P314
Classical Mechanics
Sp 2007 Final
Complete the exam in 3 hours/ EACH PROBLEM IS WORTH 25 POINTS 1. Consider a uniform rod mounted on a horizontal frictionless axis through its center. The axle is supported by a turntable rotation about its ve
March 28, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 8, Due: 3/13/13, 10AM
Question 1: Harmonic oscillator and Friction
Consider a system with one DOF and friction such that we modify the EOM to be
d
dt
L
q
L
f
April 2, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 9, Due: 3/27/13, 10AM
Question 1: Validity of perturbation theory
Consider a particle with mass m in the following potential
V (x) =
Kx2
+
,
2
|x|
K > 0.
(1)
W
April 11, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 10, Due: 4/3/13, 10AM
Question 1: Moments of inertia: Displaced Axis Theorem
The components of the moment of inertial tensor I of a rigid body depend on the c
Ap r i l 2 2 , 2 0 1 3
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 11, Due: 4/10/13, 10AM
Question 1: Double ladder
Consider the system in the picture below. Each rod has a uniform density and mass m.
The system has o
April 25, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 12, Due: 4/17/13, 10AM
Question 1: Hamiltonians
Find the Hamiltonian for a single particle in a cylindrical coordinates.
Answer: In cylindrical coordinates we
April 25, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 13, Due: 4/24/13, 10AM
Question 1: Schrodinger and Hamilton-Jacobi equations
In this question we would like to take the classical limit of the Schrodinger equ
May 1, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 14, Due: 5/1/13, 10AM
Question 1: Harmonic oscillator phase space
Consider a harmonic oscillator with
H=
m 2 x2
p2
+
2m
2
(1)
In class we show that the area in p
Feb. 19, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, 1st prelim
You have two hours for the exam. If you have a question, please ask! You can use your
notes and any book that you like.
Question 1: Atwood Machine
Consid
April. 4, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, 2nd prelim
You have 2 hours for the exam. If you have a question, please ask! You can use your notes
and any book that you like.
Question 1: Central potential
Cons
Feb. 18, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, review questions for 1st prelim
Question 1: (2012) Crazy Lagrangian
Consider a particle moving in three dimensions with the following Lagrangian
L=
mv 2
V (r ),
2
Question 1: A hole in a table
Consider a table with a hole such that we have mass m1 on the table connected by a
string of length to another mass, m2 below the table. m2 can move up and down and
also to the side. The system has three DOFs: The distance of
Mar. 31, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, review questions for 2nd prelim
Question 1: General relativity correction
Here we study the famous perihelion shift of Mercury.
1. For motion in a central potential
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, Optional HW # 1, Due: 1/23/13,
10AM
Question 1: Polar Coordinates
Choosing appropriate coordinates will greatly simply the analysis of many common problems.
Here we work a bit wi
January 23, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 2, Due: 1/30/13, 10AM
Question 1: DOFs
For each of the following systems determine the number of DOFs
1. A particle in a three dimensional potential, V (~r)
March 25, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 7, Due: 3/5/13, 10AM
Question 1: Springs
Consider two beads of mass m each that are bounded to move on a ring or radius R.
They are connected by two springs o
March 4, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 6, Due: 2/27/13, 10AM
Question 1: The Laplace-Runge-Lenz vector
Consider a 1/r potential and the LRL vector that are dened as
r
AvL ,
r
V = ,
r
> 0.
(1)
1. Sh
March 4, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 5, Due: 2/20/13, 10AM
Question 1: Reduced mass
Consider two particles in 3d with masses m1 and m2 . We dene
r = r1 r2 ,
R=
m1 r1 + m2 r2
m1 + m2
(1)
Perform th
PHYS 3314
Spring '11
HW 1 Solutions
HW 1 Solutions
Qn 1: Gradients and Laplacians
a) First note that for Cartesian coordinates r = xi ei and r/xi = xi /r. Then applying the chain rule and the definition of the gradient operator in Cartesian coordinates =
PHYS 3314
Spring '11
HW 2 Solutions
HW 2 Solutions
Qn1: Hanging Chain
a) The tangent vector to the curve, using x as a parameter, is by definition dr = (1, y (x) = dr = ex + y (x)e)y dx . dx The differential length of the chain dr dr/dx dx =
L a
(1)
1 + y
PHYS 3314
Spring '11
HW 3 Solutions
HW 3 Solutions
Qn 1: Vertically Driven Pendulum
Let's first draw a picture with our choice of coordinates.
y
a) Since the ceiling position oscillates as y = a sin t, then the location of the bob at any given time is, in
PHYS 3314
Spring '11
HW 4 Solutions
HW 4 Solutions
Qn 1: Spherical Pendulum
a) As usual, it's useful to draw a picture first.
In spherical coordinates, the kinetic energy is just T = m 2 /2(2 + sin2 2 ), and the potential U = -mg cos . Hence the Lagrangi
PHYS 3314
Spring '11
HW 5 Solutions
HW 5 Solutions
Qn 1: Inverse Square Potential
a) For a particle of energy T0 and impact parameter b, the energy conservation relation is T0 = 1 2 L2 1 k 1 = r2 + 2 (k + T0 b2 ) . r + 2 + 2 2 r 2r 2 r (1)
For a minimum,
PHYS 3314
Spring '11
HW 6 Solutions
HW 6 Solutions
Qn 1: Double Spring
a) In the coordinates provided, the kinetic and potential energy for this system are respectively 1 m x2 + x2 1 2 2 1 U = k x2 + (x1 - x2 )2 . 1 2 T = Hence we have Lagrangian L= and e
PHYS 3314
Spring '11
HW 7 Solutions
HW 7 Solutions
Qn 1: M&T 13-2
In lecture, we found that the general solution for massive string of length L is (the real part of) q(x, t) =
r
r eir t sin rx/L)
(1)
with normal frequencies r = (r/L) /, r Z. If the string
PHYS 3314
Spring '11
HW 8 Solutions
HW 8 Solutions
Qn 1: Artillery on the Earth's Surface
a) As shown in lecture, the acceleration of a particle in a uniformly rotating frame, with stationary origin, is aR = aF - ( r) - 2 v . (1)
In terms of spherical pol
PHYS 3314
Spring '11
HW 9 Solutions
HW 9 Solutions
Qn 1: Parallel Axis Theorem
a) As derived in lecture in frame O, Iij =
m ij xi, xi, - xi xj .
(1)
b) Under change of coordinates x = x - a, Iij =
m ij (x + a)2 - (xi, + ai )(xj, + aj )
= Iij + M ij a2 - a
Physics 3314 Spring 2011 Final
Problem #1 Triple Play 1. Write the Lagrangian for this system 2. Derive the coupled equations of motion. 3. Determine the natural frequencies (normal mode frequencies). 4. What is the shape of each mode of vibration assoc
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, Optional HW # 1, Due: 1/23/13,
10AM
Question 1: Polar Coordinates
Choosing appropriate coordinates will greatly simply the analysis of many common problems.
Here we work a bit wi
February 6, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 2, Due: 1/30/13, 10AM
Question 1: DOFs
For each of the following systems determine the number of DOFs
1. A particle in a three dimensional potential, V (r)
January 30, 2013
Cornell University, Department of Physics
PHYS 3314, Intermediate Mechanics, HW # 3, Due: 2/6/13, 10AM
Question 1: Different Kinetic Terms
We will explain in class that there are good reasons to use
T =
mx 2
2
(1)
as the kinetic term. Yet