Physics 105
Mechanics
Autumn 2009
Homework Assignment #9
Due Friday, December 4 by 4:00 pm.
Note: the final exam will be in the usual classroom on Monday, December 7 at 8:00 am. A
practice exam with solutions is posted near the end of the WebCT course materials page. Be
sure to take a close look at it to be aware of what to expect on one of my exams. I will not
present any new material in the last lecture, Friday December 4, but will use that time for
examples and review. Solutions to this assignment will be posted by 5:00 pm on December 4.
1. If a particle is projected vertically upward to a height h above a point on Earth’s surface at a
northern latitude λ, show that it strikes the ground at a point
cos 8 ⁄ to the west.
Consider only small vertical heights (so that g is approximately constant and 2nd-order effects
can be neglected), and neglect air resistance.
2. A warship fires a projectile due south at latitude −50° (i.e. in the southern hemisphere) at a
target that is due south (i.e. with no correction for the Coriolis effect). If the shells are fired
at 37° elevation with a muzzle velocity of 800 m/s, by how much do the shells miss their
target and in what direction? Ignore air resistance and the variation of g with altitude.
Calculate only to first order in ω of the Earth.
x3 = x3′
3. Calculate the principal moments of inertia I1 I 2 I 3 for a
R h
homogeneous cone of mass M whose height is h and whose base
CM
x2′
has a radius R . Choose the x3 axis to be along the axis of
symmetry of the cone (this insures that all off-diagonal terms in the x1′
x2
inertia tensor are zero). Choose the origin to be at the apex of the
cone for this calculation. Then, make a transformation such that the x1
center of mass of the cone becomes the origin, and find the
corresponding principal moments of inertia I1 I 2 I 3 for that new origin (see Eqn. 11.49).
4. A three-particle system consists of masses mi and coordinates ( x1 , x 2 , x3 ) as follows:
m1 = 3m, (b,0, b )
m2 = 4m,
(b, b,−b )
(− b, b,0)
m3 = 2m,
Find the inertia tensor and the principal moments of inertia. You have to solve a cubic
equation. One of the eigenvalues (principal moments) is exactly 10mb 2 , so if you factor that
out, then you can solve the quadratic equation for the other two (which are irrational). Then,
find the principal axis for only the 10mb 2 eigenvalue.
5. A thin uniform rod of length b stands vertically upright on a rough floor and then tips over
without slipping. What is the rod’s angular velocity when it hits the floor?
z
6. Determine the principal axes and principal moments of inertia of a uniform
solid hemisphere of radius b and mass M with respect to an origin at the
y
center of the spherical surface, as indicated in this figure.
x
Physics 105
1
M
Mechanics
Autumn 2009
7. Find eigenvectors and eigenvvalues for thhe problem discussed
d
in Section 12.22 and in Exaample
12.1 for the case that the maasses are diffferent: m2 = 2m1 . Assuume that thee spring consstants
are alll equal. No
ormalize the eigenvectorss according to
t Eqn. 12.558.
8. A bloock of masss m is connnected to a massless
m
sprring of
spring constant k and unexteended lengthh b such thaat it can
oscilllate horizon
ntally. Froom the masss is suspennded a
simplle plane pen
ndulum of length l and mass m . Assume
A
that all
a of the motion takes pllace in a singgle plane.
a) Using
U
as gen
neralized coordinates thhe displacem
ment of
thhe block frrom equilibbrium and the angle of the
pendulum, deerive the Laggrangian forr this system
m. Note
thhat I did this one in lectuure a few weeks ago.
b) Expand
E
the kinetic
k
and potential ennergies abouut the equiliibrium pointt to arrive at
a the
t
t
1
1
coonstant tenso
ors m and A , such that T = ∑ m jk q& j q& k andd U = ∑ A jk q j q k .
2
j ,k
2
j ,k
c) Solve for the two eigenfreequencies ussing the form
malism of Seection 12.4.
d) Find the two correspondinng eigenvectors. Normaalization is not
n required.