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A form of energy associated with the presence of conservative interactions such as gravity or a spring. |
Motivation for Concept
Conservative interactions like gravity have the ability to "store" kinetic energy. Consider an object thrown up to a high roof. If the object is thrown perfectly, the force of gravity will slow it to a stop just as it reaches the roof. The object will then remain at rest on the roof until disturbed. But, as it falls, gravity will restore all the speed that was removed on the way up. Because we can perfectly recover the kinetic energy removed by gravity, we can consider the total energy to be constant in such a situation if we can associate some energy with the object's height. That energy is the gravitational potential energy.
Mathematical Definition
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Finding Potential Energy From Force
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The work done by a general force is given by:
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| Wiki Markup |
{excerpt}A form of energy associated with the presence of [conservative|conservative force] interactions such as gravity or a spring.{excerpt} h4. Motivation for Concept [Conservative interactions|conservative force] like [gravity|gravity (near-earth)] have the ability to "store" [kinetic energy]. Consider an object thrown up to a high roof. If the object is thrown perfectly, the force of gravity will slow it to a stop just as it reaches the roof. The object will then remain at rest on the roof until disturbed. But, as it falls, gravity will restore all the speed that was removed on the way up. Because we can perfectly recover the kinetic energy removed by gravity, we can consider the total energy to be constant in such a situation if we can associate some energy with the object's height. That energy is the gravitational potential energy. h4. Mathematical Definition h6. Finding Potential Energy From Force The [work] done by a general force is given by: {latex}\begin{large}\[ W = \int_{\rm path} \vec{F}\cdot d\vec{r}\]\end{large}{latex} The [ |
The Work-Kinetic
...
...
...
tells
...
us
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that
...
when
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work
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is
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done
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on
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a
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system,
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the
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system's
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kinetic
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energy
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will
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change:
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}\begin{large}\[ K_{i} + W = K_{f}\]\end{large}{latex} |
Suppose
...
that
...
we
...
consider
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a
...
system
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acted
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upon
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by
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a
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single,
...
...
...
.
...
If
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we
...
want
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to
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define
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a
...
potential
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energy
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U
...
to
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represent
...
this
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interaction
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in
...
such
...
a
...
way
...
that
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the
...
...
...
of
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the
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system
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is
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conserved,
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we
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must
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take:
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}\begin{large}\[ U_{i} - U_{f} = W^{\rm cons} \]\end{large}{latex} |
With
...
this
...
definition,
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the
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work-energy
...
theorem
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takes
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the
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form:
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}\begin{large}\[ K_{i} + (U_{i}-U_{f}) = K_{f} \]\end{large}{latex} |
which
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is
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equivalent
...
to
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the
...
conservation
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of
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mechanical
...
energy:
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}\begin{large}\[ K_{i} + U_{i} = K_{f} + U_{f}\]\end{large}{latex} |
The
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definition
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we
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have
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arrived
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at
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expresses
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potential
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energy
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in
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terms
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of
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force
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through
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the
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application
...
of
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a
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path
...
integral:
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}\begin{large}\[ U_{f} - U_{i} = - \int_{\rm path} \vec{F}^{\:\rm cons}\cdot d\vec{r}\]\end{large}{latex} |
it
...
is
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important
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to
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note,
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however,
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that
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the
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work
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done
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by
...
...
...
is,
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by
...
definition,
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path
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independent.
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Thus,
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the
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integrals
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can
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be
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done
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using
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the
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most
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advantageous
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path,
...
and
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the
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value
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will
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depend
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only
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upon
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the
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initial
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and
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final
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positions
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of
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the
...
system.
...
We
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can
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therefore
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write:
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}\begin{large}\[ U_{f} - U_{i} = - \int_{\vec{r}_{i}}^{\vec{r}_{f}} \vec{F}^{\:\rm cons}\cdot d\vec{r} \] \end{large}{latex} |
Note
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also
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that
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the
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expression
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we
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have
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found
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is
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only
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useful
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for
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computing
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potential
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energy
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differences.
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The
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formula's
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validity
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does
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not
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depend
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upon
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the
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precise
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value
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of
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U
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f or
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U
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i,
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but
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instead
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upon
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the
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difference.
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That
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means
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that
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an
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arbitrary
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constant
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can
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be
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added
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to
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the
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potential
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energy
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without
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affecting
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its
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usefulness.
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In
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problems
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involving
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potential
...
energy,
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then,
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it
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is
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customary
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to
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specify
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a
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zero
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point
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for
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the
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potential
...
energy
...
(
...
r
...
0)
...
such
...
that:
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}\begin{large}\[ U(\vec{r}_{0}) = 0 \]\end{large}{latex} h4. Finding Force From Potential Energy h6. Mathematically Taking the componentwise derivative of the above definition of potential energy with respect to position yields the three expressions: {latex} |
Finding Force From Potential Energy
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Mathematically
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Taking the componentwise derivative of the above definition of potential energy with respect to position yields the three expressions:
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\begin{large}\[ -\frac{\partial U}{\partial x} = F^{\rm cons}_{x} \]
\[ -\frac{\partial U}{\partial y} = F^{\rm cons}_{y} \]
\[ -\frac{\partial U}{\partial z} = F^{\rm cons}_{z} \]\end{large}{latex}
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Thus,
...
given
...
information
...
about
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the
...
dependence
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of
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the
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potential
...
energy
...
on
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position,
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the
...
force
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acting
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on
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the
...
system
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subject
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to
...
that
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potential
...
energy
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can
...
be
...
determined.
...
Diagrammatically
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A potential energy curve is a graphical representation of a system's potential energy as a function of position. This can be done for any system, but it is most often drawn for a system confined to move in one dimension (since multidimensional graphs are difficult to draw and interpret). The graph can be useful in furthering both qualitative and quantitative understanding of the system's behavior.
Common Conservative Forces
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Near-Earth Gravity
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One conservative force which is often encountered in introductory mechanics is near-earth gravity. The customary form of the gravitational potential energy near the earth's surface is:
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h6. Diagrammatically A *potential energy curve* is a graphical representation of a system's potential energy as a function of position. This can be done for any system, but it is most often drawn for a system confined to move in one dimension (since multidimensional graphs are difficult to draw and interpret). The graph can be useful in furthering both qualitative and quantitative understanding of the system's behavior. h4. Common Conservative Forces h6. Near-Earth Gravity One [conservative force] which is often encountered in introductory mechanics is near-earth [gravity|gravity (near-earth)]. The customary form of the [gravitational potential energy|gravitation (universal)#negpe] near the earth's surface is: {latex}\begin{large}\[ U(y) = mgy \]\end{large}{latex} |
assuming
...
that
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the
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y
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direction
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is
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taken
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to
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point
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upward
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from
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the
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earth's
...
surface.
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...
Springs
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Springs whose interaction is well described by Hooke's
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are
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another
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example
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of
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a
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commonly
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encountered
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...
...
.
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The
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customary
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form
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of
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the
...
...
...
...
associated
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with
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a
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spring
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is:
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}\begin{large}\[ U(x) = \frac{1}{2}kx^{2} \]\end{large}{latex} |
where
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one
...
end
...
of
...
the
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spring
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is
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fixed
...
and
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the
...
other
...
end
...
is
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constrained
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to
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stretch
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or
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compress
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only
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in
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the
...
x
...
direction,
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and
...
the
...
coordinates
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have
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been
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defined
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such
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that
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the
...
free
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end
...
of
...
the
...
spring
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provides
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zero
...
force
...
when
...
it
...
is
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at
...
the
...
position
...
x
...
=
...
0.
...