Set - 4

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Set - 4

• Explaining Motion

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What is the difference between speed and velocity?

1. The two are the same.
2. Velocity relates to instantaneous speed, but not
   to average speed.
3. Velocity is the speed and the direction the
   object is traveling.
4. Velocity relates to invisible objects like atoms,
   while speed relates to visible objects like cars.

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What is the difference between average speed and
instantaneous speed?

1. Average speed is the speed of an average runner
   and instantaneous speed is the speed of a very
   fast runner.
2. Instantaneous speed is the average speed of a
   very small portion of the trip.
3. Average speed is calculated over many trips, but
   instantaneous speed is calculated during one
   trip.
4. They're the same thing.

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Why do we care about motion?

• Because we all move in various ways. Our cars
  move and they move us. Our friends move.
• Music is sound and sound moves through the air.
  We need these concepts to really understand what
  music is and how it works.

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The next slide gives an example of why we need
this stuff to understand music.
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The Ear

• Responds to Pressure
• A force on the membrane
• A movement inside the ear
• Translation into the brain
• Music !

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If you drop a piano out of a 20 story building

• How long will it take to hit the ground?
• How fast will it be moving?
• After it hits the ground .. how difficult will it
  be to play it?

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This chapter

• Mostly physics
• Needed for understanding of many concepts in
  music.

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IntroductionExplaining Motion

• In physics, to explain something means to create
  a model that can predict the outcome of
  experiments.
• Motions appear to be reproducible that is, if we
  start out with the same conditions and do the
  same thing to an object, we get the same
  resulting motion.
• The same motion occurs regardless of
• when the experiment is done, and
• where the experiment is done.
• This reproducibility is a necessary condition for
  attempting to search for a set of rules that
  nature obeys. Rules of nature might be difficult
  to find, but they do not change.

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Translation

• Under identical and repetitive conditions, the
  same outcome will always occur.
• Physical motion is PREDICTABLE based upon certain
  laws of motion.
• There are three of them that are referred to as
  NEWTONS LAWS.

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Do Objects Tend to Rest?

• Give your book a brief push across a table.
• Although the book starts in a straight line at
  some particular speed, it quickly slows and
  stops.
• If you epeat this book-pushing experiment on a
  surface covered with ice.
• The book would travel a much greater distance
  before coming to rest.
• The ice is slicker than the desktop. Different
  surfaces interact with the book with different
  strengths.
• What would happen to the book if the surface were
  perfectly slick?
• The book would not slow down at all it would
  continue in a straight line at a constant speed
  forever.

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Galileo
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The man Galileo of Pisa

• Born in Pisa (1564)
• Thought the Church had become sterile and began
  to translate it nto modern music.
• His work began the development that culminated in
  ITALIAN OPERA!
• Smart Dude!

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Galileo

• Enrolled in medicine but switched to Mathematics.
• At the age of 25, he was appointed Chair of
  Mathematics at Padula.
• In 1610 Developed the telescope
• Observed
• Mountains on the moon
• Moons of Jupiter
• Phases of Venus

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Galileo

• From his OBSERVATIONS, he favored the Copernican
  world view that the Earth orbited around the sun
  rather than the other way around.
• This created big conflicts with the church.
  Eventually, he was confined to his home where he
  died in 1642.

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Galileos Thought Experiment

• Galileo noted that a ball rolling down a slope
  speeds up. Conversely, if the ball rolls up the
  slope, it naturally slows down.
• The ball experiences an interaction on the
  falling slope that speeds it up and an
  interaction on the rising slope that slows it
  down.
• Now, Galileo asked himself, what would happen to
  the ball if it were placed on a level surface?
  Nothing. Because the surface does not slope, the
  ball would neither speed up nor slow down the
  ball would continue its motion forever.

FOREVER ???
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The importance of TIME in Galileos Experiments
The red dots represent the positions after equal
time intervals. What can we say about this motion?
Pendulum
Acceleration . WHY??
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The Tower??

• Aristotle claimed that a heavier object will hit
  the ground sooner than a light object dropped at
  the same time.
• Galileo did the experiment and proved Aristotle
  wrong.
• There is no evidence to prove that Galileo
  actually did this experiment.
• But he may have!

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What he would have found had he done the
experiment

• The longer the object fell, the faster it went.
• The weight (to be more carefully defined shortly)
  of the object didnt matter.
• The size of the object MAY have mattered.

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Timing
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More importantly

• For every second a falling object falls
  (vertically), its speed increases by about 10
  meters per second (or 32 ft/sec).
• This only works if air resistance can be
  neglected.
• The object is said to be UNIFORMLY ACCELERATING.

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The acceleration of gravityg
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g is a VECTOR!!
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Galileos BIG Contribution
OBSERVE
MODEL
Change Something
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Acceleration

• If an object starts with a velocity v0 and ends
  with a velocity vf after a time t, then the
  average acceleration is said to be
• Units velocity/time (m/sec)sec or m/s2

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Some algebra
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• An object is thrown into the air with a velocity
  of 20 m/s. How long will it take to get to the
  top of the trajectory? How long back to the
  thrower?

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No Surprise Instantaneous Acceleration
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AGAINAcceleration of gravity

• 10 m/s2
• Actually 9.8 m/sec2
• 32 ft/sec2

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An object is THROWN vertically down from the top
of a building with an initial speed of 20 meters
per second. 2 seconds later it will have a speed
of

1. 30 m/s
2. 40 m/s
3. 50 m/s
4. 60 m/s

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A woman throws a ball straight up into the air at
a speed of 30 m/s. After how many seconds will
the ball return to her hands?

• 3
• 4
• 5
• 6
• None of these

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Look at the graph
Vf
v
Area v x Dt distance traveled
V0
Dt
Time t
Distance v0t (1/2) at2
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Reaction Time Experiment

• 2 People
• One holds the meter stick
• The second has hand at the bottom
• When the first person releases the meter stick,
  the second catches it as fast as possible.
• From the distance, we can calculate reaction time.

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Reaction Time DROP THE RULER
Distance v0t (1/2) at2
Start from rest so Distance (1/2) at2
D (1/2) gt2
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Lets do it again!
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REACTION TIME
Minus Signs Are Important
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The Modern Explanation

• Galileo was the first to suggest that
  constant-speed, straight-line motion was just as
  natural as at-rest motion.
• Natural motion is one in which the speed and
  direction are constant.
• An interaction with an external agent is required
  to cause an object to change its velocity.
• Objects at rest tend to remain at rest. Objects
  in motion tend to remain in motion.
• This property of remaining at rest or continuing
  to move in a straight line at a constant speed is
  known as inertia.

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INERTIA Is Important!
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Note-

• This resistance to changes in motion was later
  quantified by Sir Isaac Newton
• This is the next topic

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The Modern Explanation

• There is more to inertia than getting things
  moving. If something is already moving, it is
  difficult to slow it down or speed it up.
• An example is drying your wet hands by shaking
  them. When you stop your hands abruptly, the
  water continues to move and leaves your hands.
• In a similar way, seat belts counteract your
  bodys inertial tendency to continue forward at a
  constant speed when the car suddenly stops.

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The Modern Explanation

• All objects do not have the same inertia.
• Imagine trying to stop a baseball and a
  cannonball, each of which is moving at 150
  kilometers per hour (about the speed a
  major-league pitcher throws a baseball).
• The cannonball has more inertia and, as you can
  guess, requires a much larger effort to stop it.
• Conversely, if you were the pitcher trying to
  throw them, you would find it much harder to get
  the cannonball moving.

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Fun Trick with Inertia
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Newton and Galileos Legacy

• Although Galileo did not fully explain motion, he
  did take the first important step and, by doing
  so, radically changed the way we view the motion
  of objects.

• His work profoundly influenced Isaac Newton, the
  originator of our present-day rules of motion.

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Sir Isaac Newton

• Born in 1642
• One of the GREATS of physics (on a par with
  Einstein!)
• Strange dude rumor is that he died a virgin.

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Newton

• Born in England and he was supposed to return and
  look after the family farm.
• At age 17 he returned from boarding school he
  proved to be a total failure at farming.
• Went to Trinity College, paying his way by
  waiting on tables and cleaning rooms for faculty
  and wealthy students.
• Many students do this today as well.

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Newton

• In1667 he published a treatise on infinite
  series.
• Developed a reflecting telescope and was elected
  membership in the Royal Society.
• Decided that Christianity had deviated from the
  original teachings of Christ and refused to take
  the colleges holy orders.
• He was excused .. The only person ever to have
  received this allowance.

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More Newton

• In 1686 he published his Principia .. Which
  established the laws of motion in a mathematical
  way.
• He developed the calculus while Leibnitz
  developed a different approach. For years they
  fought over who was the real inventor!
• He showed that white light was a mixture of all
  of the colors of the rainbow.
• He developed the math to show that the planets
  and comets rotated around the sun in elliptical
  orbits.
• He died in 1727.

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Newtons First Law

• Newtons first law of motion. It is also referred
  to as the law of inertia
• -The velocity of an object remains constant
  unless it is acted upon by an external force.
• For the velocity of an object to remain constant,
  its speed and its direction must both remain
  constant.

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Newtons First Law

• Remember
• Velocity can be ZERO
• Therefore Newtons first law also says that an
  object at REST (not moving) will stay at rest.

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Newtons First Law Force??

• The first law incorporates Galileos idea of
  inertia and introduces a new concept, force.
• A book sliding across the table slows down and
  stops because there is a force (called friction)
  that opposes the motion.
• Similarly, a falling rock speeds up because there
  is a force (called gravity) acting on it.
• .

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What Are Forces?

• Casually speaking, a force is a push or a pull.
• We dont actually see forces. We see objects
  behave in a certain way, and we infer that a
  force is present.

• The direction of the force is as important as its
  size, in the way they make objects behave.

Therefore, we treat forces as vectors.
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Whats Up??
Bye, Bye!
No force UP!
Supermans WEIGHT
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A guy is pulling his girlfriend on a sled at
constant velocity.
Lets discuss the FORCES that are acting.
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Question 1 What object are we talking about?
The Guy? The Girl?

• We must apply our ideas to only ONE object at a
  time, or
• an appropriate combination of objects that are
  functioning as a single body.

We choose the girl sled
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The (Girl Sled) since they move together!
Something NEW The force the earth pushes up
with! We call it the NORMAL FORCE
The Pull
Friction
Weight of the girl AND the sled
This is called a FREE BODY DIAGRAM
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Normal?
The two are equal but opposite in direction.
W
Gravity
N
NW
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FREE BODY DIAGRAM of the (Girl Sled) since they
move together!
EQUAL
components
Equilibrium Constant Velocity
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Balanced Unbalanced Forces

• Remember that Newtons first law refers to the
  unbalanced force.
• In many situations there is more than one force
  on an object.

There is an unbalanced force only if the sum of
the forces is not zero.
When two forces of equal size act along a
straight line but in opposite directions, they
cancel each other.
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Balanced Unbalanced Forces

• When we observe an object with no acceleration,
  we infer that there is no unbalanced force on
  that object.
• If you see a car moving at a constant speed on a
  level, straight highway, you infer that the
  frictional forces balance the driving forces.
• What is the net force acting on an airplane in
  level flight flying at 500 mph due east?
• Because the speed and direction are constant,
  there is no acceleration, and the net force must
  be zero.

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Adding Vectors

• Mathematicians have developed rules for combining
  vector quantities such as displacements,
  velocities, accelerations, or forces.
• In this chapter you will learn to combine
  vectors using a graphical method and the scale
  shown in the next slide.

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Vector Shorthand

• In texts, vectors are represented by boldface
  symbols (such as F).
• When writing by hand about vectors, you use an
  arrow over the symbol (such as F ).
• The magnitude of the vector quantity is
  represented by an italic symbol. Therefore, a
  force is written as F, and its magnitude is
  written as F.

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Drawing Vectors Accurately
We can represent any vector by an arrow its
length represents the magnitude of the quantity,
and its direction represents the direction of the
quantity.
To complete this representation, we assign a
convenient scale to our drawings. Here, one
centimeter on paper represents 20 meters on the
ground.
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Tail-to-Tip Vector Addition

• When you are given a list of directions, each
  succeeding arrow is drawn beginning at the head
  of the previous arrow.
• The arrow drawn from the tail of the first arrow
  to the head of the last arrow represents the
  vector sum.
• You can determine the direction and magnitude of
  this last vector, the sum, with a ruler and a
  protractor.

In this way the three forces acting on the ball
(a) can be added to find the net force (b). Note
that you dont have to start with F1. The order
in which the forces are added doesnt matter you
could choose F2 or F3. Try it!
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Adding Vectors
SUM
The Same Diagram
The ORDER doesn't matter
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Vectors we have used

• Position (Not too often in this class)
• Velocity
• Acceleration
• Force
• Any others???

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Summary Newtons FIRST Law

• Objects in motion tend to remain in motion unless
  acted upon by an external force.
• Objects at rest tend to remain at rest unless
  acted upon by an external force

How different are these two cases??
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Do objects at rest have a velocity??
Question
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Newtons Second Law

• The acceleration of an object is proportional to
  the net force acting on it.
• Twice the force produces twice the acceleration.
• The direction of the acceleration is always in
  the direction of the net force.

Two springs pulling side by side exert twice the
force of one spring pulled by the same amount.
Thus, they produce twice the acceleration.
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Newtons Second Law

• Mass and acceleration are inversely proportional.
  
• Inversely indicates that the changes in the two
  values are opposite each other.
• If the mass is increased by a certain multiple,
  the acceleration produced by the force is reduced
  by the same multiple.

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Newtons Second Law

• Newton put the two preceding ideas together into
  his second law of motion.
• The net force on an object is equal to its mass
  times its acceleration and points in the
  direction of the acceleration.

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Conceptual Question on the 2nd Law

• A crate falls from a helicopter and lands on a
  very deep snowdrift. The snow slows the crate and
  eventually brings it to a stop.
• During the time that the crate is moving
  downward through the snow, is the magnitude of
  the upward force exerted on the crate by the snow
  greater than, equal to, or less than the
  magnitude of the gravitational force acting
  downward on the crate?

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Answer to the Conceptual Question

• Because the crate is moving downward, its
  velocity is pointing down. Because the crate is
  losing speed, its acceleration must be pointing
  in the opposite directionthat is, up. The net
  force always points in the same direction as the
  acceleration. Therefore, the force acting upward
  on the crate must be larger than the force acting
  downward. Thus, the snow exerts the greater force.

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Defining Units of Mass and Force

• In the previous chapter, we have defined a
  measure for acceleration, but not as yet for mass
  and force.
• Historically, a certain amount of matter was
  chosen as a mass standard.
• The mass of a liter of water has a mass of 1
  kilogram.
• The force needed to accelerate a 1-kilogram mass
  at 1 (meter per second) per second is called 1
  newton (N), in honor of Isaac Newton.
• In using any rule of nature, we must use a
  consistent set of units.

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Newtons Second Law - UNITS
F m a
Newtons
Kilograms
Meters/second2
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Everything in this course is pretty much based on
Fma
Note
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Meters/second2

• Can be written as (meter/second)/second
• Consider an object accelerating at 10
  (meters/second)/second
• Every second its velocity increases by 10 m/s.

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Example

• At t0 (what does this mean???), an object is
  moving at 15 m/s and is accelerating at 10 m/s2.
• What is its velocity as a function of time??

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The results
Time (seconds) Velocity
0 15
1 25
2 35
3 45
4 55
5 65
6 75
7 85
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The Graph
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Numerical Question on the 2nd Law

• What is the net force needed to accelerate a 5-kg
  object at 3 m/s2?
• When accelerations are measured in (meters per
  second) per second, as they are here, the masses
  must be in kilograms and the forces in newtons.
• A newton can be written as kg m/s2.
• Applying the second law, we have

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Mass ? Weight

• Mass is often confused with weight.
• They are proportional to each other.
• Our weight depends on where we are.
• We compress the spring in a bathroom scale
  because Earth is attracting us.
• If we were on the Moon, our weight would be less
  because the Moons gravitational force on us
  would be less.
• Our mass, however, is not dependent on our
  location in the Universe. It is a constant
  property that depends only on how much there is
  of us.

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ConsequenceDoes the 2nd Law Apply in Space?

• Being weightless does not mean that you are
  massless.
• Imagine a huge truck in outer space hanging
  from a spring scale. Although the scale would
  read zero, if you tried to kick the truck, you
  would find that it resisted moving.
• Weight can be expressed in newtons to clear up
  misunderstandings.

A 1-kilogram mass near Earths surface has a
weight of 9.8 newtons, or about 2.2 pounds.
Therefore, a pound of butter has a weight of
4.5 newtons and a mass a little less than ½
kilogram.
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Weight as a Force

• Lets represent the acceleration due to gravity
  by the symbol g, where weve used a vector to
  indicate both the size and direction. If we then
  replace the net force Fnet by the weight W, we
  obtain

General equation of the 2nd Law
Re-stated in gravitational terms
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How Much Do You Weigh?

• (Remember, mass cannot be expressed in newtons
  weight can.)
• Calculate the weight of a child with a mass of 25
  kg.
• Obtain the mass of a dog that has a weight of 150
  N.

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Introducing Free-Body Diagrams

• Imagine that you are pulling your little sister
  on a sled and that the sled is speeding up. There
  are many forces acting on the sled.
• The rope is exerting a tension on the sled,
  pulling it forward.
• Earth is pulling down on the sled with a
  gravitational force.
• The snow is pushing up on the sled with a force
  commonly called a normal force.
• (Normal means perpendicular, and this force acts
  perpendicular to the surface between the sled and
  the snow.)
• Your sister is pushing down on the sled with a
  normal force,
• and the snow is resisting your efforts with a
  frictional force that acts parallel to the
  surface of the snow.

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Introducing Free-Body Diagrams
Actor Direction
Rope Forward
Earth Downward
Sister Downward
Snow Backward
Snow Upward

• Which of these forces do we use in Newtons
  second law?
• Fnet ma

• Fnet , the net force, is the vector sum of all
  the forces acting on the sled. It is important,
  therefore, to correctly identify all the forces
  acting on an object when analyzing its motion.
• We identify the forces by drawing a free-body
  diagram.

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Free Fall RevisitedFnet

• Objects falling on Earth dont fall in a vacuum
  but through air. Thus, in realistic situations, a
  falling object has two forces acting on it
  simultaneously
• the weight acting downward,
• and the air resistance acting upward.
• And the greater the speed, the greater the air
  resistance.

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Free Fall RevisitedTerminal Speed

• With these facts in mind, consider the downward
  motion of a falling rock.
• Initially, it falls at a low speed, and the air
  resistance is small.
• Because there is a net force, the rock
  accelerates, thus increasing its speed.
• As the rock speeds up, however, its weight
  remains constant while the air resistance
  increases.
• Thus, the net force and the acceleration
  decrease.
• The rock continues to speed up but at a
  decreasing rate.
• Eventually, the rock reaches a speed for which
  the air resistance equals the weight.
• Fnet 0. The rock stops accelerating.
• This maximum speed is called the terminal speed
  of the object.

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Terminal Speeds

• The terminal speeds of different objects are not
  necessarily equal. Factors include
• The shape of the falling object,
• Its size,
• Its weight, and
• Resistive properties of the medium.
• An object will continue to accelerate until the
  terminal force is the same size as its weight.
• The maximum speed of a skydiver near sea level is
  190 mphno matter how high up they were when they
  started!
• If they fall spread-eagled, this value is
  decreased.

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Forces Discouraging MovementStatic Friction

• The static frictional force seems a bit
  mysterious. Because it is equal to the force you
  exert
• The frictional force is small if you push with a
  small force (a).
• If you push with a large force, the frictional
  force is large (b).
• It ceases to exist when the applied force is
  removed.

91
Forces Discouraging MovementKinetic Friction

• If your applied force exceeds the maximum static
  frictional force, the crate accelerates in the
  direction of your applied force.
• Although the crate is now sliding, there is still
  a frictional force (c).
• The value of this kinetic friction is less than
  the maximum value of the static frictional force.
  Unlike air resistance, kinetic friction has a
  constant value, independent of the speed of the
  object.

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Test Static vs. Kinetic Friction

• With a simple wooden block and a long rubber
  band, you can verify the behavior of static and
  kinetic friction.
• Connect the rubber band to the block with a
  thumbtack, and slowly pull on the block.
• The stretch of the band provides a visual
  indication of the force you are applying.
• If the block does not move, the static force is
  equal but opposite in direction to the force of
  the rubber band.
• Continue to increase your pull. What happens?
• Repeat the experiment, with the block sliding
  across the table at a constant speed.
• How does the stretch of the rubber band now
  compare with its maximum stretch in the static
  situation?

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Newtons Third Law

• There is no way to push something without being
  pushed yourself.
• For every force there is always an equal and
  opposite force.
• If an object exerts a force on a second object,
  the second object exerts an equal force back on
  the first object.
• Forces always occur in pairs. And these forces
  never act on the same body.

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I dropped it and the world stood still

• Consider a ball with a weight of 10 newtons
  falling freely toward Earths surface.
• Earth exerts a force Web on the ball. According
  to Newtons third law, the ball exerts an equal
  and opposite force Wbe on Earth.
• Web Wbe 10 N.

Earths mass is so large that its acceleration
toward the ball is minuscule. Earth does
accelerateevery time we drop somethingbut we
dont notice it.
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Another Example

• When you fire a rifle, it recoils.
• Third Law Rifle pushes bullet, bullet pushes
  rifle.
• But why doesnt the rifle accelerate as much as
  the bullet?
• The force of the bullet on the rifle is the same
  size but produces a small acceleration because
  the mass of the rifle is large.
• The force of the rifle on the bullet produces a
  large acceleration because the mass of the bullet
  is small!

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Without the 3rd Law, You Cannot Walk

• In walking you must have a force exerted on you
  in the direction of your acceleration. And yet
  the force you produce is clearly in the opposite
  direction.
• As you start to walk, you exert a force against
  the floor (down and backward) the floor
  therefore exerts a force back, causing you to go
  forward (and up a little).
• If you want a clearer demonstration of this, try
  walking in a rowboat.

As the man walks to the left, he exerts a force
on the boat that causes the boat to move to the
right.
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Without the 3rd Law, You Cannot Stand

• Without the third law, paradoxical events would
  occur in the Newtonian world view.
• A person stands on a floor. Because the person is
  not accelerating, the net force must be zero.
• Q What is the force that balances the
  gravitational force?

A Earth exerts a force Wep on the person, which
causes the person to exert a force Npf on the
floor. By Newtons third law, the floor exerts an
equal and opposite force Nfp on the person.
Although Wep and Nfp are equal and opposite, they
are not an actionreaction pair they both act on
the person.