We consider a heavy symmetric top of mass M, pinned at a point P which as...

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We consider a heavy symmetric top of mass M, pinned at a point P which as at distance I from its centre of mass (as discussed in class). The principal momenta are I1, I1 and I3 and the Euler-angles are shown in the figure. fr Mg 24 You may use the Lagrangian derived in class for this problem. The top is now spun with initial conditions o(t = 0) = 0 and 0(t = 0) = 00- a) Show that obeys the equations of motion where Ueff(0) = 110 = JUeff (0) 20 w (cos - cos 00)² 211 sin² 0 + Mgl cos 0. Hint: It can be useful to substitute u = cos in the derivation of this results (however take care of chain rule and Jacobians in case you do so). b) Suppose that the top is spinning very fast, such that 1303 » √MglI₁. (7) This means that you can neglect the second term in the effective potential. Show that 0 = 00 is an equilibrium point of the resulting effective potential. Consider next small deviations from this equilibrium point and show that the top nutates with the frequency Ω = (8) We consider a heavy symmetric top of mass M, pinned at a point P which as at distance I from its centre of mass (as discussed in class). The principal momenta are I1, I1 and I3 and the Euler-angles are shown in the figure. fr Mg 24 You may use the Lagrangian derived in class for this problem. The top is now spun with initial conditions o(t = 0) = 0 and 0(t = 0) = 00- a) Show that obeys the equations of motion where Ueff(0) = 110 = JUeff (0) 20 w (cos - cos 00)² 211 sin² 0 + Mgl cos 0. Hint: It can be useful to substitute u = cos in the derivation of this results (however take care of chain rule and Jacobians in case you do so). b) Suppose that the top is spinning very fast, such that 1303 » √MglI₁. (7) This means that you can neglect the second term in the effective potential. Show that 0 = 00 is an equilibrium point of the resulting effective potential. Consider next small deviations from this equilibrium point and show that the top nutates with the frequency Ω = (8)