A red Archimedean spiral is fixed to the ground. An external motor turns a grey support clockwise at w, the support can only turn around itself. On the support there is one orange disk that doesn't turn around itself at start. A stem is on the support.
- turns like the support clockwise at w
- is attached on the spiral at one end
- has no mass
- is connected to the disk:
........... - if the spiral attrack the stem: like gears can be: no friction between the stem and the disk, no sliding
............- if the spiral block the stem, there is friction between the stem and the disk
At start w=0.
1) the spiral attack or block the stem ?
2) drawn all forces
3) find the work that the motor need to give
4) find the work needed to move the stem around the spiral
5) does the disk has a rotation around itself ?
Inertia of the support: I1
Inertia of the disk around main axis: I2
Inertia of the disk around itself: I3
There is no friction except between the stem and the disk
The Attempt at a Solution
1) The support turns clockwise so the disk turns counterclockise at -w in the support reference, I need to add 2piR/4 when the support has turned of 90°. I noted R the radius of the disk. I measured it with 3 positions:
I don't know how to calculate this, if you have a method because it's only a graphical solution. With the graphical method, the spiral block the stem.
Edit:With a numerical method I found the the spiral attrack the stem. I found for my spiral, the distance pass from 11 to 13.194 and the tangent is at 1.3274, so the radius of the disk can be 1.3274, it become 1.3274*2*pi/4=2.0850 but the spiral move to 2.194, it confirms what I found with a graphic method. I take a=1.4 for the spiral. The parametric equations are : ##x=-1.4\theta cos\theta## and ##y=1.4\theta sin\theta##.
2) The stem gives the forces F2 to the disk and receives F1. The spiral receives the force F4. The support receives the forces F3.
the stem rotates the disk clockwise.
3) the motor needs to give only the work for turn the support and the disk around the main axis, the work is :
4) The force F4 is always perpendicular to the trajectory then the work needed is 0. The stem has no mass so its kinetic energy is always 0.
5) Yes, I measured the distance for 3 positions and the stem rotates the disk around itself clockwise, and there is friction. In this case the support must receive a counterclockwise torque for keep constant the energy. So, I think I'm wrong in my forces or the spiral don't attrack or block the stem ?
With a better image:
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