Two-body motion with a central potential
Translation from general expressions:
gravitational force:
based on FW-4
particle with the
reduced mass subject to
a central force
Everything we have derived for the motion
of a particle with mass m subject to a
ce
Classical Mechanics
P 521, Fall 2013
Radovan Dermisek
Indiana University, Bloomington
Notes based on:
Fetter and Walecka, Theoretical Mechanics of Particles and Continua,
Chapters 1-6
1
Basic Principles
based on FW-1
Newtons first law:
In the primary iner
Falling particle:
solution as an expansion in !
the deflection is Eastward
for a 100m drop the equatorial deflection is 2.2cm
57
Horizontal motion:
Coriolis force deflects moving objects to
the right in the Northern Hemisphere and
to the left in the South
Homework set - 4
due on Wednesday, October 9, 11:00am
1) Problem 3.1 from Fetter and Walecka
2) Problem 3.5 from Fetter and Walecka
3) Problem 3.6 from Fetter and Walecka
4) next page
1
4) Imagine a simple pendulum consisting of a mass m attached to one e
Homework set - 9
due on Wednesday, December 4, 11:00am
1) A nasty bacterium in the shape of a spheroid with principal axes b, b, a and
uniform density is spinning in free space about its axis of symmetry
with angular velocity !r . The symmetry axis of the
Homework set - 3
due on Wednesday, September 25, 11:00am
1) Problem 2.4 from Fetter and Walecka
2) Problem 2.5 from Fetter and Walecka
3) Problem 2.1 a) from Fetter and Walecka
b) What is the trajectory of the particle with charge e in the rotating frame
Homework set - 1
due on Wednesday, September 11, 11:00am
1,2,3) Problems 1.4, 1.6 and 1.10 from Fetter and Walecka
4) Imagine a particle of mass m moving in a perturbed attractive Coulomb
potential:
k
V (r) =
r
r2
where k (> 0) and alpha are constants.
a)
Homework set - 5
due on Wednesday, October 23, 11:00am
1) Problem 3.10 from Fetter and Walecka
2) Problem 3.11 from Fetter and Walecka
3) Problem 3.15 from Fetter and Walecka
4) Problem 3.17 from Fetter and Walecka
Homework set - 2
due on Wednesday, September 18, 11:00am
1) Problem 1.7 from Fetter and Walecka
2) Problem 1.13 from Fetter and Walecka
3) Problem 1.14 from Fetter and Walecka
hint: do first the problem 1.13 and think about similarities
4) next page
1
4)
Homework set - 8
due on Wednesday, November 20, 11:00am
1) Problem 5.2 from Fetter and Walecka
2) Problem 5.5 from Fetter and Walecka
3) Calculate the moments of inertia of a spheroid with principal axes b, b, a
and uniform density.
Rotating Coordinate Systems
based on FW-6,7,8
Sometimes it is useful to analyze motion in a non inertial reference frame, e.g.
when the observer is moving (accelerating).
Inertial frame:
orthonormal coordinate
system (fixed)
Rotating frame
(body fixed):
f
Falling particle:
solution as an expansion in !
the deflection is Eastward
for a 100m drop the equatorial deflection is 2.2cm
57
Horizontal motion:
Coriolis force deflects moving objects to
the right in the Northern Hemisphere and
to the left in the South
Two-body motion with a central potential
based on FW-4
Everything we have derived for the motion
of a particle with mass m subject to a
central force:
can be directly (with small modifications) applied
to a system of two particles interacting through a
ce
Two-body motion with a central potential
based on FW-4
Everything we have derived for the motion
of a particle with mass m subject to a
central force:
can be directly (with small modifications) applied
to a system of two particles interacting through a
ce
Central Forces
Keplers second law:
based on FW-3
Motion of a particle with mass m subject
to a central force:
An imaginary line drawn from each planet to the Sun sweeps
out equal areas in equal times.
conservative
xF=0
(a simple consequence of the
conserv
Cross section:
incident flux
(number of particles crossing unit transverse area per unit time)
differential cross section
event p = elastic scattering through a
deflection angle between # and # + d#:
even
ter
events scat
gle
to solid an
ts or
differential
Cross section:
Hyperbola:
incident flux
constant difference in distance from two foci
located at x=+f and x=-f:
(number of particles crossing unit transverse area per unit time)
left branch
r
"
differential cross section
even
ts or
eccentricity:
ter
event
Rotating Coordinate Systems
based on FW-6,7,8
Sometimes it is useful to analyze motion in a non inertial reference frame, e.g.
when the observer is moving (accelerating).
Inertial frame:
orthonormal coordinate
system (fixed)
Rotating frame
(body fixed):
f
Classical Mechanics
P 521, Fall 2013
Radovan Dermisek
Indiana University, Bloomington
Notes based on:
Fetter and Walecka, Theoretical Mechanics of Particles and Continua,
Chapters 1-6
1
Basic Principles
based on FW-1
Newtons first law:
In the primary iner
Cross section:
incident flux
(number of particles crossing unit transverse area per unit time)
differential cross section
(constant of proportionality)
event p = elastic scattering through a
deflection angle between ! and ! + d!:
even
er
events scatt
gle
Central Forces
based on FW-3
Motion of a particle with mass m subject
to a central force:
conservative
xF=0
)
no torque, r x F = 0
l = r x p = const.
(motion is planar, perpendicular to l)
It is convenient to work with polar coordinates:
21
Conservation o
Homework set - 7
due on Friday, November 8, 11:00am
1) Problem 4.1 from Fetter and Walecka, only parts c) and d)
2) Problem 4.3 from Fetter and Walecka, only parts d), e) and f)
3) Problem 4.2 from Fetter and Walecka, only parts a), b), c) and d)
4) Probl