|MadSci Network: Physics|
Before I go into the equations, we must realize that the motion of the center of gravity is the only thing that can determine which direction the roller moves. This is because the force initiating its motion is gravity itself. The roller slides to the point in the track where its potential energy is the lowest, and that is where its center of gravity is lowest in height. If the track were open ended the cone could even fall out of it. Having understood that, we come to answer your last two questions: Rotation is not necessary for it to move on any given direction. if friction was low enough it would slide in the direction of lowest potential energy as well. We also understand it rolls "up the slope" because the track and roller are designed so that motion in the direction that the vertical slope increases causes the center of gravity of the roller to move down. So it is in fact rolling *down*. Also, notice that even though prescence of rotation does not determine the direction of motion, it does determine the rate of acceleration because not only is the object accelerating down, it's spin is also accelerating. This is nothing that a drop of "Magic Lube" and some wishfull assumptions cannot fix. So again, take away friction, but keep contact, and the object cannot rotate, it simply slides in the same direction. Now on to the equations of motion. For this, I have taken away friction and rotation to simplify our work, we just want to know what direction it rolls anyway, and we know friction and rotation do not make any difference. So, let's define our variables: Cone dimensions (consider only one side of the roller, assume its all symmetrical) Asume cone base radius R Cone height (distance from the tip to the base) equal to H Track dimensions (consider only one side of the tracks, assume motion is constrained appropriately, as if rolling on both tracks) Slope in the vertical plane: MV Slope in the horizontal plane (the gradual splaying of the tracks) MH Slope in the vertical plane is determined by dividing overall upward travel by the overall horizontal travel. Slope in the horizontal plane is determined by dividing overall travel in the direction away from the base of the cone and towards the tip, by the overall horizontal travel. What we aim to find is an equation of the height (y) versus horizontal position (x). Let us define horizontal position as increasing in the direction that the tracks slope "up" and splay appart. With this facts, please draw yourself a diagram to follow the equations from now on. My initial guess is that this will be a linear equation. The derivative of this equation should be a constant and should tell us the direction of motion by its sign (once we plug in real world values for the dimensions) Because the roller will move in the direction that Y decreases (as the potential energy decreases) So if that derivative turns out positive, we know the roller will move in the direction that X becomes more negative, so that the height decreases. The first vertical contributor is the vertical sloping of the tracks. The vertical motion due to vertical sloping of the tracks is: x (mv) If this was the only contributor, we would know right away that a positive slope (if mv is positive) would cause Y to increase along with X, and would mean the object rolls in the direction X becomes more negative... this would be the case for a cylinder rolling down... but there is more. The second vertical motion contributor is the cone's shape constrained by the track, as it slides outward towards the cone tip. For every unit of forward motion, the point of contact on the cone slides to the outside, where the radius is smaller, and in turn the cone moves down. Because of the conical shape, this is a linear motion. (the radius of a cone decreases lienarly with distance from the base) vertical motion due to horizontal sloping of the tracks: X (mh) (-R/H) where mh is the outward sloping of the tracks, and R/H is the base Radius of the cone divided by its height (tip to base). I plugged in the negative sign because I already know as x increases with a positive mh the cone will move down with respect to the point of contact with the track. So, the overall equation of motion for its center of gravity is as follows: y = x[mv + mh(-R/H)] Its derivative is: y' = [mv + mh(-R/H)] And right away we can see that to make this cone slide up or down we may choose the values for MV and MH(-R/H) so that one is larger than the other. If we plug in some real world values we could have something like this: y' = (0.1 + 0.1(-3/2) y' is a negative number, therefore object rolls in the direction that tracks appear to slope up. In this case, both slopes (the upward and the outward) are equal to a 10% grade, and the cone has a base that is 3 units of radius and a tip to base distance of 2 units. Have fun! -Aurelio
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