The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS 3032 (Fall 2013)
Homework 6 (Total 100 points)
Due time: 10:30am on 28 October, 2013
1.
(25 points) A cockroach crawls with constant speed in a circular path of radius b on a
phonograph turntable rotating with constant angular speed . The circular pa
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
FULL marks for good TRIALS
ZERO score for COPYING
PHYS 3032 (Spring 2016)
Homework 9
Due time: 10:30 am on April 27, 2016
1. Two particles whose masses are m1 and m2 are connected by a massless spring of
unstressed length l and spring constant k. The syst
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Solution
Problem 1:
(a)
F
r
note, potential energy is time dependent.
L T (qi , qi ) V (qi , t )
V
k
So
F
V Fdr e t
r
r
1
1
T mv v m r 2 r 2 2
2
2
p
L
pr
mr
r r
r
m
p
L
p
mr 2
2
mr
p 2 k t
pr 2
H pi qi L pr r p
e
2m 2mr 2 r
Substituting for r and
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS 3032 (Spring 2016) Tutorial 4 Classwork
7, 10 March 2016
1.
Prove that   = 3 , where is the radius of curvature of the path of a moving
particle.
2.
A particle moves in a spiral orbit given by = . If increases linearly with , is the
force a central
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS 3032 (Spring 2016) Tutorial 9 Classwork
25, 28 April 2016
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
(a) With what
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
FULL marks for good TRIALS
ZERO score for COPYING
PHYS 3032 (Spring 2016)
Homework 8
Due time: 10:30am on 20 April 2016
1. Find the differential equations of motion for an elastic pendulum: a particle of mass
attached to an elastic string of stiffness and
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS 3032 (Spring 2016)
Homework 7
Due time: 10:30 am on April 13, 2016
1. (25 points)
Calculate the integral
= [ , , , , ]
for the simple harmonic oscillator. Follow the analysis presented in Section 10.1 in textbook.
Show that is an extremum at = 0.
2.
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Homework Solution 7
Problem 1:
x t x 0, t t
x t x 0, t t
x 0, t sin t and x 0, t cos t
where
T
so:
1 2
mx
2
J
1
1
V kx 2 m 2 x 2
2
2
t
m2 2
x 2 x 2 dt
2 t1
t
m2
2
2
cos t 2 sin t dt
2 t1
J
2
m2 2
2m 2 2
2
2
cos
sin
cos
sin
t
t
dt
m
t
t
dt
2 2 dt
2
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
FULL marks for good TRIALS
ZERO score for COPYING
PHYS 3032 (Spring 2016)
Homework 10
Due time: 10:30 am on May 4, 2016
1.
[20 points]A circular hoop of radius a swings as a physical pendulum about a point on the
circumference. Find the period of oscillat
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS 3032 (Spring 2016)
Homework 5 Solution
Problem 1:
1
1
r r cos
du
sin
d r cos 2
u
d 2u 1 1
2sin 2
d 2 r cos cos3
1
r cos
2 2cos 2
1
cos 2
1 2
1
2
r cos cos
d 2u
u 2r 2u 2 1 2r 2u 3 u
2
d
Substituting into equation 6.5.10
1
2r 2u 3 u u 2
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Tutorial 1 Classwork Solution
Problem 1:
a. If F(x, v) = f(x) g(v):
m
dv
dv
mv f x g v
dt
dx
mvdv
f x dx
g v
By integration, get v v x
dx
dt
b. If F(x, t) = f(x) g(t):
m
d 2x
d dx
m f x g t
2
dt
dt dt
This cannot, in general, be solved by integrat
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Problem 1:
sin 0
2
( cos ) = 0
2
Integrating:
2
2
cos or 2 2 cos cos
0
T 4
0
d
2 cos cos
1
2
.
Time for pendulum to swing from 0 to is T/4.
Now, substitute sin
sin
2 so at and use the identity cos 1 2sin 2
2
2
sin
2
T 4
0
d
And after some algebr
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Homework Solution 11
Problem 1:
When two men hold the plank, each supports
mg
.
2
mg R mxcm and R
When one man lets go:
From table 8.3.1,
I cm
l
I cm
2
ml 2
12
Rl 12 6 R
2 ml 2 ml
l
3R
xcm
2
m
3R
mg R m
3R
m
mg
R
4
6 R 6 mg
xend l l
ml m 4
3
xe
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Problem 1:
a) Locate center of coordinate system at C.M.
The potential is independent of the center of
mass coordinates.
Therefore, they are
ignorable.
L T V
1
1
1
2
m1 r12 r1212 m2 r2 2 r2 2 2 2 k r1 r2 l where l is the length of the
2
2
2
relaxed spring
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS 3032 (Spring 2016) Tutorial 5 Classwork
14, 17 March 2016
1. A comet is going in a parabolic orbit lying in the plane of Earths orbit. Regarding
Earths orbit as circular of radius a, show that the points where the comet intersects
Earths orbit are gi
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Problem 1:
Problem 2:
Period of a simple pendulum:
Period of real pendulum:
T 2
T 2
a
g
I
Mgl
Where I moment of inertia
l Distance to CM of physical pendulum
a Distance to CM of bob
b Radius of bob
Location of CM of physical pendulum:
l
m
a b
M m a
m
2
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS 3032 (Spring 2016) Tutorial 8 Classwork
18, 21 April 2016
1. A particle of mass m is subject to a central, attractive force given by
(, ) = 2
Where k and are positive constants, t is the time, and r is distance to the centre of force.
(a) Find the H
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Tutorial 4 Classwork Solution
Problem 1:
The figure below shows the unit vectors n and which are normal and tangent to
the curve. The vectors are shown at 2 different points along the trajectory. Since
the particle is moving tangentially to the curve and
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Tutorial 6 Classwork Solution
Problem 1:
We choose the Cartesian coordinate axes with x due east, y due north, and z vertically up. The picture
shows the hoop as seen from above.
Consider first a small segment of the hoop subtending an angle with polar an
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Chapter 4
General Motion of a Particle in
Three dimensions
Chapter 1: vector analysis
Particle motion described by vectors
Position of a point particle can be precisely defined
by the position vector r (t )
r =
r (t + t ) r (t )
Hence the velocity
r (t
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
PHYS3032
Classical Mechanics
http:/teaching.phys.ust.hk/phys3032/
Introduction
Review of Newtons laws
Classical Mechanics is the subject of studying why
an object (macroscopic) moves in the way it does.
Newtonian mechanics (Vectorial mechanics)
Good for
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
(PHYS221)[2010](f)quiz~2678^_10048.pdf downloaded by clcheungac from http:/petergao.net/ustpastpaper/down.php?course=PHYS221&id=7 at 20150301 05:53:53. Academic use within HKUST only.
PHYS221
Intermediate Classical Mechanics
Solution of Quiz 1
P
N
N
mg
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Chapter 5
Noninertial reference systems
Rotating Coordinate Systems
In some problems, it is convenient to analyse the
motion of an object using a rotating coordinate.
A rotating frame is not an inertial frame.
need to add fictitious forces to restore
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Chapter 9
Motion of Rigid Bodies in 3D
Angular Momentum for an Arbitrary Angular Velocity
v i mi ri ( ri )
The angular momentum L mi ri =
for an arbitrary axis passing through a fixed point
An arbitrary angular velocity = ( x , y , z ).
A double cross
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Chapter 4
General Motion of a Particle in
Three dimensions
Chapter 1: vector analysis
Particle motion described by vectors
Position of a point particle can be precisely defined
by the position vector r (t )
r =
r (t + t ) r (t )
Hence the velocity
r (t
The Hong Kong University of Science and Technology
Classical Mechanics
PHYS 3032

Spring 2015
Chapter 7
Dynamics of Systems of N Particles
The Center of Mass
The motion of the Centre of
Mass is a simple parabola.
(just like a point particle)
The motion of the entire
object is complicated =
(motion of the CM) +
(motion of points
around the CM)
vide