Motion along a three-dimensional plane

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The discussion focuses on applying Newton's laws to motion along a frictionless incline in a three-dimensional space. The user has derived equations for a two-dimensional scenario and seeks assistance in extending these equations to a three-dimensional coordinate system. The key equations derived include the normal force and acceleration in the x-direction. The user emphasizes the importance of selecting an appropriate coordinate system for accurate results. The goal is to establish a similar set of equations that can effectively describe motion in a 3D context.
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Hi, I am working on an example that demonstrates Newtons laws in 3-dimensions (this is NOT homework). In my example I focus on incline (frictionless) motion, which can be represented as below in 2-dimensions.

http://imageshack.us/scaled/landing/855/212pxfreebodysvg.png


From Newtons second law (with proper choice of coordinate system) i get:
F_y:= 0 = N - mg cos(alpha)
F_x:= m a_x = f - mg sin(theta)
which then gives me N and a_x.

Can anybody help me with deriving a similar set of equations for a 3-dimensional system (xyz-system).
 
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For three dimensions you can always find a two-dimensional slice.
 

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