The law of conservation of mechanical energy

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The law of conservation of mechanical energy
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TOPIC The law of conservation of mechanical
energy
How is kinetic energy defined?
Kinetic energy is defined as Ek 1/2 mv2
How is gravitational energy is defined?
Gravitational potential energy is defined as
Epg mgh
How is elastic potential energy is defined?
Kinetic energy is defined as Eps1/2kx2
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1 ? The concept of mechanical energy
Mechanical energy is total of Ek and Ep.
We can express it as EEkEp
E1/2mv2mgh1/2kx2

1. ?The kinetic energy and potential energy
   converted into each other

For example
(1) When an object is free fall
How does energy convert?
(2) When an object is thrown upward ,
what is the energy conversion?
(3) When the arrow is shot out by the bow,
what is the energy conversion?
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Conclusion kinetic energy and potential
energy can be converted into each other.
3? The concept of conservative forces
An object moves from point P to point Q along
path1 or path2,
The work done by gravity is the same.
WG mgh
The work done by elastic force is the same.
Ws 1/2kx22-1/2kx12
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What conclusion can we come to according to the
examples above?
A force is conservative force, if the work done
by the force on an object moving between two
points is independent of the path taken.
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4? The law of conservation of mechanical energy
According to the theorem of kinetic energy ,
the work done by the resultant
force equals the change of kinetic energy.
W?Ek (1)
The resultant force is only gravity
W mg (hi-hf ) (2)
The change of kinetic energy
?Ek 1/2 mvf2 -1/2mvi2 (3)
Hence mg (hi-hf )1/2mvf2 -1/2mvi2
Or 1/2mvi2mghi1/2mvf2mghf
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Illustration We get this equation by free fall
motion. Similarly, either an object is in
rectilinear motion or in curvilinear motion, only
the work was done by gravity. We can always get
the same conclusion.
This equation can be given
EEkiEpgi Ekf Epgf
This is the law of conservation of mechanical
energy.
It states that the total mechanical energy of a
system remains constant if only gravity does
works.
Similarly, if only the elastic force does work,
the law of conservation of mechanical energy can
be given
1/2mvi2 1/2kxi21/2mvf21/2kxf2
or EEki EpsiEkfEpsf
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Generally, if only conservative forces
do work,
the law of conservation of mechanical
energy states that
the total mechanical energy of a
system remains constant if the only force
that does work is conservative force.

if the kinetic energy of
a conservative system increases ( or decreases)
by some amount , the potential energy must
decrease ( or increase) the same amount.
This is equivalent to the statement that
It can be given E Eki?EpiEkf ?Epf
Ekf - Eki ?Epi - ?Epf
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5 ?Application
Example problem A pendulum consists of a ball of
mass 100g attached to a light cord of length
50cm. The ball is released near nose of a
person from rest when the cord makes an angle
60 with the vertical ( ignoring air
resistance)
A PredictObserveExplain(POE)methods
  (1)When the ball comes back, you predict what
will happen? ( Can the ball
hit the persons nose?)
(2)Do this experiment and observe carefully
the result of this experiment. What is the result
of this experiment?
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(3) Explain the result of this experiment with
the law of conservation of mechanical energy.
In the motion of the ball, two forces act on the
ball. One is gravity and the other is the force
of tension. Only gravity does work ( the
direction of tension is perpendicular to the
direction of the balls velocity, tension doesnt
do work), the mechanical energy is conserved. Its
initial kinetic energy and its final kinetic
energy are zero. The energy is entirely
gravitational energy, so the ball comes back to
original position a. As a result, the ball cant
hit the nose of the person.
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B Calculation What is the speed of the ball
when it is at the lowest point b ?
Solution
We choose to take the zero potential energy at
point b.
Known
Unknown
M100g0.1kg
Eka0
L50cm0.5m
Vb?
Epb0
? 600
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According to the law of conservation of
mechanical energy
EkaEpaEkbEpb
hab L Lcos?
0mghab1/2mvb20
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6 Activity of students --------Ask and answer the
questions studied with each other .
Content Formula
kinetic energy Ek 1/2 mv2
gravitational potential energy Epg mgh
elastic potential energy Eps1/2kx2
mechanical energy. E1/2mv2mgh1/2kx2
the theorem of kinetic energy W?Ek
the law of conservation of mechanical energy EEki?EpiEkf?Epf Ekf - Eki ?Epi - ?Epf
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