x3 m1 r1 r2 x2 m2
In a uniform gravitational field, the gravitational acceleration is everywhere constant. Suppose the gravitational field vector is in the x1 direction; then the masses m1 and m2 have the g
Due: Feb 18 2011
1) A communication satellite is in a circular orbit around the Earth at radius R and velocity
v . A rocket accidentally res and gives the satellite an outward radial velocity v in addition to
its original velocity.
Due: Feb 25 2011
1) Consider a family of orbits in a central potential for which the total energy is a constant.
Show that if a stable circular orbit exists, the angular momentum associated with this orbit is
larger than that for any o
Due: Mar 4 2011
1) a) A system of particles, forming a cylinder of uniform mass density , length l and radius
R, rotates around the axis of the cylinder with angular velocity (t). Calculate the energy of
this system, i.e. evaluate the
Due: Mar 11 2011
1) Taylor 10.25
2) Taylor 10.33
3) Taylor 10.35
4)Determine the principal axes of inertia and the moment of inertia tensor for a tetratomic
molecule which is an equilateral based tetrahedron. The equilateral triangle h
Due: Apr 1 2011
1) Prove that A (B C ) = B (AC ) C (AB ). This is done in Boas and I have done it in
class as well, but I want you to go through the summations carefully. (See Boas, problem 9, p.
2) Prove that the cross product of
Due: Apr 8, 2011
1) ei , i = 1, 2, 3 are orthogonal unit vectors. Prove the following identities:
ej ek = ijk ei
ijk ej ek = 2i
ijk ej ek = 0
2) Prove the following identities:
(A B ) = B ( A) A( B )
(A B ) = (B )A (A )B + A( B ) B
Due: Apr 15, 2011
1) Taylor 11.12
2) Taylor 11.14
3) Repeat Taylor 11.14 with three pendula. Assume the setup is the same as in 11.14, but
the middle pendulum is coupled to both of its neighbors. Find and describe the normal modes.
Due: Apr 29, 2011
1) Consider a loaded string consisting of 3 identical masses connected with identical springs.
At t = 0 the middle mass is lifted a distance D (and consequently the two other masses are to
D/2) and then the system is
Due: Feb 4 2011
This problem set has 2 pages!
1) A bead of mass m slides without friction on a wire that has the shape
The wire is forced to rotate with constant angular velocity 0 around the z axis.
There is uniform gravita
Due: Jan 28 2011
1) Taylor 7.8. I will discuss this problem in class, but nish it as a hw.
2) Taylor 7.15
3) Taylor 7.15 but assume that the string is massive, has a linear mass density .
4) A pendulum of mass m1 and length l is suspen
Due: Jan 14 2011
1) Taylor 6.16. Follow the Hint by putting point 1 to the north pole of the sphere. (What is
there? What is ? )
2) Taylor 6.19 - The soap-bubble problem. I have discussed this in class, but go through the
steps and ll
Dynamics of a System of Particles
9-1. Put the shell in the z > 0 region, with the base in the x-y plane. By symmetry, x = y = 0 .
=0 2 =0 =0
r = r1 2 r2
zr 2 dr sin d d r 2 dr sin d d
= 0 r = r1
Using z = r
Motion in a Noninertial Reference Frame
10-1. (1) (2)
The accelerations which we feel at the surface of the Earth are the following: :
980 cm/sec 2
Due to the Earths rotation on its own axis:
2 rad/day r = 6.4 10 c
Dynamics of Rigid Bodies
The calculation will be simplified if we use spherical coordinates:
x = r sin cos y = r sin sin z = r cos
Using the definition of the moment of inertia,
I ij = ( r ) ij
m1 = M k1 x1 k12
m2 = M k2
The equations of motion are
Mx1 + ( 1 + 12 ) x1 12 x2 = 0 Mx2 + ( 2 + 12 ) x2 12 x1 = 0 We attempt a solution of the form x1 ( t ) = B1 e it x2 ( t ) = B2 e it Su
Physics 3210, Spring 2010 HW#1 - due Wednesday, Jan. 20
For full credit, you must show all your work and explain your reasoning in complete sentences.
1) (Practice with the calculus of variations) Find the curves in the x-y plane that make the