Answer Happy

5 Steady Precession for a Symmetric Top in a Gravitational Field The Lagrangian for a symmetric spinning top in a gravitational field with one point fixed is 2 =(? sin? 0 + 0°) +3543 (scos 0 + (j) ? – mngh cos e + 0 (1) (a) Using the Euler-Lagrange equations for and y, show that if the inclination angle o is constant, the top's body axis precesses around the vertical with steady angular velocity 0; let 2 = 0 denote this steady precession rate. (b) Evaluate the Euler-Lagrange equation for the coordinate. (c) To find the precession rate 1 for fixed 0, use the Euler-Lagrange equation for the coordinate, and make the replacements o = 2 and W3 = o cos 6 + to obtain the quadratic equation 1,22 cose - 13w3.2 + mgh = 0 (2) (d) Show that for a rapidly spinning top with large 13w3 the quadratic equation above has two real-valued roots with one root being much larger than the other. (e) In the limit of large 13w3, verify that the two real-valued roots are mgh Ωs 13w3 (3) and 133 Ωs I cose