Kinetic Energy: Energy associated with motion K mv2

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Potential Energy and Energy Conservation
Kinetic Energy Energy associated with motion K
½ mv2 Work done by a force on an object is that
forces contribution to DK may only depend upon
initial and final position of object Conservative
Forces Work expressed as change in Potential
Energy
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Gravitational Potential Energy Work done by
gravity force portion of displacement along
force force is always vertical gt work weight
height lifted dW F . dl -mg dy accumulated
work by force of gravity W mg(y1 y2) mgy1
mgy2 Dmgy Note work done does not depend
upon path! Identify potential energy U ? mgy W
DU Work comes at the expense of potential
energy.
dl
Dy
w
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Conservation of Mechanical Energy (with
gravity) In the absence of any other forces W
DK (work energy theorem) now W - DU so
DK - DU or DK DU 0 Define total
Energy E K U so DE 0 or E1
E2 In the absence of other forces, total
energy is conserved! With other forces, the other
forces still do work DE Wother
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Example A .145 kg baseball is thrown straight
up into the air with an initial speed of 20 m/s.
Determine kinetic, potential and total energy at
the balls initial height as well as when the
ball reaches its maximum height.
Example, Projectile Motion Show that two balls
launched at different angles, but with the same
initial speed, will have the same speed at a
given height h. Use energy methods to relate the
maximum height of a projectile to its initial
speed and direction.
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Example, Projectile Motion Use energy methods
to relate the maximum height of a projectile to
its initial speed and direction.
Example An object of mass m slides down a
semicircular frictionless ramp of radius R
starting from rest. Find the speed and the
normal force at the bottom of the ramp
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Example A 12 kg crate is slid up a 2.5 m long
ramp inclined at 30. The crate is given an
initial velocity of 5.00 m/s up the ramp, but
only makes it 1.6m up the ramp before sliding
downward. How much work is done by friction on
the upward slide? How fast is the crate moving as
it slides back to the bottom of the ramp?
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• Elastic Potential Energy
• from last chapter work done on spring W ½
  kx22 - ½ kx12
• work done by spring Wel ½ kx12 - ½ kx22
• elastic potential energy W - DU
• U ½ kx2
• note x is always measured from equilibrium!
• another contribution to total energy
• DE Wother

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Example A 0.200 kg mass rides on a horizontal
frictionless surface, attached to a spring with a
force constant of 5.00 N/m. The mass is pulled
.100m from equilibrium. What is the speed of the
mass when it is .080 m from equilibrium? What is
the speed of the mass when it is at equilibrium?
With the same mass-spring system above, a
constant force of .610 N is applied to the mass
which is initially at rest at the equilibrium
position. What is the gliders speed when it
has reached x .100 m? If the force is turned
off at x .100 m, how much further does the mass
go?
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Example A 2000 Kg elevator is falling at a
speed of 25.0 m/s when it bottoms out on a
spring at the bottom of the elevator shaft. The
spring is supposed to stop the elevator,
compressing 3.00 m to bring it to rest. During
this breaking process, a safety clamp provides a
constant 17,000 N frictional force on the
elevator. What is the spring constant? What is
the upward acceleration of the elevator just
after it comes to a rest?
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For a Conservative Force Work can always be
expressed as a change in potential energy Is
reversible (Wab -Wba) Is independent of the
path of the object Does no work when the initial
and final positions of the object are the same
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Example A 40.0 kg object with a coefficient of
kinetic friction of 0.200 may be dragged across a
room by two paths a direct 2.5m path, and dogleg
path of 2.00m on the first leg and 1.50m along
the second leg. How much work is done dragging
the object across each path?
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In a closed system, energy is conserved! loss
(or gain) of energy associated with the change of
state of the materials is due to changes in
Internal Energy change of temperature,
melting/freezing, etc. (Thermodynamics) can be
understood in terms of microscopic kinetic and
potential energy (Statistical Mechanics) DE DK
DU DUint 0
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Force and Potential Energy
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Force and Potential Energy
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2-d example
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Interpreting Energy Graphs force is negative of
slope of U force is down hill equilibrium
zero force for 2-d use contour maps
stable equilibrium
unstable equilibrium