Homework set # 14
Problem 1 (Chapter 8 - problem 11)
The moment of inertia I cm of a ring with radius a and mass m rotating around an
axis going through the center is
I cm ma 2
a) For a rotation around a pivot point on the circumference and the axis of
ro

Homework set # 5
Chapter 3 - Problem 11
Fx kx c dx F 0 cos t
dt
with
c 3m F 0 mA
k 17 2 m
2
This is a driven, damped harmonic oscillator with
k
2 m 17 2 0 17
0
2
2
c
2m
3m
2m
3
2
a)
The resonance frequency r , where the amplitude is maximal, is given by

Homework set # 13
Problem 1 (Chapter 8.1 - a, c, e)
a)
Use coordinate system indicated in figure.
Assume mass of each segment with length L is m M/3.
The CM of each of the three segments are:
R CM,1 b ,
2
b
2
R CM,2 0, 0
R CM,3 b ,
2
so
b
2
i mixi
i mi

Homework set # 12
Problem 12 (Chapter 7 - Problem 14 16)
The given quantities are the mass m p and initial velocity p,0 0 of the proton and the
v
v
mass of the helium m He 4m p and its initial velocity v He,0 0. The collision is elastic
( 1) and the scatt

Homework set # 11
Problem 1 (Chapter 7 - Problem 1 & 2)
Given m 1 m 2 m 3 1
1 1, 1, 0
r
2 k 0, 1, 1
r
3 k
r
0, 0, 1
mi 1 1 1 3
m
1 2
v
2, 0, 0
2
v
0, 1, 0
3 k 1, 1, 1
v
1a)
cm
r
1
3
r r
r 1 2 3 1 ,
3
2
3
,
2
3
cm
v
1
3
v v
v 1 2 3 1,
2
3
,
1

Homework set # 7
Problem 1 (chapter 4 - problem 6)
a) In example 2.3.2 it was shown that the gravitational potential energy of the
Earth is
2
e
Vz mg r rz mgr e
e
1
1 rze
where r e is the radius of the Earth and z is the height above the surface. We will

Homework set # 10
Problem 1 (Chapter 6 - Problem 18)
From example 6.10.2 we find the following expressions for the velocity at the
perigee v p and the the apogee v a (derived from energy conservation)
v2
p
v2
a
GM 1
a 1
GM 1
a 1
so
v2v2
p a
GM 1 GM 1
a

Homework set # 9
Problem 1 (Chapter 6 - Problem 10)
a)
r r 0 e k
u
1
r
1
r0
e k
We will use the differential equation of the orbit (6.5.10b) to find fu 1
d2u
d2
u ml1u 2 fu 1
2
We need to find
du
d
d2u
d2
fu 1 ml 2 u 2 d 2 u u
d
2
d2u
d2
k k
r0 e
kk

Homework set # 8
Problem 1 (chapter 4 - problem 20)
The initial position of the electron q e is
0 0, 0, 0
r
and the initial velocity is
0 v 0 , 0, 0
v
The Lorentz force is
v
F qE B
with E E and B Bk
According to example (4.5.1) for B and formula (4.5.2b

Problem set no. 6
Problem 4.1b
Vx, y, z x 2 y 2 z 2 C
F V x y k z x 2 y 2 z 2 C
Fx, y, z 2x 2y 2zk
Problem 4.1d
Vr cr n
In spherical coordinates
F V r
r
cr n
Fr cnr n1 r
Problem 4.2b
F y x kz 2
y
y
z
y
k
x
F
x
z2
z2
z
x
z
y
0 0 0 0 k1 1
x
z2

Chapter 2 - Problem 2 + 3
1
For all sub-problems: t 0 = 0 x0 = 0 v0 = 0 T0 = mv0 2 = 0 E =T0 + V0 =V0
2
We will be using the following method for calculating v( x) and V ( x) by combining
problem 2 and 3 :
1) Use (2.3.1) and (2.3.2):
v
dv
=
=
F (= mx mv m

Homework set No. 4
Chapter 3 - Problem 3
Simple Harmonic Motion.
f 10 Hz 2f 62. 832 s 1
Initial conditions: t 0 x 0 0. 25 m
xt A cost 0
v 0 0. 1 m/s
vt A sint 0
During the lectures it was derived that
(or follow the text book for the x A sint case)
v
0.

Chapter 2 - Problem 11
Fv cv 3/2
vx 0 v 0 0
Show, that x max 2mv 1/2 /c.
0
v
Fv m dv cv 3/2 dt m v 3/2 t m v 3/2 dv
c
c
dt
v0
t
2m
c
1
v
m
c
2v 1/2 v 0
v
1
v0
Solve for vt
ct
1
10 2m
v
v
v
1
v0
1
ct
2m 2
dx
dt
t
xt
t
0
1
v0
dt
ct
2m 2
x max lim t
1

Chapter 1 - Problem 2
A 2 +
= i j
B= + k
i
C = 4
j
a ) A( B + C ) = 2 + + k + 4 = 2 + 4 = 6
i j
i
j
(
(
b)
A+ B C =
)
)
(3i + j + k ) ( 4 j ) =
4
A( B C ) = 2 + ( 0 4 ) + ( 0 0 ) + ( 4 0 ) k
i j
i
j
( )(
)
i j
i
=( 2 + ) ( 4 + 4k ) =8 + 0 +

Physics 311 - 2012 - Fall semester
Test no 1 - September 28, 2012
Name of Student:
(please write your full name so it is clearly readable!)
This test contains 3 problems. Partial credit will be given so please show all your work.
Indicate the final answer