Physics 105B Spring 2015
Assignment 2: Chapter 9
Due in lecture Wednesday, April 8
1. A particle in the eld U (x) = F x (F is a constant force) moves from point x(t =
0) = 0 to x(t = ) = a: Use Hamilton principle of least action to determine trajectory
s
John Miller
PHYS 105B, Assignment 2 Solutions
1. We begin by requiring that the trajectory x(t) satisfy the boundary conditions:
x(0) = C = 0 ,
x( ) = A 2 + B = a = B =
a
A ,
so that as a function of the coecient A alone, our trajectory is
x(t; A) = At2
Physics 105B Spring 2015
Assignment 3: Chapter 9
Due in lecture Wednesday, April 15th
1. Follow our discussion in class to re-derive Euler-Lagrange equations for a particle of
mass m and charge q moving in an electromagnetic eld described by the scalar V
John Miller
PHYS 105B, Assignment 3 Solutions
1. We start by calculating the variation of the action and setting it equal to zero
S =
=
=
=
L(r + r, r + r, t) dt
L(r, r, t) dt
L
L
ri +
ri dt
ri
ri
L
d L
ri
ri dt
ri
dt ri
L
d L
ri dt = 0 ,
ri dt ri
fro
Physics 105B Spring 2015
Assignment 4: Chapter 9
Due in lecture Wednesday, April 22nd
1. Thornton & Marion, Exercise 9-35
2. Thornton & Marion, Exercise 9-53
3. Consider elastic scattering of a particle from an impenetrable sphere; the potential is
given
John Miller
PHYS 105B, Assignment 4 Solutions
1. We can express the conservation of momentum and energy as
u1 = v1 +
m2
v2
m1
2
and u2 = v1 +
1
m2 2
v
m1 2
respectively. We can use these both in order to obtain the two expressions
m2
m1
2
2
v2 =
m2 2
2
u
Physics 105B Fall 2015
Assignment 5: Chapter 9, 11
Due in lecture Wednesday, April 29th
1. Consider a particle with E > 0 moving in the central potential U (r) =
Find the dierential cross-section of scattering
d
d
, if
r2
.
> 0.
For < 0 there is minimal i
John Miller
PHYS 105B, Assignment 5 Solutions
1. (a) From assignment 3 problem 5(b), we have that
= =
1+
which we invert to nd
2 =
,
E2
( )2
.
E 2 ( )2
From here, we take a derivative to nd
d
2 3
d
=
2 =
.
d
d
E2 ( 2)3
(b) From assignment 3 problem 5(b
John Miller
PHYS 105B, Week 6 Discussion
1. Crude Model of a Hamster Wheel. Consider a thin uniform-density hoop of radius R
and mass m that is xed to rotate freely in the xy-plane with a uniform gravity eld
g ey . In this hoop a thin uniform-density disk
Physics 105B Fall 2015
Assignment 6: Chapter 11, 10, 12
Due in lecture Wednesday, May 6
1. Recall your algebra class and refresh the proofs of mathematical statements in Thornton & Marion, Exercise 11-22&23. You do not need to write solutions for these tw
James Stankowicz - 105B - Due: May 6, 2015
Homework #6 Solutions
Problem 2 - Composite Disk
Question 2 [MT 11-25]. Consider a thin disk composed of two homogeneous halves connected along a diameter of
the disk. If one half has density and the other has de
Physics 105B Fall 2015
Assignment 7: Chapter 12
Due in lecture Wednesday, May 13
1. Thornton & Marion, Exercise 12-13
2. A ring of mass M and radius R is supported in the eld of gravity from a pivot located
at one point of the ring, about which it is free
Physics 105B Fall 2015
Assignment 8: Chapter 12, 13
Due in lecture Wednesday, May 20
1. Find and describe the normal modes and coordinates of the frictionless oscillations of
four point masses, m1 = m3 = m and m2 = m4 = M on the ring. Masses are attached
John Miller
PHYS 105B, Assignment 1 Solutions
1. From gure 1 we use trigonometry to nd
x = cos x sin y , y = sin x + cos y , z = z ,
or in terms of a matrix equation we have
x
cos sin 0
x
y = sin cos 0 y .
z
0
0
1
z
The rotation of the coordinate sy
Physics 105B Fall 2015
Assignment 1: Review of 105A
Due in lecture Friday, April 3rd
1. Consider two vectors ~ = (ax ; ay; az ) and ~ = (bx ; by; bz )
a
b
How will components of these vectors change, if the coordinate system is rotated
by angle ' in (x; y
Physics 105B
Spring 2017
Midterm # 1
1) A bomb of mass M is at rest on a train. The train moves with constant velocity V for
an observer on the ground, along a straight line that we call the x axis. The bomb
explodes, splitting in two pieces, of mass M/3
Practice Midterm 2 Solutions
Spring 2017
(1) Problem 1
(a) The proper generalized coordinates in this problem are the angles 1 and 2 . The
kinetic and the potential energy of the system can be easily expressed in terms of
these angles. We make the assumpt
Physics 105b Practice Final Solutions
1. (a) The moment of inertia around a fixed axis is
X
I=
mi ri2
where ri in this case is the distance to axis. Instead of writing out explicitly
the position vectors of each hydrogen atom, we just need their distance
James Stankowicz - 105B Discussion 1 - Apr. 7, 2015
Logistics
TA = JJ (or James) Stankowicz
Other TA = John Miller
We switch roles half way through the year
There's also a grader
Discussion attendance = 5% Extra Credit
Quiz grade = 5% Regular Credit
Try t
The topics on the exam:
1. Coupled oscillators (Example problem - see below)
2. Scattering cross section (Example problem - see below)
3. Special theory of relativity - electromagnetic wave
4. Rocket motion
5. Rigid bodies
6. String motion
7. Special theo
James Stankowicz - 105B Discussions Week 4 - Apr. 27, 2015
Question - Integrals of Motion
eld. (L
q in a constant magnetic eld B = Bez
= 1 mr 2 + q r A q).
2
a) Prove
A = 1 yBex + 1 xBey
2
2
Consider a particle of charge
is a valid vector potential for
an
John Miller
PHYS 105B, Week 9 Discussion
1. Resonant Scattering. Consider an innite string with tension . A point mass m is
attached to a spring with natural frequency 0 and dampening coecient b at x = 0.
The other end of the spring is xed such that the s
James Stankowicz - 105B Discussions Week 3 - Apr. 13 and 14, 2015
Question - Lagrangian and Ropes (T&M 9.15):
A smooth rope is placed above a hole in a table. One end of the rope falls through
the hole at
t = 0,
pulling steadily on the remainder of the ro
James Stankowicz - 105B Discussions Week 4 - Apr. 27, 2015
Question - Linear Coupled Spring - Mass System
Consider the following system:
m
m
k
k
(Note: this is the simplied case from discussion in which the two masses are set
equal.)
Longer Method.
From t
Kramers Rule1 :
Suppose you have a system of equations that can be represented in matrix form
as
a1,1 a1,2 a1,3
x
A
a2,1 a2,2 a2,3 y = B
a3,1 a3,2 a3,3
z
C
Kramers rule says
A a1,2 a1,3
B a2,2 a2,3
C a3,2 a3,3
x=
a1,1 a1,2 a1,3
a2,1 a2,2 a2,3
a3,1 a3,2
Bases in Vector Space
The set of vectors cfw_ei is said to be a basis in some space if any vector in
that space can be constructed from a superposition of the vectors in the
set.
V =
ai ei
i
The set of vectors cfw_i is said to be an orthonormal basis
Fourier Transformation
Another Form for the Fourier Series
If f (t) is a periodic function (of period ), it may be written in the form. . .
f (t) =
a0
2
2
+
an cos n
t + bn sin n
t
2
n=1
where
an =
2
2
f (t) cos n
2
2
t dt
and
bn =
2
2
f (t) sin n
2
2
t