PHY 3003 SPRING 2015
Solutions 2
(1) A particle of mass m moves subject to a one dimensional potential
x2 x2
0
U (x) = U0
x2 + x2
0
2
(here U0 is a constant with dimension of energy and x0 is a constant with dimension
of length).
(a) Sketch U (x), identif
0.4
0.2
0.0
-0.2
-0.4
0
1
2
3
4
5
Figure 1. Potential for U0 = U1 = 1
PHY W3003: Midterm Examination I Feb 18, 2015
Please answer all questions, and show your work as partial credit will be given.
You may bring to the exam one sheet (both sides) of paper.
SOLUTIONS: PROBLEM SET 6
(1) Consider a particle of mass and angular momentum L moving in the attractive
1/r potential U (r) = k/r with k > 0. Bound orbits have energies E in the range
k 2 /2L2 E 0. For energies in this range compute the orbital period as
PHY W3003; SOLUTIONS, ASSIGNMENTS 5
(1) 8.16
Rewrite the dening equation as
c = r + r cos =
x2 + y 2 + x
Subtracting x from both sides and squaring gives
c2 2c
x+
2
x2 = x2 + y 2
Rearranging gives
c2 = (1
2
)x2 + 2c
x + y 2 = (1
2
) x2 +
2c
(1
or
c2 2
PHY 3003 SPRING 2015
Solutions 3
(1) 3.12
(a) For a single stage rocket
v = v0 + vex ln
m0
m
Here v0 = 0 and m0 /m = 1/.4 = 2.5 so
v
= ln2.5 0.92
vex
(b) In the rst stage we have
1
0.35vex
0.7
In the second stage, the initial mass minit = 0.6m0 and the v
PHY 3003 SPRING 2015
Solutions 1
(1) 1.46
Denote the radius of the turntable as R In the frame S the trajectory of the puck
is
r(t) = R vt; (t) = 0 t < R/v
R
r(t) = v t
; (t) = t > R/v
v
In the frame S the trajectory of the puck is
r(t) = R vt; (t) = t t
3003 PROBLEM SET 4: SOLUTIONS
Consider a particle of mass moving in the half-line r > 0 and subject to the potential
U (r) =
L2
U0
2
2
2r
(r + a2 )
(1) Please nd, in terms of a, L and the values of U0 for which stable circular motion
is possible and the
PHYSICS W3003: MATHEMATICAL BASICS
1. Differential equation basics
1.1. Example. Newtons equation for the motion of a particle,
d2 R(t)
= F(R, t)
dt2
is an example of a dierential equation, an equation relating one or more derivatives of
a desired functio
PHYSICS W3003: PHASE SPACE
1. Introduction
Consider a particle moving in one dimension. In basic physics courses we represent the
motion as the trajectory X(t) and dene the velocity as the derivative V (t) = dX/dt. In
this view the trajectory X(t) is fund
Life at Low Reynolds Number E.M. Purcell Lyman Laboratory, Harvard University, Cambridge, Mass 02138 June 1976 American Journal of Physics vol 45, pages 3-11, 1977. Editor's note: This is a reprint of a (slightly edited) paper of the same title that appea
PHYSICS W3003: MECHANICS
SYLLABUS AND SCHEDULE SPRING 2014
Class meeting time and place: Monday and Wednesday, 11:40am-12:55pm 329 Pupin
Course web page: http:/phys.columbia.edu/millis/3003/3003.html
Professor: A. J. Millis
Oce: 618 Pupin (this may change