2. To discuss vector components3. To add and subtract vectors.

Section 1: Vectors

What does this symbol mean?

What does this symbol mean?

No or Prohibited

What does this symbol mean?

What does this symbol mean?

Other Signs and symbols we encounter everyday

• Daily situations use abstractions as well as math everywhere we go.

• Can symbols be used to express an idea or concept?

a)

b)

c)

1. What does these arrows mean?2. What two properties of arrows being illustrated by these examples?

Two properties: Size (magnitude) or direction

a)

b)

c)

Remember…A vector is a quantity that has both magnitude and direction.

Example of Vector Quantities: Displacement, velocity, acceleration, and force

Vector Components

Angled Vectors Have Two Components

Adding and Subtracting vectors

Vectors: Distance and DisplacementsTotal Distance

Total Displacement

5 steps south

5 steps north

10 steps west

10 Steps north

10 steps west

10 Steps north

5 steps south

5 steps north

5 steps south

10 steps west

5 steps north

10 Steps north

10 steps west

5 steps north

10 Steps north

5 steps south

5 steps south

10 Steps north

5 steps north

10 steps west

Vectors: Distance and DisplacementsTotal Distance

Total Displacement

5 steps south

5 steps north

10 steps west

10 Steps north

30 5

10 steps west

10 Steps north

5 steps south

5 steps north

30 5

5 steps south

10 steps west

5 steps north

10 Steps north

30 5

10 steps west

5 steps north

10 Steps north

5 steps south

30 5

5 steps south

10 Steps north

5 steps north

10 steps west

30 5

Free-body diagrams

SIMPLE EXERCISE

A book is at rest on a tabletop. Diagram the forces acting on the book.

SIMPLE EXERCISE

A book is at rest on a tabletop. Diagram the forces acting on the book.

As you watch… • Identify the forces acting on the cup.

Displacement and Force in Two DimensionsCopyright © McGraw-Hill Education

Displacement and Force in Two Dimensions

Follow UpMake a force diagram for each of the following situations:• The car is at rest.• The car accelerates and the cup remains on the roof.• The car accelerates and the cup falls off the roof.

Copyright © McGraw-Hill Education

Exercise on Free-body diagram

Lab Activity 2: Adding Vectors

ANSWER TO THE LAB ACTIVITY

• Does the sum of the forces measured by the two spring scales equal the weight of the hanging object? Is the sum greater than the weight? Less than the weight?

• The sum of the weight on both scales is apparently greater than the weight of the hanging object.

ANSWER TO THE LAB ACTIVITY

• Critical Thinking Draw an equilateral triangle, with one side vertical. If the two sides of the triangle are 2.0 N, explain the size of the third side. How is it possible that 2 N 2 N 2 N?

ANSWER TO THE LAB ACTIVITY• Critical Thinking Draw an equilateral triangle, with one

side vertical. If the two sides of the triangle are 2.0 N, explain the size of the third side. How is it possible that 2 N 2 N 2 N?

• Because vector addition accounts for both direction and magnitude, 2 N 2 N can equal 2 N if the two forces are added as vectors. One way to think about the net force acting on an object is to imagine replacing all of the individual forces with a single force that has the same effect. The purpose of vector resolution is to determine the magnitude and direction of that net force.

How do we add them?

For right angles

Quick exercise

Quick exercise

Using Trigonometry to Determine a Vector's Direction

MNEMONIC: SOH CAH TOA (to remember the sine, cosine and tangent functions)

Quick Exercise• Eric leaves the base camp and hikes 11 km,

north and then hikes 11 km east. Determine Eric's resulting direction of displacement.

Quick Exercise• Eric leaves the base camp and hikes 11 km,

north and then hikes 11 km east. Determine Eric's resulting displacement.

Lab Activity 1: Walking Vectors(Displacement)

Purpose: To let you experience one example of a vector.

Practical ApplicationGroup work: • Research on one practical application of vector in

real life. Be ready to share your output in class next meeting.

• Use of visual materials are highly encouraged (powerpoint, videos, animations, actual objects, etc.)

Practical Application of vectors• Architecture and Engineering: computing for forces so that the

building will be stable and will not collapse. Example: Isolation technique or ball bearings for earthquake proof buildings

http://imaginationstationtoledo.org/content/2011/03/can-you-build-an-earthquake-proof-building/

Practical Application of vectorsMeteorology• Wind direction• Typhoon formation• Tornado formation

Practical Application of vectorsNavigation

Practical Application of vectorsSports• Movements and direction of players/ball can be

represented by vectors or arrowsExample: A quarter pass

Practical Application of vectorsSports• a billiard game• with an understanding of vectors, billiards players can

predict where both balls will go following a collision, allowing them to sink more target balls while keeping the cue ball safely on the table.

Practical Application of vectors• Projectile motion of bombarding rockets• Flight testing

Lesson 2: Friction

Simple Demonstration1. Push the ice cube on the table. 2. Wipe the table with tissue to make it dry3. Push the eraser with the same force you applied

towards the ice cube.

A. What do you notice about the behavior of the two objects?B. Give possible reasons for their difference.C. What is the first thing that enters your mind when you hear

the word friction?FRICTION is a type of force between two touching objects.

TYPE OF FRICTION1. Kinetic or Dynamic Friction - force exerted on one surface by another when the two surfaces rub against each other because one or both surfaces are moving.2. Static Friction - the force exerted on one surface by another when there is no motion between the two surfaces. (You need to push harder)

TYPE OF FRICTION• An applied force is balanced by static friction

up to a maximum limit. When this limit is exceeded, the object begins to move.

On what does a friction force depend?

Example: Rubber shoes and concrete floorSocks and polished wood

How much Friction is there?

How is Coefficient of Kinetic Friction Determined?

Sample Problem

• A book is resting on the table. The coefficient of Kinetic friction between the table and the book is 13.5. If the book weighs 500 N, what horizontal force is needed to push the book at a constant speed?

Page 133

Critical Thinking

• Which one is better, a low friction between surface or high friction between surfaces?

- it depends on the situation. Low friction situations include pistons in engines and skis on snow. High friction is desired include erasers on paper or a flag twirler’s gloves gripping the flagpole.

Critical Thinking

• When it is good enough to have high coefficient of friction between two surfaces?

- running shoes on a track, sandpaper on a wood, car breaks on wheels, and car tires on roads.