Physics 507
Midterm Exam
Fall 2012
Monday, November 12, 12 - 1:20 pm, ARC 203
All problems carry the same weight.
1. (from homework) Two mass points of mass m1 and m2 are connected by a string passing
Physics 507
Solutions to Homework Set 1
Fall 2012
1. Goldstein Ch. 1, problem 12.
v=
mv 2
Mm
G
= 0 escape condition
2
R
M
= 2gR 9.8 6.4 106 m/s = 11.2 km/s,
2G
R
where we used mg = GmM/R2 , which impl
Physics 507
Solutions to Homework Set 2
Fall 2012
1. Goldstein Ch. 2, problem 12.
t2
S=
t1
t2
S =
t1
L(qi , qi , qi , t)dt
L
L
L
qi +
qi +
qi dt = 0
qi
qi
qi
Integrate the rst term by parts once a
Physics 507
Solutions to Homework Set 3
Fall 2012
1. Goldstein Ch. 3, problem 13.
a) Let f (r) be the force and V (r) the corresponding potential. The component of the force
towards the center provide
Physics 507
Solutions to Homework Set 4
Fall 2012
1. Goldstein Ch. 3, problem 18.
Impulse S means an instantaneous change in particles linear momentum
p = S r
The corresponding changes in the energy,
Physics 507
Solutions to Homework Set 5
Fall 2012
1. Goldstein Ch. 4, problem 6.
Let A and B be rotations in parts a) and b), respectively and let Rz be a regular rotation about
the current z -axis. T
Physics 507
Solutions to Homework Set 6
Fall 2012
1. Goldstein Ch. 5, problem 13.
Let be the angle the left rod makes with the x-axis. The coordinate system is as shown in the
gure in Goldstein. Initi
Physics 507
Solutions to Homework Set 7
Fall 2012
1. Goldstein Ch. 6, problem 4.
The stable equilibrium position is 1 = 2 = 0. Expand the potential and kinetic energies
around this
V = m1 gl cos 1 m2
Physics 507
Solutions to Homework Set 8
Fall 2012
1. Goldstein Ch. 8, problem 14.
The Lagrangian is in the form of Eq. (8.23) of Goldstein, which diers from what we considered
in class by a linear in
Physics 507
Solutions to Homework Set 9
Fall 2012
1. Goldstein Ch. 9, problem 11.
It is sucient to check the Poisson brackets of Qi , Pi .
[Q1 , P1 ]q,p =
1
1
Q1 P1 Q1 P1
= q2
+ q1
= 1,
qi pi
pi qi
q2
Physics 507
Final Exam
Fall 2012
Monday, December 17, 12 - 2 pm, ARC 108
All problems carry the same weight.
1. Two pendulums of equal length l and masses m1 and m2 are connected by a spring with
spri
Physics 507
Solutions to Final Exam
Fall 2012
Monday, December 17, 12 - 2 pm, ARC 108
1. Let 1,2 be the angles (in the same direction) pendulums make with the vertical. Note that
there is a normal mod
Physics 507
Solution to the Midterm
Fall 2012
1. Convenient coordinates are e.g.: x the length of the segment of the string on the table and
the angle that segment makes with any xed line in the pla