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Energy

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Title: Energy


1
Energy
2
  • Analyzing the motion of an object can often get
    to be very complicated and tedious requiring
    detailed knowledge of the path, frictional
    forces, etc.
  • There has to be an easier way
  • It turns out that there is it is done by
    analyzing the objects energy.

3
Energy
  • The something that enables an object to do work
    is energy.
  • Energy is measured in Joules (J).
  • Forms of Energy
  • Mechanical (kinetic and potential)
  • Thermal (heat)
  • Electromagnetic (light)
  • Nuclear
  • Chemical

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5
Mechanical Energy
  • Mechanical energy is the form of energy due to
    the position or the movement of a mass.

6
Kinetic Energy
  • Kinetic energy is the energy of motion it is
    associated with the state of motion of an object.
  • The faster an object is moving, the greater its
    kinetic energy an object at rest has zero
    kinetic energy.
  • For an object of mass m, we will define kinetic
    energy as
  • The SI unit of kinetic energy is the Joule (J).

7
Kinetic Energy
  • If we do positive work on an object by pushing on
    it with some force, we can increase the objects
    kinetic energy (and thereby increasing its
    speed).
  • We can account for the change in kinetic energy
    by saying that the force transferred energy from
    you to the object.
  • If we do negative work on an object by pushing on
    it with some force in the direction opposite to
    the direction of motion, we can decrease the
    objects kinetic energy (and decrease its
    speed).
  • We can account for the change in kinetic energy
    by saying that the force transferred energy from
    the object to you.

8
Kinetic Energy
  • Whenever we have a transfer of energy via a
    force, we say that work is done on the object by
    the force.
  • Work W is energy transferred to or from an object
    by means of a force acting on that object.
  • Energy transferred to the object is positive
    work.
  • Energy transferred from the object is negative
    work.
  • Work is nothing more than transferred energy it
    therefore has the same units as energy and is
    also a scalar quantity.
  • Note that nothing material is transferred.
  • Think of it like the balance in two bank
    accounts when money is transferred the number
    for one account goes down by some amount and the
    number for the other account goes up by the same
    amount.

9
Work-Energy Theorem
  • Suppose we have an bead which is constrained to
    move only along the length of a frictionless
    wire.
  • We then supply a constant force F on the bead at
    some angle ? to the wire.
  • Because the force is constant, we know that the
    acceleration will also be constant.

10
Work-Energy Theorem
  • But because of the constraint, only the force in
    the x direction matters, thus Fx max where m
    is the beads mass.
  • We can relate the beads velocity at some
    distance down the wire to the acceleration using

11
Work-Energy Theorem
  • Solving for ax, substituting into the Fx
    equation, multiplying both sides by d, and
    distributing the ½m throughout the equation

12
Work-Energy Theorem
  • But we can see that the right side of the
    equation is no more than the kinetic energy after
    the force has been applied minus the kinetic
    energy before the force was appliedand that
    by definition is the work done

13
Work-Energy Theorem
  • When calculating the work done on an object by a
    force during a displacement, use only the
    component of the force that is parallel to the
    objects displacement.
  • where ? is the angle between the force F and the
    horizontal.
  • The force component perpendicular to the
    displacement does no work.

14
Work-Energy Theorem
  • Work-Energy Theorem the net work done on an
    object is equal to the change in kinetic energy
    of the object.
  • W Kf Ki Fd 0.5m(vf2 - vi2)
  • A net force causes an object to change its
    kinetic energy because a net force causes an
    object to accelerate, and acceleration means a
    change in velocity, and if velocity changes,
    kinetic energy changes.

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17
Gravitational Potential Energy
  • Potential energy (U) an object may store energy
    because of its position. Energy that is stored
    is called potential energy because in the stored
    state it has the potential to do work.
  • Work is required to lift objects against Earths
    gravity.
  • Potential energy due to elevated positions is
    gravitational potential energy.
  • The amount of gravitational potential energy
    possessed by an elevated object is equal to the
    work done against gravity in lifting it.
  • Ug Fwh mgh

18
Work and Potential Energy
  • When we throw a tomato up in the air, negative
    work is being done on the tomato which causes it
    to slow down during its ascent.
  • As a result, the kinetic energy of the tomato is
    reduced eventually to zero at the highest
    point.
  • But where did that energy go???

19
Work and Potential Energy
  • Where it went was into an increase in the
    gravitational potential energy of the tomato.
  • The reverse happens when the tomato begins to
    fall down.
  • Now the positive work done by the gravitational
    force causes the gravitational potential energy
    to be reduced and the tomatos kinetic energy to
    increase.

20
Work and Potential Energy
  • From this we can see that for either the rise or
    fall of the tomato, the change ?U in the
    gravitational potential energy is the negative of
    the work done on the tomato by the gravitational
    force
  • In equation form we get

21
  • Only changes in potential energy have meaning it
    is important that all heights be measured from
    the same origin.
  • In many problems, the ground is chosen as the
    zero level for the determination of the height.
  • As the ball falls from A to B, the potential
    energy at A is converted to kinetic energy at B.
  • The amount of potential energy of the ball at
    point A will equal the amount of kinetic energy
    of the ball at point B.

22
Elastic Potential Energy
  • Stretching or compressing an elastic object
    requires energy and this energy is stored in the
    elastic object as elastic potential energy.
  • The work required to stretch or compress a spring
    is dependent on the force constant k.
  • The force constant will not change for a
    particular spring as long as the spring is not
    permanently distorted (which occurs when the
    elastic limit of the spring is exceeded).
  • F kx

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24
Elastic Potential Energy
  • The force required to stretch/compress an elastic
    object is not a constant force. The work needed
    varies with the amount of stretch/compression.
  • W 0.5kx2
  • The potential energy of an elastic object is
    equal to the work done on the elastic object.
  • Ue 0.5kx2
  • In actual practice, a small fraction of the work
    in stretching/compressing an elastic object is
    converted into heat energy in the spring.

25
Work and Elastic Potential Energy
  • If we give the block a shove to the right, the
    kinetic energy of the block is transferred into
    elastic potential energy as the spring
    compresses.
  • The work done in compressing the spring is the
    negative of the change in the blocks kinetic
    energy.
  • And of course the reverse happens when the spring
    stretches back out potential energy gets
    transformed back into kinetic energy.

26
Conservation of Mechanical Energy
  • Conservative forces all the work done is stored
    as energy and is available to do work later.
    Example gravitational forces, elastic forces.
  • Nonconservative (dissipative) forces the force
    generally produces a form of energy that is not
    mechanical.
  • Friction is a nonconservative (dissipative) force
    because it produces heat (thermal energy, not
    mechanical).
  • The total amount of energy in any closed system
    remains constant.

27
Conservation of Mechanical Energy
  • The sum of the potential and kinetic energy of a
    system remains constant when no dissipative
    forces (like friction) act on the system.
  • Law of Conservation of Energy energy cannot be
    created or destroyed it may be changed from one
    form to another or transferred from one object to
    another, but the total amount of energy never
    changes.

28
Conservation of Mechanical Energy
  • In a closed system in which gravitational
    potential energy and kinetic energy are involved,
    the potential energy at the highest point is
    equal to the kinetic energy at the lowest point.
  • mgh 0.5mv2

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31
  • Notice that the sum of the potential energy (PE)
    and kinetic energy (KE) at every point is 40000
    J. Energy is conserved.

32
  • The velocity at the lowest point can be
    determined by the height

33
Conservation of Mechanical Energy
  • The change in velocity due to a change in height
    can also be determined
  • The height can be determined by the initial
    velocity (vf 0 m/s)

34
Conservation of Mechanical Energy
  • In a closed system in which elastic potential
    energy and kinetic energy are involved, the
    potential energy at the maximum distance of
    stretch/compression is equal to the kinetic
    energy at the equilibrium (rest) position.

35
Conservation of Mechanical Energy
  • Conservation of Energy Equation
  • W done (by applied force) Ugravitational before
    Uelastic before K before Ugravitational
    after Uelastic after K after W done
    (usually by friction)
  • Fappliedd m g hi 0.5 k xi2 0.5 m vi2
    m g hf 0.5 k xf2 0.5 m vf2 FFd

36
  • If there is no change in height m g hi and m
    g hf drop out of the equation.
  • If there is no spring or elastic object 0.5
    k xi2 and 0.5 k xf2 drop out of the equation.
  • If there is no change in velocity 0.5 m vi2
    and 0.5 m vf2 drop out of the equation.
  • If there is no applied force (a push/pull that
    you supply) Fappliedd drops out of the
    equation.
  • If there is no friction FFd drops out of the
    equation.

37
Helpful Online Links
  • Work Energy Theorem
  • The Work-Energy Theorem
  • Elastic Constant k
  • Hookes Law Applet
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