PHY 3003 SPRING 2013
Solutions to Problem Set 4
1
(a) The energy is
m2
x + x2 + x2 + mgz
2
In polar coordinates referenced to the suspension point, with = meaning the mass is
directly below the suspension point, and noting that the cord length L is time-i

PHYSICS W3003: PHASE-PLANE PLOTS
1. Introduction and simple examples
1.1. The basic idea. Phase-plane plots are a very useful way of understanding qualitative
features of the solution of ordinary dierential equations, especially equations for a single
var

PHYSICS 3003: CONSERVATION OF ANGULAR MOMENTUM AND
MOTION OF RIGID BODIES
AJM
Because the discussion today in class was a little rushed I amplify my remarks here.
1. Generalities
Rigid Bodies: In a rigid body the position of every particle is xed if we sp

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 (in
this case the se

SOLUTIONS, MIDTERM 1, SPRING 2014
Problem 1
a) U (x) = 2k x. The plot of U is shown in the left panel
(b) all orbits are bound, there are no separatrices, unbound orbits or xed points.
2
3.0
2.5
1
2.0
0
1.5
1.0
1
0.5
2
2
4
6
8
0.0
10
0.5
1.0
1.5
2.0
Probl

SOLUTIONS PROBLEM SET 6
W3003 SPRING 2014
(1) This oscillator is overdamped (natural frequency 0 = 1 and damping
general homogeneous solution xh is thus
xh (t) = A+ et/+ + At/
e
with
1
1
1
2
=
0 = 2 3
2
while a particular solution (as you can verify by s

PHY W3003 SPRING 2013
Midterm II solutions
(1a) Kinetic energy
(1)
KE =
m2
x + z2
2
but if z = kx2 then
(2)
z = 2kxx
so
(3)
KE =
m2
x + 4k 2 x2 x2
2
Potential energy = mgz = mgkx2 so
m2
(4)
L=
x + 4k 2 x2 x2 mgkx2
2
(1b)
(5)
m
d
x + 4k 2 x2 x + 2mgkx 4mk

Addition mistake!
I=5Ma^2/4
Thus w=(Sqrt[3]/10)(v/a)
Solution: problem 5, set 2
(a)
(i) Dene the y axis to the axis coming out of the plane of the gure so the component
of angular momentum in question is Ly . This is not conserved because the force is alo

SOLUTIONS PROBLEM SET 7
W3003 SPRING 2014
(1) (a) Using g = 10m/sec2 and h = 51 03 m we have v 2.3m/s. The distance across
the Pacic from say Tokyo to San Franciso 8000km so the time is about 36, 000
sec or about 10 hours.
(b) The numerator is the arc len