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PE s = 1 2 kx 2 , size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {}

where k size 12{k} {} is the spring’s force constant and x size 12{x} {} is the displacement from its undeformed position. The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance x size 12{x} {} . The potential energy of the spring PE s size 12{"PE" rSub { size 8{s} } } {} does not depend on the path taken; it depends only on the stretch or squeeze x size 12{x} {} in the final configuration.

An undeformed spring fixed at one end with no potential energy. (b) A spring fixed at one end and stretched by a distance x by a force F equal to k x. Work done W is equal to one half k x squared. P E s is equal to one half k x squared. (c) A graph of force F versus elongation x in the spring. A straight line inclined to x axis starts from origin. The area under this line forms a right triangle with base of x and height of k x. Area of this triangle is equal to one half k x squared.
(a) An undeformed spring has no PE s size 12{"PE" rSub { size 8{s} } } {} stored in it. (b) The force needed to stretch (or compress) the spring a distance x size 12{x} {} has a magnitude F = kx size 12{F= ital "kx"} {} , and the work done to stretch (or compress) it is 1 2 kx 2 size 12{ { {1} over {2} } ital "kx" rSup { size 8{2} } } {} . Because the force is conservative, this work is stored as potential energy ( PE s ) size 12{ \( "PE" rSub { size 8{s} } \) } {} in the spring, and it can be fully recovered. (c) A graph of F size 12{F} {} vs. x size 12{x} {} has a slope of k size 12{k} {} , and the area under the graph is 1 2 kx 2 size 12{ { {1} over {2} } ital "kx" rSup { size 8{2} } } {} . Thus the work done or potential energy stored is 1 2 kx 2 .

The equation PE s = 1 2 kx 2 size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {} has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of potential energy    is energy due to position, shape, or configuration. For shape or position deformations, stored energy is PE s = 1 2 kx 2 size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {} , where k size 12{k} {} is the force constant of the particular system and x size 12{x} {} is its deformation. Another example is seen in [link] for a guitar string.

A six-string guitar is placed vertically. The left-most string is plucked in the left direction with a force F shown by an arrow pointing left. The displacement of the string from the mean position is d. The plucked string is labeled P E sub string, to represent the potential energy of the string.
Work is done to deform the guitar string, giving it potential energy. When released, the potential energy is converted to kinetic energy and back to potential as the string oscillates back and forth. A very small fraction is dissipated as sound energy, slowly removing energy from the string.

Conservation of mechanical energy

Let us now consider what form the work-energy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is

W net = 1 2 mv 2 1 2 mv 0 2 = Δ KE. size 12{W rSub { size 8{"net"} } = { {1} over {2} } ital "mv" rSup { size 8{2} } - { {1} over {2} } ital "mv" rSub { size 8{0} rSup { size 8{2} } } =Δ"KE" "." } {}

If only conservative forces act, then

W net = W c , size 12{W rSub { size 8{"net"} } =W rSub { size 8{c} } } {}

where W c is the total work done by all conservative forces. Thus,

W c = Δ KE. size 12{W rSub { size 8{c} } =Δ"KE"} {}

Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, W c = Δ PE size 12{W rSub { size 8{c} } = +- D"PE"} {} . Therefore,

Δ PE = Δ KE size 12{ - Δ"PE"=Δ"KE"} {}


Δ KE + Δ PE = 0 . size 12{Δ"KE"+Δ"PE"=0} {}

This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,

KE + PE = constant     or KE i + PE i = KE f + PE f } (conservative forces only),

where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy    principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy    , ( KE + PE ) size 12{ \( "KE"+"PE" \) } {} . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between KE size 12{"KE"} {} and the various types of PE size 12{"PE"} {} , with the total energy remaining constant.

Questions & Answers

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the law is universally proved. The principal depends on certain conditions.
state Faraday first law
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Practice Key Terms 5

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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