CH 6: Forces and Motion

CH 6

How can we distinguish between energy, force and motion?

Big Ideas:

How are energy, force and motion related?

What role do energy and force play in motion?

falling rhino

falling rhino slip and slide

Consider the energy of, and forces on, falling objects:

physics of falling

What is the cause of motion?

Do forces always involve two objects?

Do forces always require contact?

Can forces involve more than 2 objects?

Forces are associated with motion; they allow us to move and stop moving.

A force is always an interaction between two objects, energy is in an object.

More simply: a force is an interaction between 2 (or more objects) that influences the motion between the objects.

In physics, a force is any influence that

causes an object to undergo a change in

speed, a change in direction, or a change

in shape.

In other words, a force is that which can cause an object

with mass to change its velocity, i.e., to accelerate, or

which can cause a flexible object to deform.

A Push Or

Pull

What is a force?

Recognizing forces and motion interactions

Which of the following are generally true of force and motion?

♦ come up with a confirming example – where it is true

♦ can you find at least one single example that shows it false?

• A net force it will speed up.

• A net force in the same direction it is moving it will

speed up.

• An increasing net force it will speed up.

• A net force in a direction opposite its motion it will

slow down.

• A decreasing net force it will slow down.

• No net force it must be at rest.

• No net force it will slow down.

• No net force it will continue moving at constant speed

(or remain stationary).

A force has both magnitude and direction, making it a vector quantity.

quick vocab

system: the object

environment: the “world” around

the object exerting

force on it

A force represents the interaction of an object

(system) with its environment (the forces exerted

on the system) by an agent (the cause of the force)

A force causes CHANGE in the MOTION of an object.

quick review of forces

all forces have agents* and objects**

*A specific and identifiable cause.

These can be animate or inanimate –

if you can’t name an agent – it’s not a force!

Ex: gravity – earth’s mass Ex: friction – microscopic roughness of

surfaces

**the victim of the force.

two main types of forces

• contact

• long-range

(at-a-distance/field)

tension applied spring

friction air resistance

normal (support) – perpendicular to

a surface

Nuclear (strong)

magnetic

electrical

Gravitational

(weight)

strongest

Range (meters)

Strong 2x10-15

Electromagnetic infinity

Weak 10-18

Graviational infinity

hank green strong force

strong force part 2

gravitational force

How Does an External Force Affect Speed and Direction?

Check your Understanding

Forces and Engineering

ex: designed for cornering

suspension design

10 tallest buildings in the world

421 meters (1,380 ft)

Jin Mao tower,

China, 1998 #10

#9 Trump Tower,

Chicago 2009

423 meters (1,388 ft)

Guangzhou International Finance Center,

China, 2010 #8

440.2 meters tall (1,444 ft)

#6

Nanjing Greenland Financial Center,

China, 2010

450 meters (1,480 ft)

#5

Petronas Towers, Malaysia,

1998

451 meters or 1,482 ft

#7 Willis Tower,

Chicago, 1974

527 meters (1,730 ft)

International Commerce Center,

Hong Kong, 2010

484 meters (1587 ft)

#4

#3

Shanghai World Financial Center,

China, 2008

494.4 m (1,622.0 ft)

#2

509.2 m (1,670.6 ft)

Taipei 101, Taiwan, 2004

World Trade Center, New York,

1970 (N. tower) 1972 (S. tower)

In order to create the 16-acre World Trade Center site, five streets were closed off and

164 buildings were demolished. Construction required the excavation of more than 1.2

million cubic yards of earth, which was used to create 23.5 acres of land along the

Hudson River, now part of Battery Park City in lower Manhattan. During peak

construction periods, 3,500 people worked at the site. A total of 10,000 people worked

on the towers; 60 died during its construction.

• Became the world’s tallest buildings • Had its own zip code • Each tower – 110 floors • 1,368 ft.

In 1993 terrorists drove a truck packed with 1,100 lbs of explosives into the

basement parking garage at the World Trade Center. The blast left a crater 22 ft

wide and 5 stories deep—only 6 people were killed and 1,000 injured. The towers

were repaired, cleaned, and reopened in less than a month.

Not the first attack…

In Feb. 2003, architect Daniel

Libeskind's design was

chosen for rebuilding the 16-

acre site of the former World

Trade Center. 1 World Trade,

which will be a symbolic

1,776 feet tall from the

ground to the top of its spire

is scheduled to open in 2013

(making it the 2nd tallest

building in the world).

#1 tallest building in the world (completed 2010)

The tallest man made structure sits in the heart of UAE, Dubai. Its former name was Burj

Dubai but was changed when the president of Abu Dhabi (Khalifa Bin Zayed) loaned

Dubai $10 billion since Dubai was drastically effected by the recent recession. The name

was changed in his honor. Getting and office would set you back a staggering $ 43,000

per sq m.

• Height: 828 m (2,717 ft)

• Floors: 163

• Completion: 2010

• Floor area: 517,240 m sq

• Estimated cost: USD $ 1.5 billion

• Designed by: Adrian Smith

Burj Khalifa, Dubai United Arab Emirates

SKYSCRAPER CHALLENGE Build it Big

http://www.engineeringinteract.org/resourc

es/parkworldplot/flash/concepts/allaboutfor

ces.htm

Newton explained how forces influence motion.

youtube

Newton’s 1st Law of Motion

Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

Galileo’s “Law of Inertia”.

How are force, mass and acceleration related?

Want to know the

secret to the

Force….

youngling,

younglings gather

round…

use equations

you must

How does acceleration depend on force?

How does acceleration depend on the object?

Can you write a rule from your PHET experimentation?

The relationship between an object’s mass m,

its acceleration a,

and the applied force F, is

m F = a

Newton’s 2nd Law of Motion

The relationship between an object's mass m, its

acceleration a, and the applied force F is F = ma.

Acceleration and force are vectors; in this law the direction of the force vector is the same as the direction of the acceleration vector.

The original version: When a force acts on an object, the object accelerates in the direction of the force. If the mass of an object is held constant, increasing force will increase acceleration. If the force on an object remains constant, increasing mass will decrease acceleration. In other words, force and acceleration are directly proportional, while mass and acceleration are inversely proportional.

F = ma – a quick history

Technically, Newton equated force to the differential change

in momentum per unit time. Momentum is a characteristic of

a moving body determined by the product of the body's

mass and velocity.

To determine the differential change in momentum per unit

time, Newton developed a new type of math -- differential

calculus. His original equation looked something like this:

F = (m)(Δv/Δt)

Because acceleration is defined as the instantaneous

change in velocity in an instant of time (Δv/Δt), the equation

is often rewritten as:

F = ma How Stuff Works

This equation form of Newton's second law allows us to specify a

unit of measurement for force. Because the standard unit of mass is

the kilogram (kg) and the standard unit of acceleration is meters per

second squared (m/s2), the unit for force must be a product of the

two -- (kg)(m/s2). This is a little awkward, so scientists decided to use

a Newton as the official unit of force. One Newton, or N, is

equivalent to 1 kilogram-meter per second squared. There are 4.448

N in 1 pound.

How Stuff Works

F = ma lets you quantify

motion of every variety

Let's say, for example,

you want to calculate

the acceleration of the

dog sled shown here.

modify the force equation to get a = F/m. When you plug in the numbers for force (100

N) and mass (50 kg), you find that the acceleration is 2 m/s2.

Units

Fnet = m a

1 N = 1 kg m/s2

The SI unit of force is the Newton*.

A Newton is about a quarter pound.

1 lb = 4.45 N

*Amount of force acting on a 1 kg mass to produce acceleration of 1 m/s/s)

Now let's say that the mass of the sled stays at 50 kg and that

another dog is added to the team. If we assume the second dog

pulls with the same force as the first (100 N), the total force would be

200 N and the acceleration would be 4 m/s2

Notice that doubling the force by adding another dog doubles the acceleration. Oppositely,

doubling the mass to 100 kg would halve the acceleration to 2 m/s2.

Finally, let’s imagine that a second dog team is attached to the sled

so that it can pull in the opposite direction.

If two dogs are on each side, then the total force pulling to the left (200 N) balances

the total force pulling to the right (200 N). That means the net force on the sled is zero,

so the sled doesn’t move.

Newton's second law is concerned with net forces.

We could rewrite the law to say: When a net force acts on an object, the

object accelerates in the direction of the net force.

Now imagine that one of the dogs on the left breaks free and runs away.

Suddenly, the force pulling to the right is larger than the force pulling to the

left, so the sled accelerates to the right.

What's not so obvious is that the sled is also

applying a force on the dogs. In other words, all

forces act in pairs. This is Newton's third law…. How Stuff Works

What is Net Force?

When more than one

force acts on a body,

the net force

(resultant force) is the

vector combination of

all the forces, i.e., the

“net effect.”

F1

F2

F3

Fnet

Net Force & the 2nd Law For a while, we’ll only deal with forces that are

horizontal or vertical.

When forces act in the same line, we can just

add or subtract their magnitudes to find the

net force.

2 kg

15 N 32 N

Fnet = 27 N to the right

10 N

a = 13.5 m/s2

? What is the acceleration of the object ?

A karate chop can break a 3.8 cm thick concrete block by moving the hand at 11 m/s, creating 3069 N force.

The bones of the hand can

withstand 40X that force.

check your understanding…

Let’s assume that the wheels of a 5-kg car apply 10 N of

force. What is the net force if friction and drag are

negligible?

Mass = 5 kg

The net force would equal 10 Newtons.

What is the acceleration of the car?

F = ma 10 = 5a

acceleration = 2m/s2

What is the net force if the wheels of the

5-kg car apply 10 Newtons but a 1-kg

parachute applies 3 Newtons in the

other direction?

The net force would equal 3 N

(the total mass = 6 kg)

What is the acceleration of the car?

a = F/m a = 3/6

acceleration = 0.5 m/s2

A rocket is added to the car and

applies an additional force of 10

Newtons. The wheels still apply 10

N. What is the net force if the

parachute continues to apply 7

Newtons in the other direction?

The total mass of the car, rocket

and parachute is 10 kg.

The net force would equal 13 Newtons. The total mass = 10 kg.

What is the acceleration of the car?

a = F/m a = 13/10

acceleration = 1.3 m/s2

Newton’s 3rd Law of Motion

For every action, there is an equal and opposite reaction.

OR “The mutual actions of two bodies upon

each other are always equal, and directed to

contrary parts”.

An example from nature…

The size of the force on the water equals the size of the force on the fish; the

direction of the force on the water (backwards) is opposite the direction of the

force on the fish (forwards). For every action, there is an equal (in size) and

opposite (in direction) reaction force.

Consider the propulsion of a fish through the water.

A fish uses its fins to push

water backwards. But a

push on the water only

accelerates the water.

Since forces result from

mutual interactions, the

water must also be pushing

the fish forwards, propelling

the fish through the water.

I love

Newton’s 3rd

Law

Check your understanding:

While driving down the road, a firefly

strikes the windshield of a bus in front

of the face of the driver. This is a clear

case of Newton's third law of motion.

The firefly hit the bus and the bus hits

the firefly. Which of the two forces is

greater: the force on the firefly or the

force on the bus?

Practice problems:

#2-6, pg. 122

how strong is your force?

NEWTON’S 3 LAWS OF MOTION

A closer look

A force represents the interaction of an object

(system) with its environment (where the force is

being exerted on the system) by an agent (the cause

of the force)

A net force causes CHANGE in the MOTION of an object.

Newton’s 1st Law

Galileo (1630’s) concluded “it is not the nature of an

object to stop once it is set in motion; rather, it is an

object’s nature to maintain its state of motion.”

Newton (1687) further developed this

concept to become his First Law of Motion.

“An object in motion tends to stay in motion;

an object at rest tends to stay at rest unless it

experiences a net external force.”

jocks on physics

Now for something more

“serious”….

1st Law

• A moving body will continue moving in the same direction with the constant speed until some net force acts on it.

• A body at rest will remain at rest unless a net force acts on it.

• Summing it up: It takes a net force to change a body’s velocity.

Inertia Example 1

An astronaut in

outer space will

continue drifting

in the same

direction at the

same speed

indefinitely, until

acted upon by an

outside force.

Inertia Example 2 If you’re driving at 65 mph and have an

accident, your car may come to a stop in

an instant, while your body is still moving

at 65 mph. Without a seatbelt, your inertia

could carry you through the windshield.

1st Law: Law of Inertia

The tendency of an object not to

accelerate.

“In the absences of forces, a body will preserve its state of

motion.”

The egg drop

challenge

“When the net external force on an object is zero, its acceleration (or the change in its motion) is zero.”

Myth Busters

Determining Net Force

When more than one

force acts on a body,

the net force

(resultant force) is the

vector combination of

all the forces, i.e., the

“net effect.”

F1

F2

F3

Fnet

consider…

Fresistance

vector Fforward represents forward force of car, Fresistance acts in opposite direction,

backward force air exerts on car to resist motion. The vector Fgravity represents

downward force of gravity and vector Fnormal, represents the upward force the ground

exerts on the car.

Fforward

Fgravity

All 4 forces are external forces acting on the car. The net external

force* is the vector sum of all the forces acting on a body. Defined as

“the total force resulting from a combination of external forces on an

object”.

*sometimes called the resultant force, found using methods for finding resultant vectors.

consider…

If both teams pull on the rope with equal but opposite force, the

net external force is zero. A knot in the center would remain at

rest even though forces are pulling in opposite directions.

If one side increases the force, the knot experiences a net

external force equal to the difference in the forces – it moves in

the direction of the greater pull.

1st law – inertia is an object’s resistance to change in

velocity

The force that brings an accelerating object to equilibrium must be = and opposite to the force causing the acceleration.

An object is in

equilibrium if it

is at rest OR

moving at

constant velocity

Fnet = 0

mass and inertia

The inertia of an object is directly proportional to its mass.

From 1st law: an object with no net external

forces is in a state of equilibrium.

How much does a

known force affect the

motion of an object?

Newton’s 2nd Law of Motion

F=ma

more free body diagram practice

practice problems 7-11, pg. 124

worksheet

review what we know so far

common misconceptions

pg. 125

• When a ball has been thrown, the force of the

hand that threw it remains on it.

• A force is needed to keep an object moving.

• Inertia is a force.

• Air does not exert a force.

• The quantity ma is a force.

Graph of F vs. a

In the lab various known forces are applied—

one at a time, to the same mass—and the

corresponding accelerations are measured.

The data are plotted. Since F and a are

directly proportional, the relationship is linear.

F

a

What does the slope of the line represent?

Slope

F

a

Since slope = rise / run = F / a, the slope is

equal to the mass.

(Or, think of y = m x + b, (like in algebra). y corresponds

to force, m to mass, x to acceleration, and b (the y-

intercept) is zero.)

F

a

2nd Law: Fnet = m a • The acceleration an object undergoes is directly

proportion to the net force acting on it.

• Mass is the constant of proportionality.

• For a given mass, if Fnet doubles, triples, etc. in size, so does a.

• For a given Fnet if m doubles, a is cut in half.

• Fnet and a are vectors; m is a scalar.

• Fnet and a always point in the same direction.

• The 1st law is really a special case of the 2nd law (if net force is zero, so is acceleration).

Net Force & the 2nd Law dealing with forces that are horizontal or vertical is straight

forward:

When forces act in the same line, we can just add

or subtract their magnitudes to find the net force.

2 kg

15 N 32 N

Fnet = 27 N to the right

10 N

a = 13.5 m/s2

? What is the acceleration of the object ?

What about when they are not?

1st law – inertia (if net force is 0 so is acceleration)

2nd law – the acceleration of an object is proportional to the

net force acting on it

The force that brings an accelerating object to equilibrium must be = and opposite to the force causing the acceleration.

(applying Newton’s 3rd law)

An object is in

equilibrium if it is at

rest OR moving at

constant velocity

Fnet = 0

Experimenting with Newton’s 2nd Law

Newton’s 3rd Law For every action, there is an equal and opposite reaction.

You know: Force is exerted on an object when that object interacts with some other object.

Forces always act in pairs.

“Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force

on the first object.

In other words: if there is a Force F on the surface of

an object such as the golden ball above, then there is

an equal and opposite force exerted back at the same

point. This equal and opposite force is often given by

negative F (-F).

when the boxer kicks the golden ball

with a force F, there is an equal and

opposite force (-F) being applied back

at his foot.

• If you hit a tennis ball with a racquet, the force on the ball due to the racquet is the same as the force on the racquet due to the ball, except in the opposite direction.

• If you catch a ball, the force of the ball on the glove is the same as the force of the glove on the ball.

Examples:

Action - Reaction • If you drop an apple, the Earth pulls on the

apple just as hard as the apple pulls on the Earth.

• If you fire a gun, the bullet pushes the gun backwards just as hard as the gun pushes the bullet forwards.

“For every action there’s an

equal but opposite reaction.”

Action/Reaction on different masses

Same Force

More Inertia

Same Force

Less Inertia

Earth / Apple How could the forces on the tennis ball, apple, and bullet, be

the same as on the racquet, Earth, and rifle? The 3rd Law

says they must be, the effects are different because of

the 2nd Law!

Earth

apple

3.92 N

3.92 N

0.40 kg

5.98 1024 kg

A 0.40 kg apple weighs 3.92 N

(W = mg). The apple’s weight is

Earth’s force on it. The apple

pulls back just as hard. So, the

same force acts on both bodies.

Since their masses are different,

so are their accelerations (2nd

Law). The Earth’s mass is so

big, it’s acceleration is negligible.

Earth / Apple (cont.)

a = m m a

Apple’s big

acceleration

Apple’s

little mass Earth’s little

acceleration

Earth’s

big mass

The products are the same, since the forces are the same.

Demolition Derby

When two cars of

different sizes collide,

the forces on each are

the SAME (but in

opposite directions).

However, the same

force on a smaller car

means a bigger

acceleration!

Swimming Due to the 3rd Law, when you swim you push the water

(red), and it pushes you back just as hard (blue) in the

forward direction. The water around your body also

produces a drag force (green) on you, pushing you in the

backward direction. If the green and blue cancel out, you

don’t accelerate (2nd Law) and maintain a constant velocity.

Note: The red vector is a force on the water, not the on

swimmer! Only the green and blue vectors act on the swimmer.

action-reaction pairs

The two forces

act at the same

time. Don’t necessarily

result in

equilibrium (can be

unbalanced).

Force of the

wood on the nail

Force of hammer

on the nail

ACTION: She’s pushing the

wall

REACTION: The wall

pushes back.

Will she accelerate?

YES! The force from wall

will push her to the left.

What is the ACTION-REACTION pair here?

Newton’s 3rd Law

How you touch the world is how the world touches you.

(you cannot touch without being touched)

question this…

Could a sky diver fall with enough force to launch a

child off a see-saw 7 stories into the air?

Identify the action-reaction pairs in the following:

a. A person takes a step

b. A snowball hits someone in the back

c. A person rowing a boat

Person pushes on ground, ground pushes on the person.

Snowball on the back, back on the snowball.

Oar in the water, water on the oar.

Quick review: forces song

A 6.0 kg object undergoes an acceleration of

2.0 m/s2.

a. What is the magnitude of the net external

force acting on it?

b. If this same force is applied to a 4.0 kg

object, what acceleration is produced?

12 N F=ma F = (6.0)(2)

3 m/s2 F=ma 12 = (4)(x)

A phonebook is resting on the table. We know gravity is pulling on it with 30 N, what must be the magnitude of the table’s force (FN) holding up the phonebook?

FNet = FN - Fg

= (30 N)(9.8 m/s2) = 293 N

A child pulls a wagon with a horizontal force, causing it

to accelerate. Draw a free body diagram showing the

acceleration of the wagon.

F pulling

F friction

The forces on a sailboat are 390 N north and 180 N east.

If the boat and crew has a mass of 270 kg, what are the

magnitude and direction of their acceleration?

270 kg

180 N

390 N

Fnet

θ

Θ = tan-1 390 = 65°

180

Fnet = √(390)2 + (180)2

= 429.5

a = 429.5 = 1.6 m/s2

270

1.6 m/s2 at 65° N of E

WEIGHT GRAVITY and ACCELERATION

fearofphysics.com

WEIGHT THE MAGNITUDE OF THE FORCE OF GRAVITY ON AN OBJECT

Remember that “scientifically

speaking”, mass isn’t related to size.

Mass is related to how much an object

resists changes to its state of motion.

Weight is caused by gravitational acceleration.

Gravitational force

The equation for gravitational force is:

FG = Mass x Gravity = mg. Or W = mg

The value of “g” (the strength of the gravitational field) is unique to each planet.

While g here on Earth may be ~10 m/s2, on Jupiter, g ~25 m/s2 and on the Moon,

g is only ~ 1.6 m/s2.

All masses near Earth feel a

gravitational force proportional

to their mass: the bigger the

mass, the bigger the

gravitational force.

W = mg

• Weight = mass acceleration due to gravity.

• This follows directly from F = m a.

Near the surface of the

Earth,

g = 9.8 m/s2.

Fg = mg

Gravitational force is associated with acceleration

in the direction of that force.

Simply put, an object subject to a gravitation force will

“fall” in the same direction in which the force is acting.

Weight will change based on whether there

are forces acting that increase or

reduce the “upward” force

necessary to balance out the

“downward” gravitational force;

for instance the buoyant force that

helps an object float in water,

making it “weightless”.

The force it takes to counteract and balance out Fg is the object’s

“weight.”

Weight = mass acceleration due to gravity

If an object is pushed upwards so that it accelerates up, it’s

weight on the surface pushing it up will increase.

Alternatively, if the object is allowed to fall, it’s weight will be

reduced.

when you’re in an elevator your weight changes because you are experiencing

an upward or downward acceleration, so your weight does not completely offset

the gravitational force.

Another example: an object’s weight changes due to it’s acceleration

relative to the gravitational field.

your weight is the net force

required to counteract the

downward force such that you

experience a certain

acceleration, and that value

can change even though the

gravitational force remains the

same

do the math…

Example problem pg. 128

Your mass is 75 kg. You stand on a bathroom scale in an

elevator going UP. Starting from rest, the elevator

accelerates at 2.0 m/s2 for 2.0 s, then continues at a

constant speed. What is the scale reading (your weight)

during the acceleration?

Draw the free body diagram.

Which direction is the net force?

Calculate…..

What do you know? m = 75 kg a = +2.0 m/s2

Fnet is sum of the Fscale on you and the Fgravity acting in the opposite direction Fnet = Fscale - Fgravity

Fscale = Fup + Fgravity Fscale = ma + mg Fscale = m(a + g) = 75(2.0 + 9.8) = 890 N

In a lecture delivered in Kyoto in 1922,

Einstein described that moment of

epiphany he had in 1907: "I was

sitting in the patent office in

Bern when all of a sudden a

thought occurred to me: If a

person falls freely, he won't

feel his own weight. I was

startled. This simple thought

made a deep impression on

me. It impelled me toward a

theory of gravitation."

When astronauts are in the space station, their mass is the same

as it is on Earth.

The gravitational force on the space station is only slightly less

than the gravitational force on Earth.

The space station, and everything in it, is subject to Earth’s gravity

(that’s what keeps it in orbit).

Because the station (and

everything on it) moves together

around Earth, the space station

and its contents are constantly

Falling towards Earth; they are

In free fall.

Weightlessness

The station and its contents are

weightless since no force is

exerted to counterbalance the

gravitational force.

Based on what it means for

something to have weight, this

explains why – despite having

mass and despite being subject

to a gravitational force – the

astronauts are weightless.

They never fall to Earth, since the curvature of Earth

exactly matches the shape of the orbit, but they are

constantly falling, nonetheless.

NASA - mass vs weight intro

Demonstrating acceleration based on mass in microgravity.

Accelerating Mass - NASA

Tools such as hammers, and levers are designed to make work easier.

Sometimes, the environment in which a person works dictates whether or

not a certain tool will get the job done. In the microgravity environment of

space, astronauts are often required to do experiments or simple repairs.

On Earth, gravity helps anchor a person to the ground, making tightening a

bolt a rather simple activity. Engineers have had to develop different types

of bracing systems — a whole new set of tools — to assist astronauts

working in space.

Using Gravity

Using tools in space

Forces and the motion they create are fundamental concepts in the fields of civil,

materials and mechanical engineering. Engineers must account for forces when

designing roadways and buildings, since these structures must not fail under their

own weight of construction materials, the weight of people and equipment on them,

and environmental loads (vibrations, wind drag, etc.).

Galloping Gertie, Tacoma, Nov. 7, 1940

galloping gertie

lateral forces

Lateral forces are those directed at the side of the

structure. These forces include those generated by

things such as the wind, earthquakes, and

explosions.

As a system, a structure must be designed so that it can resist all forces

to which it is subjected.

Great Science Project Ideas, J. VanCleave

tension gravity forces (weight)

compression

a force that pushes materials together

a force that pulls materials apart

the weight of the bridge as well as the weight of the car causes the beam to

bend. The top edge of the beam has shortened because the compression

forces, squeezing the materials together. The bottom edge of the beam has

lengthened due to tension forces, stretching the material.

Stress and Strain

The term stress (s) is used to express the loading in terms of force applied

to a certain cross-sectional area of an object.

From the perspective of loading, stress is the applied force or system of

forces that tends to deform a body.

From the perspective of what is happening within a material, stress is the

internal distribution of forces within a body that balance and react to the

loads applied to it.

The stress distribution may or may not be uniform, depending on the nature

of the loading condition.

Stress is often represented as a vector quantity for many engineering

calculations and for material property determination. For example, the

stress in an axially loaded bar is simply equal to the applied force divided

by the bar's cross-sectional area.

Some common measurements of stress are:

Psi = lbs/in2 (pounds per square inch)

ksi or kpsi = kilopounds/in2 (one thousand or 103 pounds per square

inch)

Pa = N/m 2 (Pascals or Newtons per square meter)

kPa = Kilopascals (one thousand or 103 Newtons per square meter)

is the response of a system (object) to an applied stress.

When a material is loaded with a force, it produces a stress, which then

causes a material to deform.

Engineering strain is defined as the amount of deformation in the direction

of the applied force divided by the initial length of the material.

For example, the strain in a bar that is being stretched in tension is the

amount of elongation or change in length divided by its original length.

Strain

If the stress is small, the material may only strain a small amount and

the material will return to its original size after the stress is released.

This is called elastic deformation, because like elastic it returns to its

unstressed state.

If a material is loaded beyond it elastic limit, the material will remain in

a deformed condition after the load is removed. This is called plastic

deformation.

Why giant moles can’t exist

the mole people

Biophysics

F

e

m

u

r

The strength of a bone, like a femur, is proportional to

its cross-sectional area, A. But an animal’s weight is

proportional to its volume.

Giant ants and moles from sci-fi movies couldn’t exist

because they’d crush themselves!

Here’s why: Suppose all dimensions

are increased by a factor of 10. Then

the volume (and hence the weight)

becomes 1000 times bigger, but the

area (and hence the strength) only

becomes 100 times bigger.

A real life example:

Basketball players, because of their

height, tend to suffer many stress

fractures;

and elephants have

evolved proportionally

bigger femurs than deer.

THE ATWOOD MACHINE

The Atwood machine invented in 1784 by Rev. George Atwood as a laboratory experiment to verify the mechanical laws of motion with constant acceleration.

The ideal Atwood Machine consists of two

objects of mass m1 and m2, connected by

an inextensible massless string over an

ideal massless pulley.

When m1 = m2, the machine is in neutral

equilibrium regardless of the position of the

weights.

When m1 ≠ m2 both masses experience

uniform acceleration.

Solving Atwood Machine Problems

requires that you calculate the

acceleration of the system of

weights.

Using Newton’s 2nd law: F = mass X acceleration.

The difficulty of Atwood machine

problems lies in determining the

tension force on the string.

m1

FT

Fg

heavier

weight

lighter

weight

Both weights have

tension forces pulling up.

Both weights have gravity

forces pulling down.

The force of gravity is equal to the mass ("m1" for weight 1 and "m2" for weight 2)

of the weight times "g" (equal to 9.8). Therefore, the gravitational force on the

lighter weight is m1*g, and on the heavier weight is m2*g.

m1

m2

Calculate the net force acting on the lighter weight:

Fnet = FT – (m1*g)

(since they pull in opposite directions)

m1

FT

Fg

Calculate the net force acting on the heavier weight:

Fnet = (m2*g) - FT

On this side, tension is subtracted because the direction of tension

is opposite on opposite sides of the pulley. This makes sense if

you consider the weights and string laid out horizontally -- the

tension pulls in opposite directions.

m2

FT

Fg

Substitute (FT - m1*g) in for the net force in the equation

Fnet = ma (Newton's 2nd law)

FT - m1*g = m1*a or Tension = m1*g + m1*a.

Substitute this equation into the equation for m2 : Fnet = (m2*g) - FT

Fnet = (m2*g) - (m1*g + m1*a)

By Newton's 2nd law, Fnet = m2*a

By substitution, m2*a = (m2*g) - (m1*g + m1*a)

Find the acceleration of the system by solving for a:

a*(m1 + m2) = (m2 – m1)*g

so, a = (m2 – m1)*g/(m1 + m2)

In other words, the acceleration is equal to 9.8 times

the difference of the two masses, divided by the sum

of the two masses.

a = 9.8(m2 – m1)

(m2 + m1)

If you think about it, gravity is

accelerating only the difference in

the masses because otherwise, the

system is in static equilibrium.

Frictional Forces

Friction is a force that always exists

between any two surfaces in contact

with each other.

Cause

of

friction

Static friction The friction that exists between two surfaces that are

not moving relative to each other.

Kinetic friction The friction that exists between two surfaces that are

moving relative to each other.

There are two kinds of friction, based on how the two surfaces are moving relative to each other:

In any situation, the static friction is greater than

the kinetic friction.

Frictional Forces Occur When Materials are

in Contact

W

fs F

N

Surfaces in

Contact

M1

Spring Scale

F = Force Causing Motion (Pull on Scale)

Fs = Force of Static Friction (Resists Motion)

N = Force Normal Holds Surfaces in Contact

W = Weight of Object ( Mass x Gravity)

Friction is a Force That Resists Motion

W

fs F

N

Surfaces in

Contact

The pink block M1 will not move until the

force, F (pull on the scale ) exceeds the force of

Static Friction fs.

M1

Spring Scale

The relative force of static friction between 2 objects is

expressed as the quotient of the force required to

move the object (F) divided by the weight (W) of the object.

W

fs F

N

Surfaces in

Contact

M1

Spring Scale

This is the Coefficient of Friction

s

FW

= Force Required to Cause Motion

= Weight of Object

= Coefficient of Friction

W

Fs Coefficient of Static Friction

W

fs F

N

Surfaces in

Contact

M1

Spring Scale

Using an object to calculate

W

Fs

s

Record the Maximum Force (F) before the object begins to move

W

Fs

Maximum Force F = 110 g

Record the Weight

(W ) of the object

W

Fs

580 g

The coefficient of static friction between the

surface and the battery is described

algebraically:

W

Fs

= 580

= 110

s = .190

W

Fk

The coefficient of Kinetic Friction can be found

using the same technique.

Record the force required to move the battery at

a constant rate.

Friction is also proportional to the normal force, which is how we'll

be able to calculate it.

The actual formula for friction is…

Ff = μ FN

Ff = force due to friction (Newtons)

FN = normal force (Newtons)

μ = (Greek letter “mu”) coefficient of friction between two

surfaces (no units)

μs is static, μk is kinetic

Obviously, some surfaces have less

friction than others…

When we measure the coefficient of friction (μ), the smaller the number, the less the friction between the two surfaces.

You will be able to use a table of

predetermined values, or calculate it.

Ff,kinetic = μk FN

0 < Ff,static < μsFN

2 formulas:

The static friction force can vary from 0 to μsFN -

max static frictional force which must be overcome

before motion can begin.

Static friction between the

block and the floor

example:

A 12kg piece of wood is placed on top of another piece of

wood. There is 35N of static friction measured between

them. Determine the coefficient of static friction between

the two pieces of wood.

First calculate FN FN = Fg = mg

= (12kg) (9.81m/s2)

μs = Ff / FN

= (35N) / (117.7 N)

μs = 0.30

12 kg

FN = 117.7 N

Then use this answer to calculate Ff ... Ff = μs FN

A steel box (mass of 10 kg) is sitting on a steel workbench. Can the box be pushed out of the way with a force of 25 N?

try another example

Calculate the max. force due to static friction…

a) Draw a free body diagram of the box.

figure out the normal force… FN = Fg = mg = (10 kg) (9.81 m/s2) FN = 98 N

…use that to calculate the maximum static friction.

(the values for μs and μk are given)

for steel on steel, μs is 0.74 and μk is 0.57

Ff = μs FN

= 0.74 (98 N)

Ff = 73 N

So, does this mean that when the box is pushed with

Fa = 25 N, the friction will push back with 73 N?

No!

The force due to static friction can go up to a

maximum of 73 N, but can also be less.

It will be equal to whatever the Fa is, up to the maximum

calculated here.

Ff = Fa = 25 N (they just point in opposite directions) FNET = Zero With no net force acting on it, the box will not start to move.

What if the push is a force of 73 N? will

anything will happen?

This exactly equals the maximum static frictional force between these two surfaces.

Ff = Fa = 73 N (but in opposite directions!) FNET = Zero With no net force acting on it, the box will not start to move.

In order to move -- you must overcome

static friction.

If a push is a force of 100 N , determine if anything

will happen.

This applied force is greater than the static friction,

so it will start to move… but remember that we will

now be using kinetic friction!

Ff = μk FN = 0.57 (98 N) Ff = 56 N

FNET = FN + Ff = 100 + -56 FNET = 44 N

FNET = ma a = FNET / m = 44 / 10 a = 4.4 m/s2

The box will accelerate at 4.4 m/s2.

Try some problems:

A rightward force is applied to a 6-kg object to move it

across a rough surface at constant velocity. The object

encounters 15 N of frictional force. Use the diagram to

determine the gravitational force, normal force, net force,

and applied force. (Neglect air resistance.)

Since there is no vertical acceleration, the normal force

equals the gravity force.

Since there is no horizontal acceleration, Ffrict = Fapp = 15 N

Fnet = 0 N; Fgrav = 58.8 N; Fnorm = 58.8 N; Fapp = 15 N

When the velocity is constant,

a = 0 m/s2 and Fnet = 0 N

Since the mass is known,

Fgrav can be found:

Fgrav = m • g = 6 kg • 9.8 m/s/s =

58.8 N

A rightward force is applied to a 5-kg object to move it

across a rough surface with a rightward acceleration of 2

m/s2. The coefficient of friction between the object and the

surface is 0.1. Use the diagram to determine the

gravitational force, normal force, applied force, frictional

force, and net force. (Neglect air resistance.)

Fnet =10 N, right; Fgrav = 49 N; Fnorm = 49 N; Ffrict = 4.9 N; Fapp = 14.9 N

Fnet can be found using Fnet = m • a

= (5 kg) • (2 m/s/s) = 10 N, right.

Since the mass is known, Fgrav can be

found: Fgrav = m • g = 5 kg • 9.8 m/s2 =

49 N

Since there is no vertical acceleration, the normal force equals the gravity

force.

Once Fnorm is known, Ffrict can be found using Ffrict = μ • Fnorm

= (0.1) • (49 N) = 4.9 N.

Since the Fnet = 10 N, right, the rightward force (Fapp) must be 10 N more

than the leftward force (Ffrict); thus, Fapp must be 14.9 N.

In a Physics lab, Bob and Joe apply a 34.5 N rightward

force to a 4.52-kg cart to accelerate it across a

horizontal surface at a rate of 1.28 m/s2. Determine the

friction force acting upon the cart.

Ffrict = 28.7 N, left

The net force can be determined from the mass and acceleration of the sled.

Fnet = m • a = (4.52 kg) • (1.28 m/s2) = 5.7856 N, right.

Since the net force is in the direction of the applied force, then the applied

force must be greater than the friction force. The friction force can be

determined because the net force is the vector sum of all the forces. So

5.7856 N, right = 34.5 N, right + Ffrict. Therefore,

Fn

Fg

Fapp Ff

Since there is no vertical acceleration,

normal force = gravity force.

Fgrav = m • g = (4.525 kg) • (9.8 m/s2) = 44.296 N

Example Problems, pg. 131-132

Practice problems 14-16, pg. 133

Read/study pgs. 134 - 143

Finish calculations.

Answer questions – turn in!

Designing frictionless transportation.

PERIODIC MOTION

A repeated motion

Periodic motion

Back and forth over the same path.

Vibrating

Tuning fork

200

grams

200

grams

A weight on

a spring

A boy on

a swing

Simple Harmonic Motion

Simple harmonic motion (SHM) refers to a Periodic Motion which repeats itself at regular, equal intervals of time.

SHM describes the behavior of many physical

phenomena: – a pendulum – a bob attached to a spring – low amplitude waves in air (sound), water, the ground – the electromagnetic field of laser light – vibration of a plucked guitar string – the electric current of most AC power supplies

Simple Harmonic Motion

• The force causing the motion is in direct relationship to the displacement of the body.

• The displacement, velocity, acceleration and force are specific at various points in the cycle.

SHM

• Equilibrium: the position at which no net force acts on the object.

• Displacement: The distance of the object from its equilibrium position.

• Amplitude: the maximum distance the object moves from equilibrium.

SHM Systems

Period (T) = time to

complete one cycle

Springs and SHM

Attach an object of mass m to the end of a spring, pull it out to a distance A, and let it go from rest. The object will then undergo simple harmonic motion. The spring demonstrates

SHM because it vibrates

back and forth around its

unstretched position.

F elastic The direction of the

force acting on the

mass (F elastic) is

opposite the direction

of displacement from

equilibrium.

F elastic

F elastic

Direction of the spring force is always opposite the direction of the mass’ displacement.

Restoring Force

The spring force always pushes/pulls the mass

back to its original equilibrium position

(restoring force).

Robert Hooke (1678)

First defined this relationship

between mass/displacement of

a spring system.

Hooke’s Law

Felastic = -kx

Spring force = -(spring constant)(displacement)

The Simple Pendulum

• The pendulum bob is clearly oscillating as it moves back and forth – but is it exhibiting SHM?

Amplitude:

max.

displacement

from

equilibrium (m,

radian)

Period, T :

time it takes to

execute a

complete cycle

of motion (s)

Frequency, f :

number of

cycles or

vibrations per

unit of time

(Hz)

Calculations of the period and frequency are different

than the mass-spring system because they depend on

different physical factors.

(for small angles)

neither mass

nor amplitude

affect the period

θ

But changing the string length does

Also – a change in the free fall acceleration!

Galileo is credited as the first

person to notice that the motion

of a pendulum depends on its

length and is independent of its

amplitude (for small angles)

Measured the frequency by timing

the swings with his pulse.

Why does period depend on string length?

When 2 pendulums have

different length but same

amplitude, shorter

pendulum has a smaller

arc to travel.

The distance to

equilibrium is less while

the acceleration remains

the same --- so the

shorter length will have a

shorter period.

Why don’t mass and amplitude affect the period?

When the bobs differ in

mass, the heavier

provides a larger

restoring force, but also

needs larger force to

achieve the same

acceleration.

Because the

acceleration of

both is the same,

the period for both

is the same.

Period (T) of a simple pendulum in SHM

period = 2π x square root of (length/free-fall acceleration

ex. problem:

What is the period of a 3.98

m long pendulum?

What is the period of a

99.4 cm long

pendulum?

4.0 s

2.0 s

A washer on a string swings

back and forth once every

1.0 s. How long is the

string? 0.25 m

T = 1.0 s

g = 9.81 m/s/s

T√g = √L

2π T2g = L

4π2

one more….

You need to know the height of a tower, but

darkness obscures the ceiling. There is a

pendulum extending from the ceiling that

almost touches the floor and it has a period

of 12 s. How tall is the tower?

T = 12 s

g = 9.81 m/s/s

T√g = √L

2π

T2g = L

4π2

(12 s)2(9.81 m/s/s) = L

4π2

L = 36 m