Question

An important special case of the motion of a symmetric top occurs when it spins about a vertical axis. Analyze this motion as follows: (a) By inspecting the effective PE, U_eff(theta)=((L_z-(L_3)cos(theta))^2)/((2lambda_1(sin^2(theta)))+((L_3)^3)/(2lambda_3))+MgRcos(theta), show that if at any time theta=0, then L_3 and L_z must be equal. (b) Set L_z=(L_3=lambda_3)(w_3) and then make a Taylor expansion of U_eff(theta) about theta=0 to terms of order (theta)^2. (c) Show that if w_3>w_min=2sqrt(MgRlambda_1/((lambda_3)^2), then the position theta=0 is stable, but if w_3=w_min it is unstable. (In practice, friction slows the top's spinning. Thus with w_3 sufficiently fast, the vertical top is stable, but as it slows down the top will eventually lurch away from the vertical when w_3 reaches w_min.)

Answer 1