Chapter 6 Dynamics I: Motion Along a Line

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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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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 pucks 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 pucks subsequent motion?

Newtons first law!
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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?
• 20 N
• 10cos? N
• 10sin? N
• 20cos? N
• 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?
• 20 N
• 10cos? N
• 10sin? N
• 20cos? N
• 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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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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Using Newtons Second Law

• The essence of Newtonian mechanics can be
  expressed in two steps
• The forces on an object determine its
    acceleration , and
• The objects trajectory can be determined by
    using in the equations of kinematics.

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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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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
  objects 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 , Newtons 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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Weight A Measurement

• The figure shows a man weighing himself in an
  accelerating elevator.
• Looking at the free-body diagram, the
  y-component of Newtons second law is

• The mans 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

1. gt 490 N.
2. 490 N.
3. lt 490 N but not 0 N.
4. 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

1. gt 490 N.
2. 490 N.
3. lt 490 N but not 0 N.
4. 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?

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 students weight during the few second it
takes the student to plunge to his doom?

1. gt 490 N.
2. 490 N.
3. lt 490 N but not 0 N.
4. 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 students weight during the few second it
takes the student to plunge to his doom?

1. gt 490 N.
2. 490 N.
3. lt 490 N but not 0 N.
4. 0 N.

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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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, isnt moving.
• Looking at the free-body diagram, the x-component
  of Newtons 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 force has a maximum possible size
  fs max.
• An object remains at rest as long as fs lt fs max.
• The object just begins to slip when fs fs max.
• A static friction force fs gt fs max is not
  physically possible.

  where the proportionality constant µs is called
the coefficient of static friction.
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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 lt µs.

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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

1. n gt mg.
2. n mg.
3. n lt mg.
4. n 0.
5. 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

1. n gt mg.
2. n mg.
3. n lt mg.
4. n 0.
5. Not enough information to judge the size of the
   normal force.

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Example 6.10 Make Sure the Cargo Doesnt Slide
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Example 6.10 Make Sure the Cargo Doesnt Slide

• MODEL
• Let the box, which well 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 trucks 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 Doesnt Slide
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Example 6.10 Make Sure the Cargo Doesnt Slide
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Example 6.10 Make Sure the Cargo Doesnt Slide
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Example 6.10 Make Sure the Cargo Doesnt Slide
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