Let us consider a device that consists of a ball rolling on the inside rim of a hoop (11]. This model is similar to the problem of liquid fuel sloshing in a rocket. The hoop is free to rotate about its horizontal principal axis as shown in Figure E7.1. The angular position of the hoop may be controlled via the torque T applied to the hoop from a torque motor attached to the hoop drive shaft. If negative feedback is used, the system characteristic equation is
Ks(s + 4)
1+3+25 + 2 = 0.
(a) Sketch the root locus. (b) Find the gain when the roots are both equal. (c) Find these two equal roots.