Chapters 7, 8

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Chapters 7, 8 Energy
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• Energy
• What is energy?
• Energy - is a fundamental, basic notion in
  physics
• Energy is a scalar, describing state of an
  object or a system
• Description of a system in energy language is
  equivalent to a description in force language
• Energy approach is more general and more
  effective than the force approach
• Equations of motion of an object (system) can be
  derived from the energy equations

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• Scalar product of two vectors
• The result of the scalar (dot) multiplication of
  two vectors is a scalar
• Scalar products of unit vectors

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• Scalar product of two vectors
• The result of the scalar (dot) multiplication of
  two vectors is a scalar
• Scalar product via unit vectors

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• Some calculus
• In 1D case

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• Some calculus
• In 1D case
• In 3D case, similar derivations yield
• K kinetic energy

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• Kinetic energy
• K mv2/2
• SI unit kgm2/s2 J (Joule)
• Kinetic energy describes objects state of
  motion
• Kinetic energy is a scalar

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Chapter 7 Problem 31
A 3.00-kg object has a velocity of (6.00i
2.00j) m/s. (a) What is its kinetic energy at
this moment? (b) What is the net work done on the
object if its velocity changes to (800i
4.00j) m/s?
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• Workkinetic energy theorem
• Wnet work (net)
• Work is a scalar
• Work is equal to the change in kinetic energy,
  i.e. work is required to produce a change in
  kinetic energy
• Work is done on the object by a force

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• Work graphical representation
• 1D case Graphically - work is the area under
  the curve F(x)

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• Net work vs. net force
• We can consider a system, with several forces
  acting on it
• Each force acting on the system, considered
  separately, produces its own work
• Since

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• Work done by a constant force
• If a force is constant
• If the displacement and the constant force are
  not parallel

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• Work done by a spring force
• Hookes law in 1D
• From the workkinetic energy theorem

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• Work done by the gravitational force
• Gravity force is constant near the surface of
  the Earth
• If the displacement is vertically up
• In this case the gravity force does a negative
  work (against the direction of motion)

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• Lifting an object
• We apply a force F to lift an object
• Force F does a positive work Wa
• The net work done
• If in the initial and final states the object is
  at rest, then the net work done is zero, and the
  work done by the force F is

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• Power
• Average power
• Instantaneous power the rate of doing work
• SI unit J/s kgm2/s3 W (Watt)

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• Power of a constant force
• In the case of a constant force

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Chapter 8 Problem 32
A 650-kg elevator starts from rest. It moves
upward for 3.00 s with constant acceleration
until it reaches its cruising speed of 1.75 in/s.
(a) What is the average power of the elevator
motor during this time interval? (b) How does
this power compare with the motor power when the
elevator moves at its cruising speed?
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• Conservative forces
• The net work done by a conservative force on a
  particle moving around any closed path is zero
• The net work done by a conservative force on a
  particle moving between two points does not
  depend on the path taken by the particle

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• Conservative forces examples
• Gravity force
• Spring force

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• Potential energy
• For conservative forces we introduce a
  definition of potential energy U
• The change in potential energy of an object is
  being defined as being equal to the negative of
  the work done by conservative forces on the
  object
• Potential energy is associated with the
  arrangement of the system subject to conservative
  forces

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• Potential energy
• For 1D case
• A conservative force is associated with a
  potential energy
• There is a freedom in defining a potential
  energy adding or subtracting a constant does not
  change the force
• In 3D

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Chapter 7 Problem 44
A single conservative force acting on a particle
varies F ( Ax Bx2) i N, where A and B are
constants and x is in meters. (a) Calculate the
potential energy function U(x) associated with
this force, taking U 0 at x 0. (b) Find the
change in potential energy and the change in
kinetic energy of the system as the particle
moves from x 2.00 m to x 3.00 m.
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• Gravitational potential energy
• For an upward direction the y axis

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• Elastic potential energy
• For a spring obeying the Hookes law

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• Internal energy
• The energy associated with an objects
  temperature is called its internal energy, Eint
• In this example, the friction does work and
  increases the internal energy of the surface

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• Conservation of mechanical energy
• Mechanical energy of an object is
• When a conservative force does work on the
  object
• In an isolated system, where only conservative
  forces cause energy changes, the kinetic and
  potential energies can change, but the mechanical
  energy cannot change

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• Work done by an external force
• Work is transferred to or from the system by
  means of an external force acting on that system
• The total energy of a system can change only by
  amounts of energy that are transferred to or from
  the system
• Power of energy transfer, average and
  intantaneous

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Conservation of mechanical energy pendulum
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Potential energy curve
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Potential energy curve equilibrium points
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Chapter 8 Problem 55
A 10.0-kg block is released from point A. The
track is frictionless except for the portion
between points B and C, which has a length of
6.00 m. The block travels down the track, hits a
spring of force constant 2250 N/m, and compresses
the spring 0.300 m from its equilibrium position
before coming to rest momentarily. Determine the
coefficient of kinetic friction between block and
the rough surface between B and C.
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Answers to the even-numbered problems Chapter
7 Problem 2 (a) 3.28 10-2 J (b) - 3.28
10-2 J
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Answers to the even-numbered problems Chapter
7 Problem 10 16.0
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Answers to the even-numbered problems Chapter
7 Problem 46 (7-9x2y)ˆi-3x3ˆj
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Answers to the even-numbered problems Chapter
8 Problem 14 (a) 0.791 m/s (b) 0.531 m/s
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Answers to the even-numbered problems Chapter
8 Problem 28 8.01 W
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Answers to the even-numbered problems Chapter
8 Problem 34 194 m
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Answers to the even-numbered problems Chapter
8 Problem 50 (a) 0.588 J (b) 0.588 J (c)
2.42 m/s (d) UC 0.392 J, KC 0.196 J