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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
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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
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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 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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(
...
r
...
0)
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such
...
that:
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}\begin{large}\[ U(\vec{r}_{0}) = 0 \]\end{large}{latex} h4. Finding Force From Potential Energy Taking the componentwise derivative of the above definition of potential energy with respect to position yields the three |
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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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
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system
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subject
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to
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that
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potential
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energy
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can
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be
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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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\begin{large}\[ U(y) = mgy \]\end{large} |
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}\[ U(x) = \frac{1}{2}kx^{2} \]\end{large} |
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.