Editing Inclined plane (section)

==Frictionless inclined plane==
[[File:Schiefe-ebene hg.jpg|right|thumb|Instrumented inclined plane used for physics education, around 1900. The lefthand weight provides the load force <math>F_\text{w}</math>. The righthand weight provides the input force <math>F_\text{i}</math> pulling the roller up the plane.]]

If there is no [[friction]] between the object being moved and the plane, the device is called an ''ideal inclined plane''.  This condition might be approached if the object is rolling like a [[barrel]], or supported on wheels or [[caster]]s. Due to [[conservation of energy]], for a frictionless inclined plane the [[work (physics)|work]] done on the load lifting it, <math>W_\text{out}</math>, is equal to the work done by the input force, <math>W_\text{in}</math><ref name="Hyperphysics">{{cite web
  | last = Nave
  | first = Carl R.
  | title = The Incline
  | work = Hyperphysics
  | publisher = Dept. of Physics and Astronomy, Georgia State Univ. 
  | year = 2010
  | url = http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/incline.html
  | access-date = September 8, 2012}}</ref><ref name="Westminster">{{cite web
  | last = Martin
  | first = Lori
  | title = Lab Mech14:The Inclined Plane - A Simple Machine
  | work =  Science in Motion
  | publisher = Westminster College
  | year = 2010
  | url = http://www.westminster.edu/acad/sim/documents/STHEINCLINEDPLANE.pdf
  | access-date = September 8, 2012}}</ref><ref name="Pearson">{{cite book   
  | last = Pearson
  | title = Physics class 10 - The IIT Foundation Series
  | publisher = Pearson Education India
  | year = 2009
  | location = New Delhi
  | pages = 69
  | url = https://books.google.com/books?id=4Gm_PiLjwBcC&q=%22inclined+plane%22++%22mechanical+advantage%22+slope+angle&pg=PA69
  | isbn = 978-81-317-2843-7}}</ref>

:<math>W_{\rm out} = W_{\rm in} \,</math>

Work is defined as the force multiplied by the displacement an object moves. The work done on the load is equal to its weight multiplied by the vertical displacement it rises, which is the "rise" of the inclined plane
:<math>W_{\rm out} = F_{\rm w} \cdot \text{Rise} \,</math>
The input work is equal to the force <math>F_\text{i}</math> on the object times the diagonal length of the inclined plane. 
:<math>W_{\rm in} = F_{\rm i} \cdot \text{Length}  \,</math>
Substituting these values into the conservation of energy equation above and rearranging

:<math>\text{MA} = \frac{F_{\rm w}}{F_{\rm i}} = \frac {\text{Length}}{\text{Rise}} \,</math>

To express the mechanical advantage by the angle <math>\theta</math> of the plane,<ref name="Westminster" /> it can be seen from the diagram ''(above)'' that
:<math>\sin \theta = \frac {\text{Rise}}{\text{Length}}  \,</math>
So
{{Equation box 1 |indent =: |cellpadding = 0 |border = 1 |border colour = black |background colour = transparent
|equation = &nbsp;&nbsp;<math>\text{MA} = \frac{F_{\rm w}}{F_{\rm i}} = \frac {1}{\sin \theta}</math>&nbsp;&nbsp;
}}
So the mechanical advantage of a frictionless inclined plane is equal to the reciprocal of the sine of the slope angle. The input force <math>F_{\rm i}</math> from this equation is the force needed to hold the load motionless on the inclined plane, or push it up at a constant velocity. If the input force is greater than this, the load will accelerate up the plane. If the force is less, it will accelerate down the plane.