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**1. Homework Statement**

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.

That stem:

- 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 ?

Datas:

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

http://imageshack.com/a/img910/1162/77armo.png [Broken]

**2. Homework Equations**

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**3. 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:

http://imageshack.com/a/img540/3440/9WNKka.png [Broken]

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.

http://imageshack.com/a/img673/3516/M03spN.png [Broken]

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 :

##\frac{1}{2}I_1w^2+\frac{1}{2}I_2w^2##

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:

http://imageshack.com/a/img673/7249/JcXIC5.png [Broken]

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