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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
U is negative when F and x are in the same direction
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Conservative Forces and Potential Energy The conservative force is related to the
potential energy function through
The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system 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:
This is the expression for the vertical component of the gravitational force
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7.6 Potential Energy for Gravitational Forces
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
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 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 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 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
(b) Determine the amount of work done by this force in moving the particle from x = A to x = 0.
iF ˆ1222
Lx
Lkx
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(a) Show that the potential energy of the system is
(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?
222 2 LxLkLkxxU
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Exercises of chapter 7
3, 5, 9, 14, 17, 26, 33, 39, 42, 54, 62