Chapters 4, 5

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Chapters 4, 5 Force and Laws of Motion
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• What causes motion?
• Thats the wrong question!
• The ancient Greek philosopher Aristotle believed
  that forces - pushes and pulls - caused motion
• The Aristotelian view prevailed for some 2000
  years
• Galileo first discovered the correct relation
  between force and motion
• Force causes not motion itself but change in
  motion

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• Newtonian mechanics
• Describes motion and interaction of objects
• Applicable for speeds much slower than the speed
  of light
• Applicable on scales much greater than the
  atomic scale
• Applicable for inertial reference frames
  frames that dont accelerate themselves

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• Force
• What is a force?
• Colloquial understanding of a force a push or
  a pull
• Forces can have different nature
• Forces are vectors
• Several forces can act on a single object at a
  time they will add as vectors

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• Force superposition
• Forces applied to the same object are adding as
  vectors superposition
• The net force a vector sum of all the forces
  applied to the same object

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• Newtons First Law
• If the net force on the body is zero, the bodys
  acceleration is zero

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• Newtons Second Law
• If the net force on the body is not zero, the
  bodys acceleration is not zero
• Acceleration of the body is directly
  proportional to the net force on the body
• The coefficient of proportionality is equal to
  the mass (the amount of substance) of the object

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• Newtons Second Law
• SI unit of force kgm/s2 N (Newton)
• Newtons Second Law can be applied to all the
  components separately
• To solve problems with Newtons Second Law we
  need to consider a free-body diagram
• If the system consists of more than one body,
  only external forces acting on the system have to
  be considered
• Forces acting between the bodies of the system
  are internal and are not considered

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• Newtons Third Law
• When two bodies interact with each other, they
  exert forces on each other
• The forces that interacting bodies exert on each
  other, are equal in magnitude and opposite in
  direction

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• Forces of different origins
• Gravitational force
• Normal force
• Tension force
• Frictional force (friction)
• Drag force
• Spring force

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• Gravity force (a bit of Ch. 8)
• Any two (or more) massive bodies attract each
  other
• Gravitational force (Newton's law of
  gravitation)
• Gravitational constant G 6.6710 11 Nm2/kg2
  6.6710 11 m3/(kgs2) universal constant

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Gravity force at the surface of the
Earth g 9.8 m/s2
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• Gravity force at the surface of the Earth
• The apple is attracted by the Earth
• According to the Newtons Third Law, the Earth
  should be attracted by the apple with the force
  of the same magnitude

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• Weight
• Weight (W) of a body is a force that the body
  exerts on a support as a result of gravity pull
  from the Earth
• Weight at the surface of the Earth W mg
• While the mass of a body is a constant, the
  weight may change under different circumstances

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• Tension force
• A weightless cord (string, rope, etc.) attached
  to the object can pull the object
• The force of the pull is tension ( T )
• The tension is pointing away from the body

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Free-body diagrams
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Chapter 4 Problem 56
Your engineering firm is asked to specify the
maximum load for the elevators in a new building.
Each elevator has mass 490 kg when empty and
maximum acceleration 2.24 m/s2. The elevator
cables can withstand a maximum tension of 19.5 kN
before breaking. For safety, you need to ensure
that the tension never exceeds two-thirds of that
value. What do you specify for the maximum load?
How many 70-kg people is that?
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• Normal force
• When the body presses against the surface
  (support), the surface deforms and pushes on the
  body with a normal force (n) that is
  perpendicular to the surface
• The nature of the normal force reaction of the
  molecules and atoms to the deformation of material

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• Normal force
• The normal force is not always equal to the
  gravitational force of the object

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Free-body diagrams
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Free-body diagrams
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Chapter 5 Problem 19
If the left-hand slope in the figure makes a 60
angle with the horizontal, and the right-hand
slope makes a 20 angle, how should the masses
compare if the objects are not to slide along the
frictionless slopes?
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• Spring force
• Spring in the relaxed state
• Spring force (restoring force) acts to restore
  the relaxed state from a deformed state

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• Hookes law
• For relatively small deformations
• Spring force is proportional to the deformation
  and opposite in direction
• k spring constant
• Spring force is a variable force
• Hookes law can be applied not to springs only,
  but to all elastic materials and objects

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• Frictional force
• Friction ( f ) - resistance to the sliding
  attempt
• Direction of friction opposite to the
  direction of attempted sliding (along the
  surface)
• The origin of friction bonding between the
  sliding surfaces (microscopic cold-welding)

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• Static friction and kinetic friction
• Moving an object static friction vs. kinetic

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• Friction coefficient
• Experiments show that friction is related to the
  magnitude of the normal force
• Coefficient of static friction µs
• Coefficient of kinetic friction µk
• Values of the friction coefficients depend on
  the combination of surfaces in contact and their
  conditions (experimentally determined)

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Free-body diagrams
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Free-body diagrams
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Chapter 5 Problem 30
Starting from rest, a skier slides 100 m down a
28 slope. How much longer does the run take if
the coefficient of kinetic friction is 0.17
instead of 0?
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• Drag force
• Fluid a substance that can flow (gases,
  liquids)
• If there is a relative motion between a fluid
  and a body in this fluid, the body experiences a
  resistance (drag)
• Drag force (R)
• R ½D?Av2
• D - drag coefficient ? fluid density A
  effective cross-sectional area of the body (area
  of a cross-section taken perpendicular to the
  velocity) v - speed

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• Terminal velocity
• When objects falls in air, the drag force points
  upward (resistance to motion)
• According to the Newtons Second Law
• ma mg R mg ½D?Av2
• As v grows, a decreases. At some point
  acceleration becomes zero, and the speed value
  riches maximum value terminal speed
• ½D?Avt2 mg

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• Terminal velocity
• Solving ½D?Avt2 mg we obtain

vt 300 km/h
vt 10 km/h
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• Centripetal force
• For an object in a uniform circular motion, the
  centripetal acceleration is
• According to the Newtons Second Law, a force
  must cause this acceleration centripetal force
• A centripetal force accelerates a body by
  changing the direction of the bodys velocity
  without changing the speed

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• Centripetal force
• Centripetal forces may have different origins
• Gravitation can be a centripetal force
• Tension can be a centripetal force
• Etc.

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• Centripetal force
• Centripetal forces may have different origins
• Gravitation can be a centripetal force
• Tension can be a centripetal force
• Etc.

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Free-body diagram
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Chapter 5 Problem 25
Youre investigating a subway accident in which a
train derailed while rounding an unbanked curve
of radius 132 m, and youre asked to estimate
whether the train exceeded the 45-km/h speed
limit for this curve. You interview a passenger
who had been standing and holding onto a strap
she noticed that an unused strap was hanging at
about a 15 angle to the vertical just before the
accident. What do you conclude?
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Answers to the even-numbered problems Chapter 4
Problem 20 7.7 cm
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Answers to the even-numbered problems Chapter 4
Problem 26 590 N
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Answers to the even-numbered problems Chapter 4
Problem 38 5.77 N 72.3
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Answers to the even-numbered problems Chapter 5
Problem 28 580 N opposite to the motion of the
cabinet
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Answers to the even-numbered problems Chapter 5
Problem 50 110 m