# Dynamic Confinement

1. Feb 16, 2008

### carmicheal99@ya

What is the minumum angular velocity of an object to behave solid to various objects?

I've tasked myself with determining the rotational velocity (omega) for (n) vertical beams rotating a distance (r) from common axis such that particles moving at a linear velocity (v) with a radius of (p) may not pass through the area generated by revolving the beams. To simplify things; thickness, length and mass of all objects have been neglected leaving that for later engineering challenges as such this is a 2 dimensional geometry problem now.

Another conceptual way to ask this would be: how fast must a fan spin so that a ball thrown at it will always bounce back? Or possibly how fast must a ring spin before you can touch it without losing a finger.

I started with the perimeter of the circle divided by the number of open sections made by placing the beams equidistant from one another then subtracting off the width of the beam and the diameter of the particle. This leaves the open space that must be traversed by the beams in time (t).

A little rearranging yields:
phi = ((2*pi*r)/n - r*theta - 2*p)/r
where:
phi = angle traversed by the beam to prevent passage of the particle

Solving the linear velocity for time to cross this gap and realizing that the time is the same for angular velocity to move the beam into position.

t = r/v = phi/omega

Work this out to obtain:
omega = -((theta*n - 2*pi)*r + 2*n*p)*v/(n*r*p)

Where:
theta = the angle covered by the width of the beam ----> width = r*phi
n = number of beams
pi = 3.14159...
r = radius of circle generated by revolving beams
v = linear velocity of particle