Math Help Forum - View Single Post - Direction a marble would travel on a plane

earboth · #2 May 21st, 2009, 07:05 AM

Quote:

Originally Posted by epiekarc

I am trying to calculate the direction a marble would travle if placed on a tilted plane. I need a formula to calcualte this with variables b, c and d. d is defined as an angle measured from a base line created by starting at point 0,0,0 to the perpendicular intersection of planes x and y.

I will always know what b and c equals, we are solve for d.

Here are some additional details

starting on a level plane (x) i have another plane (y) intersecting at b degrees.

From plane y i have another plane (z) rotated at 90 degrees and tilted c degrees.

I need a formula to calculate the direction the marble would roll if placed on plane z. the answer needs to be an angle (of direction of travel) on plane x.

Please see the attached image for claification. email me for clarification

epiekarc at yahoo dot com

Thanks for all the help,

Eric

1. This is only an attempt and I'm quite sure that I've forgotten to take something important into account.

2. There are three forces which act on the marble:
a) the weight: $w = m \cdot g$
b) a force acting parallel to the plane z: $|\vec z| = w \cdot \sin(c)$ ......... (dark blue)
c) a force acting parallel to the plane y: $|\vec y| = w \cdot \sin(b)$ ......... (dark green)

3. As far as I understand the question you are interested in the angle between the orange base line B and the initial direction of the movement of the marble projected into the plane x(?). If so:

$\vec z \perp \vec y$

That means the vectors $\vec z$ and $\vec y$ form a rectangle parallel to the plane z. If this rectangle is projected into the X-plane (nice name for a mathhelpforum

) it becomes the rectangle which has the side length z' (light blue) and y' (light green) with:

$z'=w \cdot \sin(c) \cdot \cos(c)$

$y'=w \cdot \sin(b) \cdot \cos(b)$

4. The angle $\theta$ can be calculated by:
$\tan(\theta)=\dfrac{w \cdot \sin(c) \cdot \cos(c)}{w \cdot \sin(b) \cdot \cos(b)} = \dfrac{ \sin(c) \cdot \cos(c)}{\sin(b) \cdot \cos(b)}$