PHYS 321 Lecture #8, 09/13/2002
Central Forces
Conservation of Angular Momentum
r
dl r r
= r F
(= 0 for central forces; 0 for others)
dt
This is basically Keplers 2nd Law (equal areas in equal times).
PHYSICS 321 Final Examination (12 December 2002)
Time limit 3 hours. Answer all 6 questions.
1. You and an assistant are holding the (opposite) ends
of a long plank when oops! the butterfingered assis
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PHYS 321 Homework Assignment
Due: Friday, 6 September 2002
1. Having deduced that gravitational acceleration falls o as the inverse
square of distance, Newton wrote the following formula for the gra
PHYS 321 Midterm Exam (21 October 2002) Solutions
p. 1 of 3
Do all 4 problems. Time: 1 hour.
1. A spool of thread rests on a horizontal
table, as shown. The end of the thread is
pulled upward at an an
1
PHYS 321 Homework Assignment #9
Due: Friday, 22 November 2002 (4 probs)
1. Problem 714, B&O p. 280.
Solution:
The situation is as shown above. We take the bodyxed axes x^0 and y^0
to lie in the pl
PHYS 321 Solutions to Practice Final (December 2002).
1. Two masses, m and 2m are connected by a spring of constant k, leading to the po1 r r 2
tential V ( r ) = k ( r1 r2 ) .
2
a) What is the Lagrang
1
PHYS 321 Homework Assignment
Due: Friday, 13 September 2002
1. A monkey is hanging from a tree limb at height h above the ground. A
hunter 50 yards away from the base of the tree sees him, raises hi
1
PHYS 321 Homework Assignment
Due: Friday, 6 September 2002
1. Having deduced that gravitational acceleration falls o as the inverse
square of distance, Newton wrote the following formula for the gra
PHYS 321 Homework Assignment #5
Due: Friday, 11 October 2002
1
1. A uniform chain, of mass per unit length, hangs from vertical supports a horizontal distance L apart, and whose heights are h and 2h,
1
PHYS 321 Homework Assignment #6: Solutions
1. In class Einstein's addition formula for velocities was used to derive
the Lorentz transformation properties of the 3vector components of
momentum:
.q
1
PHYS 321 Homework Assignment
Due: Friday, 13 September 2002
1. A monkey is hanging from a tree limb at height h above the ground. A
hunter 50 yards away from the base of the tree sees him, raises hi
1
PHYS 321 Homework Assignment #3  Solutions
1. Halley's Comet has an elliptical orbit with semimajor axis 17.94175555
AU and eccentricity 0.967277241. Its last perihelion was February 9,
1986. Calc
PHYS 321 Practice Midterm (October 2002)
1. The exhaust velocity of a rocket is 3 Km/sec and its ratio of empty mass to mass
with fuel is 1:4. Can this rocket achieve lowEarth orbit (that is, outside
PHYS 321 Mechanics (by Adam Brown)
1
Lecture 4 (Sept 4th 2002)
Conservation of Linear Momentum
In any collision, linear momentum must be conserved. If an object of mass m moving at
velocity v collides
1
Lecture 3 (02 September2002)
Consequences of Newtons 2nd Law
Newtons 2nd Law of motion is most generally
written
r
dp r
=F
dt
where for
r nowr we define
p = mv .
If we have a system compsed of parti
PHYS 321 Lecture 17: 09 October 2002
Didos problem
A plane curve may be described by the parameterized vector
r
r ( ) = x ( ) x + y ( ) y
and its area is then given by
1 2
A = d x 2 ( ) + y2 ( )
2 0
1
PHYS 321 Lecture 7 (11 September 2002)
Solution of the driven, damped harmonic oscillator problem
There are several standard methods for solving odes, with or without constant coefficients. First we
PHYS 321 Lecture 10 (20 September 2002)
r(
) = r<r>
1

R 1 + cos(0)
x2 +
y2
= 1
2
R
b2
This is the equation for an
ellipse.
x = a + rcos
y = rsin
(more information is
posted on the Lecture
Notes pag
PHYS 321  Classical Mechanics
Lecture 13, 27 September 2002
Lagrangian Method
Using the Lagrange approach, the position of the kth particle is described by a set
of generalized coordinates that repla
1
Lecture 2 (30 August 2002)
Kinematics vs. dynamics
Kinematics is the mathematical description of motion. It
was first invented by Galileo (15641642). Ren Descartes (15961650) seems either to have
1
Lecture 5: 06 September 2002
Work and kinetic energy
Begin with Newton's Second Law
F~
v
:
= m d~
dt
Work is dened by
dW
= F~ d~r ;
dW
v
= m d~
d~r ;
dt
hence
and
d~v
dW = m
dt
d~r
dt
dt :
Becaus
PHYS 321 Lecture 15 (02 Oct 02)
Calculus of Variations:
You have an equation of general form:
b
F(cfw_q) =
(q(t), dq
dt ; t ) dt
a
When one wants to find an extreme point with such a function of infi
PHYS 321 Lecture 30 (13 November 2002)
Center of Percussion
Example: Bat hitting a ball
Let:
mass of bat = m
distance from hands to the centerofmass (COM) = a
distance from COM to where ball hits =
1
PHYS 321 Lecture 6 (9 September 2002)
Hookes Law is
dV
dx
corresponding to a potential energy
1
V( x) = kx 2 .
2
Where does this kind of force arise?
Matter is stable because the forces
between atom
PHYS 321 Lecture 22: 23 October 2002
The relativistic rocket
We can treat the rocket in free
space relativistically in two
ways:
1. Imagine we are in the instantaneous rest frame of
of the rocket and
1
PHYS 321 Lecture 14: 30 September 2002
Review: Solving the Orbit Problem Using the EulerLagrange equations
Express the Lagrangian L in terms of three generalized coordinates, r, , . The kinetic
ene
1
Lecture 1 (28 August 2002)
How a theoretical physicist thinks
Sir Isaac Newton (16431727) used the work of Galileo (15641642), Descartes (15961650) and Kepler (15711630), as well as the common ex
Lecture 13 27 September 2002
DAlemberts Principle
vr
v
v
dp v
= Fk = Fkapp + Fkconstra int
dt
Since the constraint force does no work, if we vary the variables just a little
v
v
Fkc o (rk ) = 0
N
v ap
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PHYS 321 Homework Assignment #7
Due: Friday, 7 November 2002
Make sure to read Chapter 7 of Barger & Olsson.
1. The system shown below consists of a block of mass M that can slide
without friction,
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PHYS 321 Homework Assignment #4
Due: Friday, 27 September 2002
1. In Einstein's theory of gravitation the planetary orbit equation becomes
du
d
2
2
mE 2GM
2
E
= `2 + `2 1 + mc2 u
u2 +
2GM u3 :
c2
(a
PHYS 321 Homework Assignment #5
Due: Friday, 11 October 2002
1
1. A uniform chain, of mass per unit length, hangs from vertical supports a horizontal distance L apart, and whose heights are h and 2h,
PHYS 321 Homework Assignment #7: Solutions
1
1. The system shown below consists of a block of mass M that can slide
without friction, in the xdirection along a horizontal air track. A
pendulum hangs
PHYS 321 Practice Final (December 2002). Time limit 3 hours.
You will be allowed to bring with you an 8.511 sheet of paper with anything written
on (both) sides that you desire.
1. Two masses, m and 2
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PHYS 321 Homework Assignment #4
Due: Friday, 27 September 2002
1. In Einstein's theory of gravitation the planetary orbit equation becomes
du 2
d
=
2mE
`2
+
2GM
`2
1+
2E
mc2
u
u2 +
2GM
c2
u3 :
(a)
PHYS 321 Homework Assignment #10
Due: Friday, 06 December 2002 (5 probs)
1
1. The lowest resonant frequency of an organ pipe of length `, closed
at both ends, is = vs /2`. Use this fact to explain why