Question

der a frictionless ball of mass m rotating in a vertical circular cone as s the following figure. The height of the oportionality constant k. Assume the gravitational constant is g Cons cone z is proportional to the radius r with Figure 1: A ball rotating in a frictionless cone a) Write the Lagrangian L in terms of variables r and the rotation angle ф. b.) Write the equations of motion from the Lagrangian. Do not solve them. c) Prove that mr2φ is a constant of motion. d.) What conserved quantity does this constant of motion refer to e) Consider a harmonic ansatz such that r(t-Aeiat and ф(t)-Belt. Apply this ansatz to just the constant of motion. What is the relationship between the frequencies a and b?

Answer 1

Is the sphere rotating in a plane inside the cone or having a more general motion?

Will provide a more general solution in terms of 3 Euler angles. If required in a certain plane, nonrequired terms can be eliminated.

Refer the figure attached

Is the sphere rotating in a plane inside the cone or having a more general motion? Will provide a more general solution in terms of 3 Euler angles. If required in a certain plane, nonrequired terms can be eliminated. Refer the figure attached \r\n Position of point A x = r sin alpha cos phi (polar angle of one sphere) y = r sin alpha sin phi z = r cos alpha Tangential component of velocity U = r^dot V = r sin alpha phi^dot equation (A) lagrangious = KE_linear + KE_Rotation - Mg z = 1/2 M(U^2 + V^2) + 1/2 I(phi^2 + 2phi psi cos theta + theta^2 + phi^2) -Mg 2 \r\n I = 5 Ma^2/2 (a rightarrow rad of sphere M rightarrow mass of sphere) phi, theta, psi rightarrow Euler angles Equations of motion constraints equation (B) U - aw middot e_1^^= 0 V + aw middot e_2^^= 0 w rightarrow ang vel unit vectors