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Vectors Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use: Force, displacement, velocity, and acceleration. A vector quantity is graphically represented by a vector.
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Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use: Force, displacement, velocity, and acceleration.

Jan 21, 2016

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Page 1: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

VectorsVectors

All vector quantities have magnitude (size) and

direction.

Vectors in physics that we will use:

Force, displacement, velocity, and acceleration.

A vector quantity is graphically represented by a

vector.

Page 2: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

An example is a force vector.

The label for the vector indicates the type of

vector.

Every vector has 4 parts:

is the point of application sometimes

referred to as the point of concurrency.

F is the label symbolizing the type of vector.

F•

Page 3: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

The arrowhead indicates the direction of the

vector.

The length of the vector indicates the magnitude of the vector.

Page 4: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Free-Body Diagrams (Force Free-Body Diagrams (Force Diagrams)Diagrams)

A free-body diagram sometimes called a force

diagram shows all the forces that are exerted on

a single object.

Usually a point is used to represent the object.

Page 5: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Examples of force diagrams:

Consistency is important!

Fw represents the weight of an object.

T represents the tension in a rope or string

and FN the normal (perpendicular) force.

•Fw

T

Fw

T

Fw

FN

Page 6: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Guidelines for Free-Body Guidelines for Free-Body DiagramsDiagrams

Only forces are to appear in diagrams.

All the forces exerted on the object but none of

those objects exerting those forces are included.

Each force is represented by a force vector.

The tail end of the arrow is placed at the point of application of the force.

The direction of the arrow indicates the direction of the force.

Page 7: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Forces in Free-Body DiagramsForces in Free-Body Diagrams

The following is a list of common forces found in

force diagrams.

Normal Force (FN)

For each surface in contact with an object,

there is a normal force which is perpendicular to the surfaces.

Page 8: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Frictional Force (Ff)

For each surface in contact with an object,

there is a normal force which is perpendicular to the surfaces.

When a surface is described as frictionless,

then Ff = 0 and is not included in the diagram.

Page 9: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Gravitational Force (Fw)

Be careful!

Even when an object is said to be in a state

of weightlessness, it still has weight.

Tension (T or FT)

For each string or rope, there is a tension.

If the string or rope is slack, then T = 0.

Page 10: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Air Resistance (FR)

Occurs when an object moves relative to the

air, i.e. a parachute.

If an object is a small dense object moving very quickly, this is usually ignored.

Net Force

The net or resultant force is the vector sum

of all the forces acting on the object.

Page 11: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

The net or resultant force is the vector sum

of all the forces acting on the object.

The net force is never drawn in the diagram!

It is the net force that determines the motion

of an object.

Page 12: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

More New VocabularyMore New Vocabulary

Forces are measured in units of newtons (N).

In dynamics, specifically Newton’s 2nd law,

we will more formally define a newton.

Right now, consider 1 N ≈ 1/5 pound.

A resultant force is the one force that combines

the effects of all the forces acting on an object.

Page 13: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

The direction in which a force acts is sometimes

referred to as bearing.

Sometimes instead of using N, E, S, and W to

designate the direction, an angular measure is

given.

000°

090°

180°

270° You start at 000° and goclockwise.

Page 14: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

A Typical Force ProblemA Typical Force Problem

Two soccer players kick a soccer ball at exactly

the same time. One player exerts a force of 55 N

north and the second player exerts a force of

75 N east. Mathematically determine themagnitude and direction of the resultant.

•F1

F2

FR

θ•

F1

F2

FRθ

Page 15: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

The force diagram can be drawn using theparallelogram method (left) or the triangle

method(right).

They give the same results and sometimes onemay be favored more than the other dependingon the application.

To solve the problem mathematically, we use the

pythagorean theorem, c2 = a2 + b2, which must be

written in terms of forces.

Page 16: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

To determine the direction (bearing), we use one

of the trigonometric functions which must bewritten in terms forces.

FR = (F12 + F2

2)1/2

FR = ((55 N)2 + (75 N)2)1/2 = 93 N

θ = sin-1

FR = 93 N, 36° N of E

F1

FR= sin-1 55 N

93 N=36°

Page 17: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Graphical SolutionGraphical Solution

We will not solve the problem graphically butrather just outline the steps.

Choose a scale of ? cm = ? N, where the ?’s

are chosen to conveniently draw the force

(vector) diagram.

The resultant vector can be measured in cm

from using either diagram.

Page 18: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

The resultant vector must have the same

point of application as the other vectors.

According to your scale, you convert the

number of cm to the number of N (if drawing

a force diagram).

The angle is measured with a protractor.

Page 19: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Velocity VectorsVelocity Vectors

A motorboat travels at 7.5 m/s W. The motorboat

heads straight across a river 120 m wide.

(a) If the river flows downstream at a rate of

3.2 m/s S, determine the resultant velocity of the boat.•θVB

VrVRVr =3.2 m/s S

VB =7.5 m/s W

Page 20: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

VR = (VB2 + Vr

2)1/2

VR = ((7.5 m/s)2 + (3.2 m/s)2)1/2 = 8.2 m/s

θ = sin-1

VR = 8.2 m/s, 23° S of W or 8.2 m/s, 247°

Vr

VR= =sin-1

3.2 m/s

8.2 m/s23°

Page 21: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

(b) How long does it take the boat to reach the

opposite side of the river?

Δs = 120 m VB = 7.5 m/s W

ΔsΔt

vave =

Δt = 120 m

7.5 m/s=16 s

Page 22: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

(c) How far downstream is the boat when it reaches the other side of the river?

vave = ΔsΔt

Δs=3.2 m/s

×16 s = 51 m

Page 23: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

The Resultant and EquilibrantThe Resultant and Equilibrant

One force of 17 N at a bearing of 090° actsconcurrently with a force of 11 N at a

bearing of180°.

(a) Determine the magnitude and bearing of the resultant.

F2 =11 N

F1 =17 NF1

F2FR

θ

Page 24: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

FR = (F12 + F2

2)1/2

FR = ((17 N)2 + (11 N2)1/2 = 20.0 N

θ = cos-1

FR = 20.0 N, 122°

F1

FR= =cos-1

17 N

20. N32°

Page 25: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

(b) Determine the magnitude and bearing of the

equilibrant.

The equilibrant is equal in magnitude and opposite in direction (180°) to the resultant.

Fnet = FR + Feq = 0 N

•F1

F2FR

θ

Feq

Feq =20.0 N, 302°

Page 26: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Resolution of ForcesResolution of Forces

Harry pushes a lawn mower along a level ground.

The handle of the lawn mower makes an angle of

20.0° with the ground as Harry pushes with a

force of 78.0 N.

(a) Determine the useful component of the force.

•θFH

FVFRθ=20.0°

FR =78.0 N

Page 27: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

.

cos θ

=FH

FR

×FH = FR cos θ

=78.0 N × 0.940 = 73.3 N

(b) Determine the wasted component of the force.

sin θ =FV

FR

FV = FR sin θ =78.0 N × 0.940 = 26.7 N ×

Page 28: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

The force diagram shows the lawn mower beingpushed from right to left. It could also be shownbeing pushed from left to right.

The results would be the same.

If a particular direction was given, it would have to appear that way in the problem.

The wasted component would be FV because thatcomponent of the force is trying to push the lawnmower straight into the ground.

Page 29: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Wrap Up QuestionsWrap Up Questions

Can the magnitude of a vector have a negative

value? Justify your answer.

Remember that vectors are measurablequantities that have a direction as well as amagnitude. Such quantities as forces,displacements, velocities, and accelerations

cannot have a magnitude below zero but can

point ina negative direction.

Page 30: Vectors All vector quantities have magnitude (size) and direction. Vectors in physics that we will use:  Force, displacement, velocity, and acceleration.

Two forces, F1 = 12.0 N and F2 = 4.0 N areconcurrent (acting on the same point).

(a) What is the minimum resultant?

The minimum resultant would be 8.0 N if the

angle between them was 180°.

(b) What is the maximum resultant?

The maximum resultant would be 16.0 N if the angle between them was 0°.