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Composition Setup
 Excerpt
hiddentrue
A standard example from work and energy.
Image Added
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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idsys
 System: 
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idsys
 Coaster as , plus the earth as a rigid body of infinite mass.
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idint
 Interactions: 
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idint
The coaster and earth create a conservative interaction of gravity, which will be treated as a  in the system. The system is also subject to a non-conservative interaction from the the electric motor.
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idmod
 Model: 
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idmod
 .
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idapp
 Approach: 
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iddiag
  Diagrammatic Representation 
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We begin with an initial-state final-state diagram and energy bar graphs.
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Initial
Final
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diag
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  Mathematical Representation 
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idmath
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
 Wiki Markup
!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

*System:*  Coaster as [point particle], plus the earth as a rigid body of infinite mass.

*Interactions:*  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] interaction from the the electric motor.  

*Model:*  [Mechanical Energy and Non-Conservative Work].

*Approach:*  We begin with an [initial-state final-state diagram] and [energy bar graphs|energy bar graph].


|!LIMcoaster2.png!|!LIMcoaster3.png!|
||Initial||Final||

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 
...
 work 
...
 is 
...
 due 
...
 to 
...
 the 
...
 motor.
...
From 
...
 the 
...
 definition 
...
 of 
...
 work, 
...
 the 
...
 work 
...
 done 
...
 by 
...
 the 
...
 motor 
...
 can 
...
 be 
...
 related 
...
 to 
...
 the 
...
 force 
...
 provided 
...
 by 
...
 the 
...
 motor:


{
 Latex
}\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 
...
 the 
...
 acceleration 
...
 zone. 
...
 Since 
...
 the 
...
 force 
...
 provided 
...
 by 
...
 the 
...
 motor 
...
 is 
...
 0.5
...
mg
...
, 
...
 we 
...
 can 
...
 write:


{
 Latex
}\begin{large}\[ \frac{1}{2}nmgd = mgH \]\end{large}{latex}


which 
...
 is 
...
 solved 
...
 to 
...
 obtain:


{
 Latex
}\begin{large}\[  n = \frac{2H}{d} = 6 \]\end{large}{latex}


which 
...
 implies 
...
 the 
...
 coaster 
...
 must 
...
 make 
...
 6 
...
 complete 
...
 passes 
...
 through 
...
 the 
...
 acceleraation 
...
 zone 
...
 to 
...
 reach 
...
 the 
...
 top 
...
 of 
...
 the 
...
 track. 
 Note
Technically this implies our 
  
{note}Technically this implies our 
 initial-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}




problem.
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