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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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{excerpt}A form of energy associated with the presence of [conservative|conservative force] interactions such as gravity or a spring.{excerpt} ||Page Contents|| |{toc:indent=10px}| h2. 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]. h2. Definition h4. 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 [work energy theorem] tells us that when work is done on a system, the system's kinetic energy will change: {latex} |
The Work-Kinetic Energy Theorem tells us that when work is done on a system, the system's kinetic energy will change:
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\begin{large}\[ K_{i} + W = K_{f}\]\end{large}{latex}
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Suppose
...
that
...
we
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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,
...
...
...
.
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If
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we
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want
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to
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define
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a
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potential
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energy
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U
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to
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represent
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this
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interaction
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in
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such
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a
...
way
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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
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this
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definition,
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the
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work-energy
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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
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to
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the
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conservation
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of
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mechanical
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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
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of
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a
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path
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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
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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
...
...
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is,
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by
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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,
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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
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system.
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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
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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
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energy
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(
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r
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0)
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such
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that:
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}\begin{large}\[ U(\vec{r}_{0}) = 0 \]\end{large}{latex} h4. |
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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Finding Force From Potential Energy Taking the componentwise derivative of the above definition of potential energy with respect to position yields the three expressions: {latex}\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} |
Thus,
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given
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information
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about
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the
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dependence
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of
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the
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potential
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energy
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on
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position,
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the
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force
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acting
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on
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the
...
system
...
subject
...
to
...
that
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potential
...
energy
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can
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be
...
determined.
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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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h4. Potential Energy Curves A [potential energy curve] is a graphical representation of a system's potential energy as a function of posittion. 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. h2. Common Types h4. Gravitational Potential Energy Near Earth Near the earth's surface, if we assume coordinates with the +{_}y_ direction pointing upward, the force of gravity can be written: {latex}\begin{large}\[ \vec{F} = -mg \hat{y}\]\end{large}{latex} Since the "natural" ground level varies depending upon the specific situation, it is customary to specify the coordinate system such that: {latex}\begin{large}\[ U(0) \equiv 0\]\end{large}{latex} The [gravitational potential energy] at any other height _y_ can then be found by choosing a path for the work integral that is perfectly vertical, such that: {latex}\begin{large}\[ U(y) = U(0) - \int_{0}^{y} (-mg)\;dy = mgymgy \]\end{large}{latex} For an object in vertical freefall (no horizontal motion) the associated potential energy curve would then be: POTENTIAL ENERGY CURVE For movement under pure near-earth gravity, then, there is no equilibrium point. At least one other force, such as a normal force, tension, etc., must be present to produce equilibrium. h4. Elastic Potential Energy Assuming an object attached to a spring that obeys Hooke's Law with the motion confined to the _x_ direction, it is customary to choose the coordinates such that _x_ = 0 when the object is in a position such that the spring is at its natural length. The force on the object from the spring is then: {latex}\begin{large}\[ \vec{F} = - kx \hat{x} \]\end{large}{latex} It is also customary to make the assignment: {latex}\begin{large}\[ U(0) \equiv 0\]\end{large}{latex} Thus, the potential can be defined: {latex}\ |
assuming that the y direction is taken to point upward from the earth's surface.
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Springs
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Springs whose interaction is well described by Hooke's Law are another example of a commonly encountered conservative force. The customary form of the elastic potential energy associated with a spring is:
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\begin{large}\[ U(x) = U(0) - \int_{0}^{x} (-kx)\:dx = \frac{1}{2}kx^{2} \]\end{large}{latex} For an object moving under the influence of a spring only, the associated potential energy curve would then be: POTENTIAL ENERGY CURVE For such a motion, then, there is one stable equilibrium point at _x_ = 0. h4. Universal Gravitational Potential Energy For two spherically symmetric objects (objects 1 and 2), it is customary to analyze the [gravitational|gravity] interaction by constructing spherical coordinates with one of the objects at the origin (if one of the objects dominates the mass of the system, its position is typically used as the origin). The force law is then: {latex}\begin{large}\[ \vec{F} = - G\frac{m_{1}m_{2}}{r^{2}} \hat{r}\]\end{large}{latex} where _r_ is the position of the object that is not at the origin. It is also customary to make the assignment that the potential energy of the system goes to zero as the separation goes to infinty: {latex}\begin{large}\[ \lim_{r \rightarrow \infty} U(r) = 0 \]\end{large}{latex} Thus, we can define the potential for any separation _r_ as: {latex}\begin{large}\[ U(r) = U(\infty) - \lim_{r_{0}\rightarrow \infty}\int_{r_{0}}^{r| \left(-G\frac{m_{1}m_{2}}{r^{2}} \;dr = - Gm_{1}m_{2} \left(\frac{1}{r}-\lim_{r_{0}\rightarrow \infty}\frac{1}{r_{0}}\right)\] \[U(r) = -G\frac{m_{1}{m_{2}}{r} \]\end{large}{latex} If the two objects are isolated from other influences, their potential energy curve is then: POTENTIAL ENERGY CURVE This potential energy curve is somewhat misleading, since the potential is spherically symmetric. Thus, although in spherical coordinates, _r_ cannot go negative, if we define a one-dimensional coordinate system by following a radial line through the origin (suppose, for instance, we chose to follow the _z_ axis where _z_ = _r{_}cosθ, we would generate a curve: POTENTIAL ENERGY CURVE which indicates the possibility of stable equilibrium when the objects' separation goes to zero. Of course, this is technically impossible for objects of finite size. |
where one end of the spring is fixed and the other end is constrained to stretch or compress only in the x direction, and the coordinates have been defined such that the free end of the spring provides zero force when it is at the position x = 0.