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A standard example from work and energy. |
Some modern roller coasters are "launched" rather than slowly pulled to the top of a hill. One recent design is a coaster on a U-shaped track like that shown above. Each time the coaster passes through the acceleration zone, it receives a kick from an electric motor. Suppose that for the track shown above, the height H is 75 m. The acceleration zone has a width d = 25 m. If a coaster of mass m = 11,000 kg experiences an accelerating horizontal force of magnitude 0.5mg while it is in the acceleration zone, how many passes through the zone must the coaster make before it will reach the top of the track? (After this, the function of the motor will be reversed to decelerate the coaster.)
Solution
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Coaster as |
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The coaster and earth create a |
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We begin with an initial-state final-state diagram and energy bar graphs.
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It is clear from the diagrammatical representations that the system has gained mechanical energy. The source of the energy is the electric motor. In equation form, we have:
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}{composition-setup} {excerpt:hidden=true}A standard example from work and energy.{excerpt} !LIMcoaster.png! Some modern roller coasters are "launched" rather than slowly pulled to the top of a hill. One recent design is a coaster on a U-shaped track like that shown above. Each time the coaster passes through the acceleration zone, it receives a kick from an electric motor. Suppose that for the track shown above, the height _H_ is 75 m. The acceleration zone has a width _d_ = 25 m. If a coaster of mass _m_ = 11,000 kg experiences an accelerating horizontal force of magnitude 0.5{_}mg_ while it is in the acceleration zone, how many passes through the zone must the coaster make before it will reach the top of the track? (After this, the function of the motor will be reversed to decelerate the coaster.) h4. Solution {toggle-cloak:id=sys} *System:* {cloak:id=sys} Coaster as [point particle], plus the earth as a rigid body of infinite mass.{cloak} {toggle-cloak:id=int} *Interactions:* {cloak:id=int}The coaster and earth create a [conservative|conservative force] interaction of gravity, which will be treated as a [potential energy] in the system. The system is also subject to a [non-conservative|non-conservative force] interaction from the the electric motor.{cloak} {toggle-cloak:id=mod} *Model:* {cloak:id=mod} [Mechanical Energy, External Work, and Internal Non-Conservative Work].{cloak} {toggle-cloak:id=app} *Approach:* {cloak:id=app} {toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color} {cloak:id=diag} We begin with an [initial-state final-state diagram] and [energy bar graphs|Diagrams and Mechanical Energy]. |!LIMcoaster2.jpg!|!LIMcoaster3.jpg!| ||Initial||Final|| {cloak:diag} {toggle-cloak:id=math} {color:red} *Mathematical Representation* {color} {cloak:id=math} It is clear from the diagrammatical representations that the system has gained mechanical energy. The source of the energy is the electric motor. In equation form, we have: {latex}\begin{large}\[ E_{i} + W^{NC} = 0 + W^{NC} = E_{f} = mgH\]\end{large}{latex} |
where
...
the
...
non-conservative
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work
...
is
...
due
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to
...
the
...
motor.
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From
...
the
...
definition
...
of
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work,
...
the
...
work
...
done
...
by
...
the
...
motor
...
can
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be
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related
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to
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the
...
force
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provided
...
by
...
the
...
motor:
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}\begin{large}\[ W^{NC}_{\rm motor} = \sum_{passes} F_{\rm motor} d = nF_{\rm motor}d\]\end{large}{latex} |
where
...
n
...
is
...
the
...
number
...
of
...
passes
...
the
...
train
...
makes
...
through
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the
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acceleration
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zone.
...
Since
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the
...
force
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provided
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by
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the
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motor
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is
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0.5
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mg
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,
...
we
...
can
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write:
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}\begin{large}\[ \frac{1}{2}nmgd = mgH \]\end{large}{latex} |
which
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is
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solved
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to
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obtain:
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}\begin{large}\[ n = \frac{2H}{d} = 6 \]\end{large}{latex} |
which
...
implies
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the
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coaster
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must
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make
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6
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complete
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passes
...
through
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the
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acceleraation
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zone
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to
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reach
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the
...
top
...
of
...
the
...
track.
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Technically this implies our {note}Technically this implies ourinitial-state final-state diagram was wrong, since we drew the coaster on the wrong side of the track. Remember that diagrams in physics problems are often useful, even if some details of the problem are not completely understood. They are meant to provide a summary of the physics concepts involved, not necessarily a mathematically precise statement of the problem. {note} {cloak:math} {cloak:app} |
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