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Chapter 2LectureOutline

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Chapter 2: Force

•Forces

•Vector Addition

•Newton’s First and Third Laws

•Gravity

•Contact Forces

•Tension

•Fundamental Forces

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§2.1 Forces

Isaac Newton was the first to discover that the laws that govern motions on the Earth also applied to celestial bodies.

Over the next few chapters we will study how bodies interact with one another.

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Simply, a force is a “push” or “pull” on an object.

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How can a force be measured? One way is with a spring scale.

By hanging masses on a spring we find that the spring stretchapplied force.

The unit of force is the newton (N).

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Vectors versus scalars:

A vector is a quantity that has both a magnitude and a direction. A force is an example of a vector quantity.

A scalar is just a number (no direction). The mass of an object is an example of a scalar quantity.

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Notation:

Vector: F

or F

The magnitude of a vector: .For or

FF

Scalar: m (not bold face; no arrow)

The direction of vector might be “35 south of east”; “20 above the +x-axis”; or….

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§2.2 Graphical Vector Addition

To graphically represent a vector, draw a directed line segment.

The length of the line can be used to represent the vector’s length or magnitude.

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To add vectors graphically they must be placed “tip to tail”. The result (F1 + F2) points from the tail of the first vector to the tip of the second vector.

For collinear vectors:

F1

Fnet

F2

F1

Fnet

F2

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§2.3 Vector Addition Using Components

Vector Addition: Place the vectors tip to tail as before. A vector may be moved any way you please provided that you do not change its length nor rotate it. The resultant points from the tail of the first vector to the tip of the second (A+B).

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Example: Vector A has a length of 5.00 meters and points along the x-axis. Vector B has a length of 3.00 meters and points 120 from the +x-axis. Compute A+B (=C).

A x

y

B

120

C

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adj

opp

cos

sintan

hyp

adjcos

hyp

oppsin

m 50.160cosm00.360cos 60cos

m 60.260sinm00.360sin60sin

BBB

B

BBB

B

xx

yy

and Ax = 5.00 m and Ay = 0.00 m

A x

y

B

12060

By

Bx

Example continued:

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The components of C:

m 2.60m 2.60 m 00.0

m 3.50m 501 m 00.5

yyy

xxx

BAC

.BAC

x

y

C

Cx = 3.50 m

Cy = 2.60 m

The length of C is:

m 36.4

m 60.2m 50.3 22

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yx CCC C

The direction of C is:

6.367429.0tan

7429.0m 3.50

m 60.2tan

1

x

y

C

C

From the +x-axis

Example continued:

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§2.4 Newton’s First Law

Newton’s 1st Law (The Law of Inertia):

If no net force acts on an object, then its speed and direction of motion do not change.

Inertia is a measure of an object’s resistance to changes in its motion.

The net force is the vector sum of all the forces acting on a body.

321net FFFFFi

i

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If the object is at rest, it remains at rest (speed = 0).

If the object is in motion, it continues to move in a straight line with the same speed.

No force is required to keep a body in straight line motion when effects such as friction are negligible.

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An object is in translational equilibrium if the net force on it is zero.

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Free Body Diagrams:

•Must be drawn for problems when forces are involved.

•Must be large so that they are readable.

•Draw an idealization of the body in question (a dot, a box,…). You will need one free body diagram for each body in the problem that will provide useful information for you to solve the given problem.

•Indicate only the forces acting on the body. Label the forces appropriately. Do not include the forces that this body exerts on any other body.

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Free Body Diagrams (continued):

•A coordinate system is a must.

•Do not include fictitious forces. Remember that ma is itself not a force!

•You may indicate the direction of the body’s acceleration or direction of motion if you wish, but it must be done well off to the side of the free body diagram.

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§2.5 Newton’s Third Law

Newton’s 3rd Law:

When 2 bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction. Or, forces come in pairs.

Mathematically: 1221 FF

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Example: Consider a box resting on a table.

F1

(a) If F1 is the force of the Earth on the box, what is the interaction partner of this force?

The force of the box on the Earth.

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F2(b) If F2 is the force of the box on the table, what is the interaction partner of this force?

Example continued:

The force of the table on the box.

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External forces:

Any force on a system from a body outside of the system.

FPulling a box across the floor

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Internal forces:

Force between bodies of a system.

Fext

Pulling 2 boxes across the floor where the two boxes are attached to each other by a rope.

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§2.6 Gravity

221

r

MGMF

Gravity is the force between two masses. Gravity is a long-range or field force. No contact is needed between the bodies. The force of gravity is always attractive!

r is the distance between the two masses M1 and M2 and G = 6.6710-11 Nm2/kg2.

M2

r

M1F21 F12

1221 FF

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Let M1 = mass of the Earth. 22M

r

GMF E

Here F = the force the Earth exerts on mass M2. This is the force known as weight, w.

.222 gMMr

GMw

E

E

N/kg 8.9 where2

E

E

r

GMg Near the surface

of the Earth

km 6370

kg 1097.5

E

24E

r

M

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What is the direction of g?

Note thatm

Fg is the gravitational force per unit mass.

This is called the gravitational field strength. It is often referred to as the acceleration due to gravity.

What is the direction of w?

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Example: What is the weight of a 100 kg astronaut on the surface of the Earth (force of the Earth on the astronaut)? How about in low Earth orbit? This is an orbit about 300 km above the surface of the Earth.

On Earth: N 980mgw

In low Earth orbit: N 890)( 2

oE

Eo

rR

GMmrmgw

Their weight is reduced by about 10%. The astronaut is NOT weightless!

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§2.7 Contact Forces

Contact forces: these forces arise because of an interaction between the atoms in the surfaces in contact.

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Normal force: this force acts in the direction perpendicular to the contact surface.

Normal force of the ground on the box

Normal force of the ramp on the box

N

w N

w

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Example: Consider a box on a table.

FBD for box

mgwN

wNFy

that So

0

This just says the magnitude of the normal force equals the magnitude of the weight; they are not Newton’s third law interaction partners.

Apply Newton’s 2nd law

N

w

x

y

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Friction: a contact force parallel to the contact surfaces.

Static friction acts to prevent objects from sliding.

Kinetic friction acts to make sliding objects slow down.

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Static Friction:

.ss Nf The force of static friction is modeled as

where s is the coefficient of static friction and N is the normal force.

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Kinetic Friction:

.kk Nf The force of kinetic friction is modeled as

where k is the coefficient of kinetic friction and N is the normal force.

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Example (text problem 2.97): A box full of books rests on a wooden floor. The normal force the floor exerts on the box is 250 N.

(a) You push horizontally on the box with a force of 120 N, but it refuses to budge. What can you say about the coefficient of friction between the box and the floor?

FBD for box

N

w

x

y

F

fs

0)2(

0)1(

sx

y

fFF

wNFApply Newton’s 2nd Law

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Example continued:

From (2): 48.0 N

FNfF sss

This is the minimum value of s, so s > 0.48.

(b) If you must push horizontally on the box with 150 N force to start it sliding, what is the coefficient of static friction?

Again from (2): 60.0 N

FNfF sss

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(c) Once the box is sliding, you only have to push with a force of 120 N to keep it sliding. What is the coefficient of kinetic friction?

FBD for box

N

w

x

y

F

fk

0)2(

0)1(

kx

y

fFF

wNFApply Newton’s 2nd Law

From 2:

48.0N 250

N 120

k

N

F

NfF kk

Example continued:

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Consider a box of mass m that is at rest on an incline. Its FBD is:

There is one long-range force acting on the box: gravity.

There is one contact force acting on the box from the ramp.

If the net force acting on the box is zero, then the contact force from the ramp must have the same magnitude as the weight force, but be in the opposite direction.

FRB

w x

y

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The force FRB can be resolved into components that are perpendicular and parallel to the ramp.

The perpendicular component is what we call the normal force.

The parallel component is the static friction force.

FRB

w

N

fs

x

y

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Example: Let the box on the ramp have a mass 2.5 kg. If the angle between the incline and the horizontal is 25, what are the magnitudes of the weight force, normal force, and static friction force acting on the box?

FRB

w

N

fs

x

y

0sin

0cos

sx

y

fwF

wNF

Apply Newton’s 2nd Law

N 31.925sinN 5.24sin

N 2.2225cosN 5.24cos

N 5.24N/kg 8.9kg 5.2

wf

wN

mgw

s

The forces are:

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An ideal cord has zero mass, does not stretch, and the tension is the same throughout the cord.

This is the force transmitted through a “rope” from one end to the other.

§2.8 Tension

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Example (text problem 2.79): A pulley is hung from the ceiling by a rope. A block of mass M is suspended by another rope that passes over the pulley and is attached to the wall. The rope fastened to the wall makes a right angle with the wall. Neglect the masses of the rope and the pulley. Find the tension in the rope from which the pulley hangs and the angle .

FDB for the mass M

x

w

T

y

MgwT

wTFy

0Apply Newton’s 2nd Law to the mass M.

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FBD for the pulley:

x

y

T

TF

0sin

0cos

TFF

TFF

y

x

Apply Newton’s 2nd Law:

sincos FFT

This statement is true only when = 45 and

MgTF 22

Example continued:

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§2.9 Fundamental Forces

The four fundamental forces of nature are:

•Gravity which is the force between two masses; it is the weakest of the four.

•Strong Force which helps to bind atomic nuclei together; it is the strongest of the four.

•Weak Force plays a role in some nuclear reactions.

•Electromagnetic is the force that acts between charged particles.

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Summary

•Newton’s First and Third Law’s

•Free Body Diagrams

•Adding Vectors

•Contact Forces Versus Long-Range Forces

•Different Forces (friction, gravity, normal, tension)

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What is the net force acting on the object shown below?

15 N 15 N

10 N

x

y

a. 40 N

b. 0 N

c. 10 N down

d. 10 N up

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wm

w

wm

wm

6

1 ,

6

1 .d

6

1 m, .c

,6

1 .b

, .a

The gravitational field strength of the Moon is about 1/6 that of Earth. If the mass and weight of an astronaut, as measured on Earth, are m and w respectively, what will they be on the Moon?