Question

Problem 1 (5 pt.) In the last class, we will consider the gyroscopie motion of a heavy symmetrical top whose lowest point is fixed. The effective potential energy of the top is given by + mgl cos ?, where M, is the constant angular momentum component along the vertical Z axis (fixed), Ms is the constant angular momentum component along the zs axis (the moving axis of symmetry of the top), i is the principal moment of inertia about the axis passing through the lowest point and parallel to the ai axis, m is the mass of the top, and I is the distance from the lowest point to the center of mass of the top (you can sce Figure 48 in the Textbook). Find the condition for the rotation of the top about a vertical axis (? = 0) to be stable. Hint: For ?= 0, the r3 and Z axes coincide, so that My-Afa. Rotation about this axis is stable if ?-0 is a minimnnn of the fiinction tar(e). To find the minimum, use the small-angle approximation in tea(0): sin ? ? and cos ? ~ 1-02/2.

Answer 1

The effective potential of the top is given by U_eff = (M_z - M_3 cos theta)^2/2 l�_1 sin^2 theta + m g l (cos theta) Now for theta = 0 M_z = M_3 To find minimum we use small angle approximation, cos theta almostequalto 1 - theta^2/2)^2/2 l�_2 theta^2 + m g l (1 - theta^2/2) = M_z theta^2/8 l�_1 + m g l (1 - theta/2) For minimum about theta = 0 partial differential^2 U_eff/partial differential theta^2 Vertical-bar _theta = 0 > 0 Now, partial differential^2 U_eff/partial differential theta^2 = M_z/4 l�_1 - m g l So, partial differential^2 U_eff/partial differential theta^2 Vertical-bar _theta = 0 > 0 implies M_z^2 = M_3^2 > 4 m g l l�_1