Maximizing Rotary Projectile Motion: Finding Distance and Path Diameter

[elmariachi]

Hi all,
the problem i have is that there is a rotating disk with nozzles on its circumference. a material is being pushed into this rotating disk/plate and its being released from the nozzles tangentially. The disk is rotating with a certain rpm such that the angular velocity of the particle just leaving the disk is w*r.
I believe that the particle from the disk will follow a circular path in the direction of the rotating disk. The rotating disk is enclosed within a room of specified length with air blown from the bottom of this enclosure such that the air resists the particles coming out of the disk.
I wanted to make sure that the particles thrown out of the disk don't go and hit the walls of the enclosure.I want to know, how can I find the longest horizontal distance , the material will travel just after leaving the disk and How big a circular path will it take (diameter).I will appreciate the help.

thanks

 

[abercrombiems02]

You need to start this problem defining two sets of unit vectors, a set that is inertially fixed and a set that is fixed to the rotating disc. In this case if we are in a horizontal plane we can neglect the effects of gravity since this disc is just traveling on the horizontal surface. I've done this problem with a rod in which there was one thruster and one end always thrusting tangential to linear velocity of the tip. If you apply F = ma, the only force you have acting is the thrust in the tangential direction. However, in your case you have a disc so there may be multiple forces, but note that if you have symmetry in these forces the disc will only rotate because while the net moment is non-zero, the net force will be zero. If you are familiar with SIMULINK i can send you my .mdl file so you can get an idea of how your EOMs should look.

Ooops forgot to add that you will also need to apply a moment equation to find a state equation for theta (or whatever you use for your angular measurement).

Unfortunately in the end you should end up with a set of non-linear DE's :(

 

[piercas]

Dear researcher,

Thank you for sharing your interesting problem with us. Your setup seems to involve the concept of rotary projectile motion, where a rotating disk is used to propel particles in a circular path. To maximize the distance and determine the path diameter of these particles, we need to consider several factors such as the rotational speed of the disk, the angle at which the particles are released, and the resistance from the air in the enclosure.

To find the longest horizontal distance that the particles will travel, we can use the equation for projectile motion:

d = v^2 * sin(2θ) / g

Where d is the distance, v is the initial velocity, θ is the angle of release, and g is the acceleration due to gravity. In this case, we can assume that the initial velocity is equal to the tangential velocity of the particle leaving the disk, which is w*r. We also need to consider the resistance from the air, which will decrease the initial velocity and therefore decrease the distance traveled. This can be taken into account by adjusting the value of g in the equation.

To determine the path diameter, we can use the equation for circular motion:

r = v^2 / a

Where r is the radius of the circular path, v is the tangential velocity, and a is the centripetal acceleration. Again, the resistance from the air will affect the tangential velocity and therefore the radius of the circular path.

In order to find the optimal values for the distance and path diameter, we can use mathematical optimization techniques or conduct experiments with different rotational speeds, angles of release, and air resistance levels.

I hope this helps in your research. Good luck!