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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

3

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:

6

Gravitational potential energy for three particles

7

Electric Potential Energy Coulomb’s Law gives the electrostatic

force between two particles

This gives an electric potential energy function of

8

9

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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