Name:Solutions
PHY 5246: Theoretical Dynamics, Fall 2012
November 5th, 2012
Midterm Exam # 2
Always remember to write full work for what you do. This will help your grade in
case of incomplete or wrong answers. Also, no credit will be given for an answer,
Name: SOLUTIONS
PHY 5246: Theoretical Dynamics, Fall 2012
October 1st, 2012
Midterm Exam # 1
Always remember to write full work for what you do. This will help your grade in
case of incomplete or wrong answers. Also, no credit will be given for an answer,
PHY 5246: Theoretical Dynamics, Fall 2012
December 5th, 2012
Assignment # 11, Solutions
1
Graded problems
Problem 1 (Goldstein 9.21)
(1.a)
We have a 1D system
H=
1
p2
2.
2
2q
D=
pq
Ht
2
We want to show that
is a constant of the motion.
D
D
pq
dD
= [D, H
PHY 5246: Theoretical Dynamics, Fall 2012
November 21st, 2012
Assignment # 10, Solutions
1
Graded problems
Problem 1
(1.a)
The Lagrangian is
1
L = m(r 2 + r 2 2 + r 2 sin2 2 ) V (r ),
2
and the conjugate momenta are
L
= mr,
r
L
= mr 2 ,
=
L
=
= mr 2 sin2
PHY 5246: Theoretical Dynamics, Fall 2012
November 28th, 2012
Assignment # 11
(Graded problems are due Wednesday December 5th, 2012)
1
Graded problems
1. Chapter 9, Problem 21 of your book.
2. Chapter 9, Problem 25 of your book.
3. Chapter 9, Problem 30 o
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 8, Solutions
1
Graded Problems
Problem 1
First we calculate the moments of inertia:
I1 = I2 = m
x3
I3 =
a
x1
ma2
.
2
(1.a)
The torque is zero! This can be seen in several ways: for
instance, from the
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 9, Solutions
1
Graded problems
1. In coordinates (1 , 2 ) (see gure), we have
1
1
m(b1 )2 + m(b2 )2
2
2
T=
V
1
= mgb(1 cos 1 ) + mgb(1 cos 2 ) + k (b sin 1 b sin 2 )2 .
2
Note the equilibrium length o
PHY 5246: Theoretical Dynamics, Fall 2012
November 16th, 2012
Assignment # 10
(Graded problems are due Wednesday November 28th, 2012)
1
Graded problems
1. Consider a particle in a central force eld.
(1.a) Obtain the Hamiltonian and the canonical equations
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 7, Solutions
1
Graded Problems
Problem 1
(1.a)
In order to nd the equation of motion of the triangle,
we need to write the Lagrangian, with generalized coordinate cfw_. The potential energy is going t
PHY 5246: Theoretical Dynamics, Fall 2012
November 7th , 2012
Assignment # 9
(Graded problems are due Friday November 16th , 2012)
1
Graded problems
1. Determine the eigenfrequencies and describe the normal mode motion for two pendula of
equal lengths b a
PHY 5246: Theoretical Dynamics, Fall 2012
October 24th , 2012
Assignment # 8
(Graded problems are due Wednesday October 31st , 2012)
1
Graded problems
1. A uniform solid cylinder of mass m, length b, and radius a is thrown up in the air; at the
instant it
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 6, Solutions
1
Graded Problems
Problem 1
The equation of the orbit for the force given in this problem is
2
2
that can be written as
1
mr 2
mk m
1
+ = 2 F (r ) = 2 + 2 ,
r
r
l
l
lr
2
2
m
1
+ 1 2
r
l
m
PHY 5246: Theoretical Dynamics, Fall 2010
Assignment # 4, Solutions
1
Graded Problems
Problem 1
(1.a)
The problem has spherical symmetry and is therefore naturally solved using spherical coordinates
(see gure). The xed length of the pendulum gives the con
PHY 5246: Theoretical Dynamics, Fall 2012
October 10th , 2012
Assignment # 6
(Graded problems are due Wednesday October 17st , 2012)
1
Graded problems
1. Discuss the motion of a particle in a central inverse-square law force eld for a super-imposed
force
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 5, Solutions
1
Graded Problems
Problem 1
(1.a)
Using the equation of the orbit or force law
d2
d2
1
mr 2
1
+ = 2 F (r ) ,
r
r
l
(1)
with r () = ke one nds
2 1
mr 2
+ = 2 F (r ) ,
r
r
l
(2)
(1 + 2 )l2
PHY 5246: Theoretical Dynamics, Fall 2012
October 17th , 2012
Assignment # 7
(Graded problems are due Wednesday October 24th , 2012)
1
Graded problems
1. Find the frequency of small oscillations for a thin homogeneous plate if the motion takes
place in th
PHY 5246: Theoretical Dynamics, Fall 2012
September 26th , 2012
Assignment # 5
(Graded problems are due Wednesday October 10th , 2012)
1
Graded problems
1. Consider a particle that moves in a logarithmic spiral orbit given by r = ke , where k and
are con
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 3, Solutions
1
Graded Problems
Problem 1
(1.a)
We use cylindrical coordinates and notice that z = r cot . We use cfw_r, as generalized coordinates
and write the kinetic energy of the bead as
1
1
1
r2
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 2, Solutions
1
Graded Problems
Problem 1
(1.a)
The coordinates can be written using spherical coordinates as:
x = R sin cos(t) ,
y = R sin sin(t) ,
z = R cos ,
z
t
y
R
x
(1)
where we have used the con
PHY 5246: Theoretical Dynamics, Fall 2012
September 19th , 2012
Assignment # 4
(Graded problems are due Wednesday September 26th , 2012)
1
Graded problems
1. A spherical pendulum consists of a bob of mass m attached to a weightless, extensionless
rod of l
PHY 5246: Theoretical Dynamics, Fall 2012
September 12th , 2012
Assignment # 3
(Graded problems are due Wednesday September 19th , 2012)
1
Graded problems
1. A particle slides on the inside surface of a frictionless cone subject to the action of the
gravi
PHY 5246: Theoretical Dynamics, Fall 2012
Assignment # 1, Solutions
Problem 1
y
cos
r
sin
r
r=
sin
cos
r
x
Starting from the position vector for a planar motion in polar coordinates
r = r ,
r
we can derive the velocity vector as
r = r + r ,
r
where w
PHY 5246: Theoretical Dynamics, Fall 2012
September 5th , 2012
Assignment # 2
(Graded problems are due Wednesday September 12th , 2012)
1
Graded problems
1. A bead of mass m slides without friction in a uniform gravitational eld on a vertical circular
hoo
PHY 5246: Theoretical Dynamics, Fall 2012
August 29th , 2012
Assignment # 1
(Due Wednesday September 5th , 2012)
1. Consider a planar motion. In Cartesian coordinates a general vector takes the form,
r = xx + y y = r cos x + r sin y .
Rewrite the same vec