Download Set 5

PHY 4222 – Problem Set 5
1) Text Problem 11.14 – orient the hemisphere with the curved side up and choose coordinates with
the origin in the center of the flat circular face and the z-axis perpendicular to the flat circular face.
It is easiest to first find the principal moments with respect to the origin of coordinates and then to
use the generalization of the parallel axis theorem (Steiner's theorem) to find the moments with
respect to the cm. Note that you will need to find the location of the cm on the z-axis.
2) Text Problem 11.16
3) A very thin circular disk with radius R and mass M lies in the xy-plane with its center at the origin.
a) Assuming that the mass of the disk is distributed uniformly over its circular area, determine the
elements of the inertia tensor. Note that the disk has azimuthal symmetry.
b) The disk is now caused to rotate with a constant angular velocity ω about an axis that lies in the xzplane and makes an angle θ with the z-axis. Derive a relation between θ and the angle α that the
angular momentum vector of the disk makes with the z-axis.
c) Show that the rotational kinetic energy of the disk is proportional to 1+cos²θ and find the constant
of proportionality.
4) Text Problem 11-27 – note that the angular velocity requested is just dϕ/dt, where ϕ is the Euler
angle identified in the text. To solve this problem, you will need to use the equations that relate the
angular velocity components to the Euler angle time derivatives for force free motion and the
equations that connect the angular momentum to the angular velocity.
5) Text Problem 11-34
6) The principal moments of inertia of a uniform plate are I1, I2>I1, and I3=I1+I2. Choose a coordinate
system with the origin at the cm of the plate. The plate rotates with an angular velocity ω about an
axis that makes an angle α with the plane of the plate such that at time t=0, ω1(t=0)=ωcos α,
ω2(t=0)=0, and ω3(t=0)=ωsin α. The rotation is force free, and I1/I2 = cos 2α.
a) Show that Euler's equations require that ω1² + ω2² = ω² cos²α at all times.
b) Show that ω2 and ω3 are related by the equation ω3² = ω² sin²α – ω2² tan²α.
c) Using the results of parts a and b in Euler's equation for dω2/dt, show that at time t,
ω2 = ω cos α tanh(ωt sin α)
Note that the integral required to obtain this result is given in appendix E of the text.