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7.4 Conservative Forces and
Potential Energy


Define a potential energy function, U,
such that the work done by a
conservative force equals the decrease
in the potential energy of the system
The work done by such a force, F, is
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DU is negative when F and x are in the
same direction
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Conservative Forces and
Potential Energy
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The conservative force is related to the
potential energy function through
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The conservative force acting between parts
of a system equals the negative of the
derivative of the potential energy associated
with that system
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This can be extended to three dimensions
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Conservative Forces and
Potential Energy – Check

Look at the case of an object located
some distance y above some reference
point:
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This is the expression for the vertical
component of the gravitational force
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7.6 Potential Energy for
Gravitational Forces
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Generalizing
gravitational potential
energy uses Newton’s
Law of Universal
Gravitation:

The potential energy
then is
Fig 7.12
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Potential Energy for
Gravitational Forces, Final
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The result for the
earth-object system
can be extended to
any two objects:
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Gravitational potential energy for
three particles
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Electric Potential Energy
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Coulomb’s Law gives the electrostatic
force between two particles
This gives an electric potential energy
function of
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7.7 Energy Diagrams and
Stable Equilibrium
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The x = 0 position is
one of stable
equilibrium
Configurations of stable
equilibrium correspond
to those for which U(x)
is a minimum
x=xmax and x=-xmax are
called the turning points
Fig 7.15
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Energy Diagrams and
Unstable Equilibrium
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Fx = 0 at x = 0, so the
particle is in equilibrium
For any other value of x,
the particle moves away
from the equilibrium
position
This is an example of
unstable equilibrium
Configurations of
unstable equilibrium
correspond to those for
which U(x) is a
maximum
Fig 7.16
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
A particle is attached between two identical springs
on a horizontal frictionless table. Both springs have
spring constant k and are initially unstressed. (a) The
particle is pulled a distance x along a direction
perpendicular to the initial configuration of the springs
as shown in Figure. Show that the force exerted by
the springs on the particle is


L
F  2 kx 1 
x 2  L2


ˆ
i


(b) Determine the amount of work done by this force
in moving the particle from x = A to x = 0.
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
(a) Show that the potential energy of the
system is

Ux  kx2  2kL L  x 2  L2
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
(b) Make a plot of U(x) versus x and identify
all equilibrium points.
(c) If the particle of mass m is pulled in a
distance d to the right and then released,
what is its speed when it reaches the
equilibrium point x = 0?
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Exercises of chapter 7

3, 5, 9, 14, 17, 26, 33, 39, 42, 54, 62
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