MATH 4051 Homework Solution 04
Exercise 1.4-1 For the initial value problem x = tx, x(0) = 1, obtain the n-th
approximation of the Picard iteration. Use mathematical induction to justify the
formula.
t
Proof. x = X (x, t) = tx, x0 (t) = x(0) = 1. U (x) =
MATH 4051 Homework Solution 03
Exercise 1.2-5 Show that the initial value problem
du
= tu + sin u,
dt
u(0) =
1.
has a unique solution.
Proof. Let U (u, t) = tu + sin u. Thus,
|U (u1 , t) U (u2 , t)| = |tu1 + sin u1 tu2 sin u2 |
t|u1 u2 | + | sin u1 sin
PHYS 3032 (Fall 2012)
Homework 4 (total 100 points)
Due time: 3:00 pm on 11 October, 2012
Problems from the textbook:
Problem 6.3, 6.4 and 6.7 (40 points)
Problem 6.12 (20 points)
Problem 6.19 (20 points)
Problem 6.20 (a) (20 points)
Homework Solution 4
Problem 6.3, 6.4 and 6.7:
GMm
6.3
F 2 er
r
The gravitational force on the particle is due only to the mass of the earth that is inside
the particles instantaneous displacement from the center of the earth, r. The net effect of
the mass
PHYS 3032 (Fall 2012)
Homework 5 (total 100 points)
Due time: 3:00 pm on 18 October, 2012
Problems from the textbook:
Problem 6.17 (20 points)
Problem 6.25 (20 points)
Problem 7.12 (20 points)
Problem 7.2 (20 points)
*You do NOT need to solve Problem 7.1.
Homework Solution 5
Problem 6.17:
6.17 From Example 6.10.1
1
v
r
2
2 2
where q
and d
1 q 2 qd sin
ve
ae
d
are dimensionless ratios of the comets speed and distance from the Sun in terms of the Earths
orbital speed and radius, respectively (q and d ar
PHYS 3032 (Fall 2012)
Homework 6 (total 100 points)
Due time: 3:00 pm on 25 October, 2012
1. A particle of mass m with initial momentum P1 collides with a particle of equal mass at
rest. If the magnitudes of the final momenta of the two particles are P1 a
PHYS 3032 (Fall 2012) Tutorial 6 Classwork
6:00 6:50pm, 13 November 2012
1. A heavy elastic spring of uniform stiffness and density supports a block of mass m. If m
is the mass of the spring and k its stiffness, show that the period of vertical oscillatio
P1
Homework Solution 6
Problem 1:
P1
P2 P2 P22 2P P2 cos which is the law of cosines.
1
1
1
Conservation of energy:
P 2 P2 P22
1
1
Q
2m 2m 2m
1
Q
P12 P12 P22 21 2P1P2 cos
2m
m
P P cos
Q 1 2
m
P2
Problem 2:
Set Fu Fg mv vm . Since v constant, v 0
m z v
Homework Solution 7
Problem 1:
(a)
r i x R cos t sin t
jR
r i R sin t R cos t
j
2
22
v R circular motion of radius R
r r v R
where r ix
jy
i R sin t R cos t k ix
j
jy
(b) r r r
i R sin t R cos t x i y
j
j
x y R sin t
y x R cos t
(c)
Let u x iy
here
PHYS 3032 (Fall 2012)
Homework 8 (total 100 points)
Due time: 3:00 pm on 15 November 2012
1. Find the differential equations of motion of a projectile in a uniform gravitational field
without air resistance.
2. Two blocks of equal mass m are connected by
Homework Solution 8
Problem 1:
V mgz
L T V
L
mx ,
x
T
1
m x2 y 2 z 2
2
1
m x 2 y 2 z 2 mgz
2
L
L
my ,
mz
y
z
d L
d L
mz
my ,
dt z
dt y
L
mg
z
L d L
Lagranges equations of motion:
0
qi dt qi
my 0
mz mg
mx 0 ,
mx const , my const
d L
mx ,
dt
Homework Solution 3
Problem 4.1 (a, d):
4.1
V V V
j
k
x
y
z
F c iyz kxy
jxz
(a) F V i
(d) F V er
V
1 V
1 V
e
e
r
r
r sin
F er cnr n 1
Problem 4.2 (a, d):
4.2
(a)
i
F
x
x
j
y
y
k
0
z
z
conservative
(d)
er
F
1
r sin
2
e r
e r sin
r
kr n
0
0
0 c
PHYS 3032 (Fall 2012)
Homework 3 (total 100 points)
Due time: 3:00 pm on 4 October, 2012
Problems from the textbook:
Problem 4.1 (a, d) (15 points)
Problem 4.2 (a, d) (15 points)
Problem 4.9 (20 points)
Problem 4.17 (25 points)
Problem 4.20 (25 points)
Problem 1:
For v the velocity of a differential mass element, dm, of the spring at a distance x below the
support
x
m
v v ,
dm dx
x
x
2
m 1
x 1 x
12
1
m
2
dm mx 2 x
T mv
dx
v
02
02
2
2
x x
1
1 m 2
T mx 2
x
2
23
m
1
2
V k x l mgx gxdm
0
2
x
1
m
2
k x
PHYS 3032 (Fall 2012) Tutorial 7 Classwork
6:00 6:50pm, 20 November 2012
1. A spherical pendulum consists of a point mass m tied by a string of length l to a fixed
point, so that it is constrained to move on a spherical surface as shown in Fig. 1.
Fig. 1
1
1
1. Let the moments of inertia of A and B be I a M a a 2 and I b M bb 2 . The
2
2
angular velocity of A is while that of B is (remember that in two
dimensions, angular velocity is the rate of change of an angle between an line or direction
fixed to the
PHYS 3032 (Fall 2012)
Homework 1 (total 100 points)
Due time: 3:00 pm on September 20, 2012.
Problems from the textbook:
2.2 (20 points)
Find the velocity as a function of the displacement for a particle of
mass , which starts from rest at
, subject to th
PHYS 3032 (Fall 2012)
Homework 2 (total 100 points)
Due time: 3:00pm on 27 September, 2012
1. (10 points) Two springs having stiffness k1 and k2, respectively, are used in a vertical
position to support a single object of mass m. Show that the angular fre
Homework Solution 2
Problem 1:
For springs tied in parallel:
Fs x k1x k2 x k1 k2 x
1
k k 2
1 2
m
For springs tied in series:
The upward force m is k eq x . Therefore, the downward force on spring k 2 is k eq x . The upward
force on the spring k 2 is k1
PHYS 3032 (Fall 2012)
Homework 9 (total 100 points)
Due time: 3:00 pm on 22 November, 2012
1. (25 points)
2. (25 points)
3. (25 points)
(Example 10.6.2 is attached.)
4. (25 points)
A uniform rod with length
with radius
and mass
is placed in a frictionless
Homework Solution 12
9.1 (a) I xx y 2 z 2 dm
dm dxdy and m 2a 2
z
I xx
y a
y 0
y
x 2 a
2
x 0
0 dxdy
a
2a y 2 dy
0
2 4 ma 2
I xx a
3
3
I yy x2 z 2 dm
y
y 0
2a
x
a
a
0
8a 3
dy
3
8a 4 4ma 2
I yy
3
3
From the perpendicular axis theorem:
5ma 2
I zz I
PHYS 3033 Assignment 5
Due: 1 Nov 2012 at begin of lecture at 12:00 pm
Problem 1.
Consider two charges, +q at (0, 0, d/2) and q at (0, 0, -d/2).
(a) By Coulombs law, write down the exact potential at r due to the two points charges.
Given that r > d, usin
PHYS 3033 Assignment 5
Due: 1 Nov 2012 at begin of lecture at 12:00 pm
Problem 1.
Consider two charges, +q at (0, 0, d/2) and q at (0, 0, -d/2).
(a) By Coulombs law, write down the exact potential at r due to the two points charges.
Given that r > d, usin