# 7.4 Conservative forces and potential energy  (Page 2/8)

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${\text{PE}}_{\text{s}}=\frac{1}{2}{\text{kx}}^{2}\text{,}$

where $k$ is the spring’s force constant and $x$ 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 $x$ . The potential energy of the spring ${\text{PE}}_{s}$ does not depend on the path taken; it depends only on the stretch or squeeze $x$ in the final configuration.

The equation ${\text{PE}}_{s}=\frac{1}{2}{\text{kx}}^{2}$ 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 ${\text{PE}}_{s}=\frac{1}{2}{\text{kx}}^{2}$ , where $k$ is the force constant of the particular system and $x$ is its deformation. Another example is seen in [link] for a guitar 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

${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{{\text{mv}}_{0}}^{2}=\Delta \text{KE.}$

If only conservative forces act, then

${W}_{\text{net}}={W}_{\text{c}}\text{,}$

where ${W}_{c}$ is the total work done by all conservative forces. Thus,

${W}_{\text{c}}=\text{Δ}\text{KE.}$

Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, ${W}_{\text{c}}=-\text{Δ}\text{PE}$ . Therefore,

$-\text{Δ}\text{PE}=\text{Δ}\text{KE}$

or

$\text{Δ}\text{KE}+\text{Δ}\text{PE}=0.$

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

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    , $\left(\text{KE}+\text{PE}\right)$ . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between $\text{KE}$ and the various types of $\text{PE}$ , with the total energy remaining constant.

#### Questions & Answers

Pls guys am having problem on these topics: latent heat of fusion, specific heat capacity and the sub topics under them.Pls who can help?
hamidat Reply
Thanks George,I appreciate.
hamidat
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Abolarin
Most especially it is the calculatory aspects that is giving me issue, but with these new strength that you guys have given me,I will put in my best to understand it again.
hamidat
you can bring up a question and let's see what we can do to it
Abolarin
the distance between two suasive crests of water wave traveling of 3.6ms1 is 0.45m calculate the frequency of the wave
Idris Reply
v=f×lemda where the velocity is given and lends also given so simply u can calculate the frequency
Abdul
You are right my brother, make frequency the subject of formula and equate the values of velocity and lamda into the equation, that all.
hamidat
lExplain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.
Christoper Reply
When light is reflected at Brewster's angle from a smooth surface, it is 100% polarizedparallel to the surface. Part of the light will be refracted into the surface.
Ekram
What is specific heat capacity?
hamidat Reply
Specific heat capacity is the amount of heat required to raise the temperature of one (Kg) of a substance through one Kelvin
Paluutar
formula for measuring Joules
Rowshan Reply
I don't understand, do you mean the S.I unit of work and energy?
hamidat
what are the effects of electric current
ADAMS Reply
What limits the Magnification of an optical instrument?
Naeem Reply
Lithography is 2 micron
Venkateshwarlu
what is expression for energy possessed by water ripple
Prabesh Reply
what is hydrolic press
Mark Reply
An hydraulic press is a type of machine that is operated by different pressure of water on pistons.
hamidat
what is dimensional unite of mah
Patrock Reply
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sharon Reply
What is Boyles law
Pascal Reply
it can simple defined as constant temperature
Muhammad
Boyles law states that the volume of a fixed amount of a gas is inversely proportional to the pressure acting on in provided that the temperature is constant.that is V=k(1/p) or V=k/p
FADILAT
what is motion
Mua Reply
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Anderson
what is escape velocity
Shuaibu Reply
the minimum thrust that an object must have in oder yo escape the gravitational pull
Joshua
what is a dimer
Mua
what is a atom
ADAMS
how to calculate tension
Deena Reply
what are the laws of motion
Mua

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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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