Transcript Chapter 6 Dynamics I: Motion Along a Line

Chapter 6 Lecture
physics
FOR SCIENTISTS AND ENGINEERS
a strategic approach
THIRD EDITION
randall d. knight
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Chapter 6 Dynamics I: Motion Along a Line
Chapter Goal: To learn how to solve linear force-and-motion problems.
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Chapter 6 Preview
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Chapter 6 Preview
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Chapter 6 Preview
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Chapter 6 Preview
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Chapter 6 Preview
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Chapter 6 Preview
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Chapter 6 Reading Quiz
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Reading Question 6.1
Newton’s first law can be applied to
A.
B.
C.
D.
E.
Static equilibrium.
Inertial equilibrium.
Dynamic equilibrium.
Both A and B.
Both A and C.
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Reading Question 6.1
Newton’s first law can be applied to
A.
B.
C.
D.
E.
Static equilibrium.
Inertial equilibrium.
Dynamic equilibrium.
Both A and B.
Both A and C.
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Reading Question 6.2
Mass is
A. An intrinsic property.
B. A force.
C. A measurement.
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Reading Question 6.2
Mass is
A. An intrinsic property.
B. A force.
C. A measurement.
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Reading Question 6.3
Gravity is
A. An intrinsic property.
B. A force.
C. A measurement.
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Reading Question 6.3
Gravity is
A. An intrinsic property.
B. A force.
C. A measurement.
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Reading Question 6.4
Weight is
A. An intrinsic property.
B. A force.
C. A measurement.
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Reading Question 6.4
Weight is
A. An intrinsic property.
B. A force.
C. A measurement.
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Reading Question 6.5
The coefficient of static friction is
A.
B.
C.
D.
Smaller than the coefficient of kinetic friction.
Equal to the coefficient of kinetic friction.
Larger than the coefficient of kinetic friction.
Not discussed in this chapter.
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Reading Question 6.5
The coefficient of static friction is
A.
B.
C.
D.
Smaller than the coefficient of kinetic friction.
Equal to the coefficient of kinetic friction.
Larger than the coefficient of kinetic friction.
Not discussed in this chapter.
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Reading Question 6.6
The force of friction is described by
A.
B.
C.
D.
The law of friction.
The theory of friction.
A model of friction.
The friction hypothesis.
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Reading Question 6.6
The force of friction is described by
A.
B.
C.
D.
The law of friction.
The theory of friction.
A model of friction.
The friction hypothesis.
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Reading Question 6.7
When an object moves through the air, the
magnitude of the drag force on it
A. Increases as the object’s speed increases.
B. Decreases as the object’s speed increases.
C. Does not depend on the object’s speed.
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Reading Question 6.7
When an object moves through the air, the
magnitude of the drag force on it
A. Increases as the object’s speed increases.
B. Decreases as the object’s speed increases.
C. Does not depend on the object’s speed.
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Reading Question 6.8
Terminal speed is
A. Equal to the speed of sound.
B. The minimum speed an object needs to escape the
earth’s gravity.
C. The speed at which the drag force cancels the
gravitational force.
D. The speed at which the drag force reaches a
minimum.
E. Any speed which can result in a person’s death.
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Reading Question 6.8
Terminal speed is
A. Equal to the speed of sound.
B. The minimum speed an object needs to escape the
earth’s gravity.
C. The speed at which the drag force cancels the
gravitational force.
D. The speed at which the drag force reaches a
minimum.
E. Any speed which can result in a person’s death.
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Chapter 6 Content, Examples, and
QuickCheck Questions
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Equilibrium
 An object on which the net force is zero is in equilibrium.
 If the object is at rest, it is in static equilibrium.
 If the object is moving along a straight line with a
constant velocity it is in dynamic equilibrium.
 The requirement for either
type of equilibrium is:
The concept of equilibrium is essential for the
engineering analysis of stationary objects such
as bridges.
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QuickCheck 6.1
The figure shows the view looking down onto a sheet of
frictionless ice. A puck, tied with a string to point P, slides on the
ice in the circular path shown and has made many revolutions. If
the string suddenly breaks with the puck in the position shown,
which path best represents the puck’s subsequent motion?
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QuickCheck 6.1
The figure shows the view looking down onto a sheet of
frictionless ice. A puck, tied with a string to point P, slides on the
ice in the circular path shown and has made many revolutions. If
the string suddenly breaks with the puck in the position shown,
which path best represents the puck’s subsequent motion?
Newton’s first law!
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Problem-Solving Strategy: Equilibrium Problems
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Problem-Solving Strategy: Equilibrium Problems
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QuickCheck 6.2
A ring, seen from above, is pulled
on by three forces. The ring is not
moving. How big is the force F?
A. 20 N
B. 10cos N
C. 10sin N
D. 20cos N
E. 20sin N
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QuickCheck 6.2
A ring, seen from above, is pulled
on by three forces. The ring is not
moving. How big is the force F?
A. 20 N
B. 10cos N
C. 10sin N
D. 20cos N
E. 20sin N
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Example 6.2 Towing a Car up a Hill
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Example 6.2 Towing a Car up a Hill
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Example 6.2 Towing a Car up a Hill
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Example 6.2 Towing a Car up a Hill
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Example 6.2 Towing a Car up a Hill
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QuickCheck 6.3
A car is parked on a hill.
Which is the correct
free-body diagram?
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QuickCheck 6.3
A car is parked on a hill.
Which is the correct
free-body diagram?
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QuickCheck 6.4
A car is towed to the right
at constant speed. Which
is the correct free-body
diagram?
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QuickCheck 6.4
A car is towed to the right
at constant speed. Which
is the correct free-body
diagram?
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Using Newton’s Second Law
The essence of Newtonian mechanics can be
expressed in two steps:
 The forces on an object determine its
acceleration
, and
 The object’s trajectory can be determined by
using in the equations of kinematics.
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Problem-Solving Strategy: Dynamics Problems
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Problem-Solving Strategy: Dynamics Problems
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QuickCheck 6.5
The cart is initially at rest. Force is applied to the cart
for time t, after which the car has speed v. Suppose the
same force is applied for the same time to a second cart
with twice the mass. Friction is negligible. Afterward, the
second cart’s speed will be
A.
1
v
4
B.
1
2
v
C. v
D. 2v
E. 4v
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QuickCheck 6.5
The cart is initially at rest. Force is applied to the cart
for time t, after which the car has speed v. Suppose the
same force is applied for the same time to a second cart
with twice the mass. Friction is negligible. Afterward, the
second cart’s speed will be
A.
1
v
4
B.
1
2
v
C. v
D. 2v
E. 4v
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Example 6.3 Speed of a Towed Car
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Example 6.3 Speed of a Towed Car
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Example 6.3 Speed of a Towed Car
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Example 6.3 Speed of a Towed Car
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QuickCheck 6.6
The box is sitting on the floor
of an elevator. The elevator is
accelerating upward. The
magnitude of the normal
force on the box is
A. n > mg.
B. n = mg.
C. n < mg.
D. n = 0.
E. Not enough information to tell.
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QuickCheck 6.6
The box is sitting on the floor
of an elevator. The elevator is
accelerating upward. The
magnitude of the normal
force on the box is
A. n > mg.
B. n = mg.
C. n < mg.
Upward acceleration
requires a net upward force.
D. n = 0.
E. Not enough information to tell.
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Mass: An Intrinsic Property
 A pan balance, shown in
the figure, is a device for
measuring mass.
 The measurement
does not depend on
the strength of gravity.
 Mass is a scalar
quantity that describes
an object’s inertia.
 Mass describes the
amount of matter in an object.
 Mass is an intrinsic property of an object.
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Gravity: A Force
 Gravity is an attractive,
long-range force between
any two objects.
 The figure shows two
objects with masses m1
and m2 whose centers are
separated by distance r.
 Each object pulls on the
other with a force:
where G = 6.67 × 10−11 N m2/kg2 is the gravitational
constant.
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Gravity: A Force
 The gravitational force between
two human-sized objects
is very small.
 Only when one of the
objects is planet-sized
or larger does gravity
become an important force.
 For objects near the surface of the planet earth:
where M and R are the mass and radius of the earth,
and g = 9.80 m/s2.
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Gravity: A Force
 The magnitude of the gravitational force is FG = mg,
where:
 The figure shows the free-body diagram of an object in
free fall near the surface of a planet.
 With
, Newton’s
second law predicts the
acceleration to be:
 All objects on the same planet, regardless
of mass, have the same free-fall acceleration!
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Weight: A Measurement
 You weigh apples in the
grocery store by placing them
in a spring scale and
stretching a spring.
 The reading of the spring
scale is the magnitude of Fsp.
 We define the weight of an
object as the reading Fsp of a
calibrated spring scale on
which the object is stationary.
 Because Fsp is a force, weight
is measured in newtons.
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Weight: A Measurement
 A bathroom scale uses compressed
springs which push up.
 When any spring scale measures
an object at rest,
.
 The upward spring force
exactly balances the
downward gravitational
force of magnitude mg:
 Weight is defined as the magnitude of Fsp when the
object is at rest relative to the stationary scale:
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QuickCheck 6.7
An astronaut takes her bathroom scales to the moon,
where g = 1.6 m/s2. On the moon, compared to at
home on earth:
A. Her weight is the same and her mass is less.
B. Her weight is less and her mass is less.
C. Her weight is less and her mass is the same.
D. Her weight is the same and her mass is the same.
E. Her weight is zero and her mass is the same.
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QuickCheck 6.7
An astronaut takes her bathroom scales to the moon,
where g = 1.6 m/s2. On the moon, compared to at
home on earth:
A. Her weight is the same and her mass is less.
B. Her weight is less and her mass is less.
C. Her weight is less and her mass is the same.
D. Her weight is the same and her mass is the same.
E. Her weight is zero and her mass is the same.
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Weight: A Measurement
 The figure shows a man weighing
himself in an accelerating elevator.
 Looking at the free-body diagram, the
y-component of Newton’s second law
is:
 The man’s weight as he accelerates
vertically is:
 You weigh more as an elevator
accelerates upward!
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QuickCheck 6.8
A 50-kg student (mg = 490 N) gets in a 1000-kg
elevator at rest and stands on a metric
bathroom scale. As the elevator accelerates
upward, the scale reads
A. > 490 N.
B. 490 N.
C. < 490 N but not 0 N.
D. 0 N.
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QuickCheck 6.8
A 50-kg student (mg = 490 N) gets in a 1000-kg
elevator at rest and stands on a metric
bathroom scale. As the elevator accelerates
upward, the scale reads
A. > 490 N.
B. 490 N.
C. < 490 N but not 0 N.
D. 0 N.
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QuickCheck 6.9
A 50-kg student (mg = 490 N) gets in a 1000-kg
elevator at rest and stands on a metric
bathroom scale. As the elevator accelerates
upward, the student’s weight is
A. > 490 N.
B. 490 N.
C. < 490 N but not 0 N.
D. 0 N.
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QuickCheck 6.9
A 50-kg student (mg = 490 N) gets in a 1000-kg
elevator at rest and stands on a metric
bathroom scale. As the elevator accelerates
upward, the student’s weight is
A. > 490 N.
B. 490 N.
Weight is reading of a
scale on which the
object is stationary
C. < 490 N but not 0 N.
relative to the scale.
D. 0 N.
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Weightlessness
 The weight of an object which accelerates
vertically is
 If an object is accelerating downward with ay = –g,
then w = 0.
 An object in free fall
has no weight!
 Astronauts while
orbiting the earth
are also weightless.
 Does this mean that
they are in free fall?
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Astronauts are weightless as they orbit the earth.
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QuickCheck 6.10
A 50-kg student (mg = 490 N) gets in a 1000-kg
elevator at rest and stands on a metric bathroom
scale. Sadly, the elevator cable breaks. What is the
student’s weight during the few second it takes the
student to plunge to his doom?
A. > 490 N.
B. 490 N.
C. < 490 N but not 0 N.
D. 0 N.
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QuickCheck 6.10
A 50-kg student (mg = 490 N) gets in a 1000-kg
elevator at rest and stands on a metric bathroom
scale. Sadly, the elevator cable breaks. What is the
student’s weight during the few second it takes the
student to plunge to his doom?
A. > 490 N.
B. 490 N.
C. < 490 N but not 0 N.
D. 0 N.
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The bathroom scale would read 0 N.
Weight is reading of a scale on which the
object is stationary relative to the scale.
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QuickCheck 6.11
A 50-kg astronaut (mg = 490 N) is orbiting the
earth in the space shuttle. Compared to on earth:
A. His weight is the same and his mass is less.
B. His weight is less and his mass is less.
C. His weight is less and his mass is the same.
D. His weight is the same and his mass is the same.
E. His weight is zero and his mass is the same.
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QuickCheck 6.11
A 50-kg astronaut (mg = 490 N) is orbiting the
earth in the space shuttle. Compared to on earth:
A. His weight is the same and his mass is less.
B. His weight is less and his mass is less.
C. His weight is less and his mass is the same.
D. His weight is the same and his mass is the same.
E. His weight is zero and his mass is the same.
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Static Friction
 A shoe pushes on a wooden
floor but does not slip.
 On a microscopic scale, both
surfaces are “rough” and high
features on the two surfaces
form molecular bonds.
 These bonds can produce a
force tangent to the surface,
called the static friction force.
 Static friction is a result of many
molecular springs being
compressed or stretched ever
so slightly.
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Static Friction
 The figure shows a person
pushing on a box that, due
to static friction, isn’t moving.
 Looking at the free-body
diagram, the x-component
of Newton’s first law requires
that the static friction force
must exactly balance the
pushing force:

points in the direction opposite to the
way the object would move if there were no static friction.
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Static Friction
Static friction acts in response to an applied force.
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QuickCheck 6.12
A box on a rough surface is
pulled by a horizontal rope
with tension T. The box is
not moving. In this situation:
A. fs > T.
B. fs = T.
C. fs < T.
D. fs = smg.
E. fs = 0.
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QuickCheck 6.12
A box on a rough surface is
pulled by a horizontal rope
with tension T. The box is
not moving. In this situation:
A. fs > T.
B. fs = T.
Newton’s first law.
C. fs < T.
D. fs = smg.
E. fs = 0.
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Static Friction
 Static friction force has a maximum possible size
fs max.
 An object remains at rest as long as fs < fs max.
 The object just begins to slip when fs = fs max.
 A static friction force fs > fs max is not physically
possible.
where the proportionality constant μs is called
the coefficient of static friction.
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QuickCheck 6.13
A box with a weight of 100 N
is at rest. It is then pulled by
a 30 N horizontal force.
Does the box move?
A. Yes
B. No
C. Not enough information to say.
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QuickCheck 6.13
A box with a weight of 100 N
is at rest. It is then pulled by
a 30 N horizontal force.
Does the box move?
A. Yes
B. No
30 N < fs max = 40 N
C. Not enough information to say.
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Kinetic Friction
 The kinetic friction force is
proportional to the magnitude
of the normal force:
where the proportionality
constant μk is called the
coefficient of kinetic friction.
 The kinetic friction direction is
opposite to the velocity of the
object relative to the surface.
 For any particular pair of
surfaces, μk < μs.
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QuickCheck 6.14
A box is being pulled to the right
over a rough surface. T > fk, so the
box is speeding up. Suddenly the
rope breaks.
What happens? The box
A. Stops immediately.
B. Continues with the speed it had when the rope broke.
C. Continues speeding up for a short while, then slows
and stops.
D. Keeps its speed for a short while, then slows and stops.
E. Slows steadily until it stops .
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QuickCheck 6.14
A box is being pulled to the right
over a rough surface. T > fk, so the
box is speeding up. Suddenly the
rope breaks.
What happens? The box
A. Stops immediately.
B. Continues with the speed it had when the rope broke.
C. Continues speeding up for a short while, then slows
and stops.
D. Keeps its speed for a short while, then slows and stops.
E. Slows steadily until it stops.
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QuickCheck 6.15
A box is being pulled to the right
at steady speed by a rope that
angles upward. In this situation:
A. n > mg.
B. n = mg.
C. n < mg.
D. n = 0.
E. Not enough information to judge the size of the normal force.
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QuickCheck 6.15
A box is being pulled to the right
at steady speed by a rope that
angles upward. In this situation:
A. n > mg.
B. n = mg.
C. n < mg.
D. n = 0.
E. Not enough information to
judge the size of the normal force.
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QuickCheck 6.16
A box is being pulled to the right by
a rope that angles upward. It is
accelerating. Its acceleration is
A. T (cosksin) – kg.
m
B. T (cosksin) – kg.
m
C. T (sinkcos) – kg.
m
You’ll have to work this
one out.
Don’t just guess!
D. T – kg.
m
E. T cos – kg.
m
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Slide 6-85
QuickCheck 6.16
A box is being pulled to the right by
a rope that angles upward. It is
accelerating. Its acceleration is
A. T (cosksin) – kg.
m
B. T (cosksin) – kg.
m
C. T (sinkcos) – kg.
m
D. T – kg.
m
E. T cos – kg.
m
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Rolling Motion
 If you slam on the brakes so
hard that the car tires slide
against the road surface,
this is kinetic friction.
 Under normal driving
conditions, the portion
of the rolling wheel that
contacts the surface is
stationary, not sliding.
 If your car is accelerating or decelerating or turning,
it is the static friction of the road on the wheels that
provides the net force which accelerates the car.
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Rolling Friction
 A car with no engine or
brakes applied does not
roll forever; it gradually
slows down.
 This is due to rolling
friction.
 The force of rolling friction can be calculated as:
where μr is called the coefficient of rolling friction.
 The rolling friction direction is opposite to the velocity
of the rolling object relative to the surface.
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Coefficients of Friction
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A Model of Friction
 The actual causes of friction involve microscopic surface
properties and molecular bonds.
 Experiments show that reasonable predictions are
produced by a model of friction — a simplification of
reality:
 Here “motion” means “motion relative to the surface.”
 Forces of kinetic and rolling friction are proportional to
the normal force of the surface on the object.
 The maximum static friction force is proportional to the
normal force of the surface on the object.
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A Model of Friction
The friction force response to an increasing applied force.
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Causes of Friction
 All surfaces are very rough
on a microscopic scale.
 When two surfaces are
pressed together, the high
points on each side come
into contact and form
molecular bonds.
 The amount of contact
depends on the normal
force n.
 When the two surfaces are
sliding against each other, the
bonds don’t form fully, but they
do tend to slow the motion.
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Drag
 The air exerts a drag force on objects as they move
through the air.
 Faster objects experience a greater drag force than
slower objects.
 The drag force on
a high-speed
motorcyclist is
significant.
 The drag force
direction is opposite
the object’s velocity.
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Drag
 For normal-sized objects on earth traveling at a
speed v which is less than a few hundred meters
per second, air resistance can be modeled as:
 A is the cross-section area of the object.
 ρ is the density of the air, which is about 1.2 kg/m3.
 C is the drag coefficient, which is a dimensionless
number that depends on the shape of the object.
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Drag
Cross-section areas for objects of different shape.
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Example 6.7 Air Resistance Compared to
Rolling Friction
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Example 6.7 Air Resistance Compared to
Rolling Friction
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Example 6.7 Air Resistance Compared to
Rolling Friction
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Example 6.7 Air Resistance Compared to
Rolling Friction
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Terminal Speed
 The drag force from the
air increases as an object
falls and gains speed.
 If the object falls
far enough, it will
eventually reach a
speed at which D = FG.
 At this speed, the net force is zero, so the object falls
at a constant speed, called the terminal speed vterm.
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Terminal Speed
 The figure shows the
velocity-versus-time graph
of a falling object with and
without drag.
 Without drag, the velocity
graph is a straight line with
ay = –g.
 When drag is included, the
vertical component of the
velocity asymptotically
approaches –vterm.
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Example 6.10 Make Sure the Cargo Doesn’t
Slide
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Example 6.10 Make Sure the Cargo Doesn’t
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MODEL
 Let the box, which we’ll model as a particle, be the object of interest.
 Only the truck exerts contact forces on the box.
 The box does not slip relative to the truck.
 If the truck bed were frictionless, the box would slide backward as
seen in the truck’s reference frame as the truck accelerates.
 The force that prevents sliding is static friction.
 The box must accelerate forward with the truck: abox = atruck.
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Example 6.10 Make Sure the Cargo Doesn’t
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Example 6.10 Make Sure the Cargo Doesn’t
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Example 6.10 Make Sure the Cargo Doesn’t
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Example 6.10 Make Sure the Cargo Doesn’t
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Chapter 6 Summary Slides
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General Strategy
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General Strategy
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Important Concepts
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Important Concepts
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