PHY 337-- Problem Set

PHY 337 Analytical Mechanics

MWF 9-9:50 AM

OPL 103

http://www.wfu.edu/~natalie/f99phy337/

Instructor: Natalie Holzwarth Phone:758-5510 Office:300 OPL e-mail:natalie@wfu.edu

Homework Assignments

Problem Set #1 (8/25/99) -- pdf
Problem Set #2 (8/27/99) -- pdf
Problem Set #3 (8/30/99) -- pdf
Problem Set #4 (9/1/99) -- pdf
Problem Set #5 (9/6/99) -- pdf
Problem Set #6 (9/8/99)-- pdf
Problem Set #7 (9/10/99)-- pdf
Problem Set #8 (9/15/99)
Problem Set #9 (9/17/99)
Problem Set #10 (9/20/99)
Problem Set #11 (9/29/99)--pdf
Problem Set #12 (10/1/99)
Problem Set #13 (10/6/99)--pdf
Problem Set #14 (10/8/99)
Problem Set #15 (10/11/99)

Aug 24, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 1

Read Chapter 6 of Marion.

Consider the integral

Á º ó
õ 1

0
æ
ú
Ö

1 + æ
ç
è dy
dx
ö
÷
ø 2

dx.
(1)

Evaluate Á (using Maple if you prefer) for:
1. y(x) = x
2. y(x) = x²
3. y(x) = x³
Which function y(x) yields the smallest value of Á and why?

File translated from T_EX by T_TH, version 2.20.
On 24 Aug 1999, 18:13.

Aug 24, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 2

Continue reading Chapter 6 of Marion.

Use Euler's equations to find the function y(x) which minimizes the integral:

Á º ó
õ 1

0
æ
ú
Ö

1 + æ
ç
è dy
dx
ö
÷
ø 2

dx,
(1)

Such that y(x = 0) = 0 and y(x = 1) = 1. Comment on your results in view of the answers in Assignment #1.

File translated from T_EX by T_TH, version 2.20.
On 24 Aug 1999, 18:13.

Aug 29, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 3

Continue reading Chapter 6 of Marion.

Consider the Brachistochrone problem where the particle starts out from the point (x₁,y₁) = (0,0) and travels along a frictionless track under the force of gravity to the point (x₂,y₂) = (h [(p)/ 2],- h). Evaluate the travel time for
1. The extremal path described by the perimetric equations:
  
  x(q) = h
  2
  (q- sin(q)) and y(q) = - h
  2
  (1 - cos(q)).
  (1)
2. A straight line path:
  
  y(x) = - 2
  p
  x.
  (2)

File translated from T_EX by T_TH, version 2.20.
On 29 Aug 1999, 16:00.

Sep 7, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 4

This problem is due Monday 9/6/99 and will be worth 40 points.

Consider a curve such as the one shown above which passes through the points (x,y) = (0,0) and (x,y) = (D,0) .

Find the equation for the curve y(x) which satisfies the two conditions:
1. Maximizes the area:
  
  A = ó
  õ D
  
  0
  y dx
2. Constrains the length of the curve:
  
  L º ó
  õ D
  
  0
  æ
  ú
  Ö
  
  1 + æ
  ç
  è dy
  dx
  ö
  ÷
  ø 2
  
  dx = pD
  2
  .
Carry out the integrals for A and L for your curve y(x).

File translated from T_EX by T_TH, version 2.20.
On 7 Sep 1999, 09:43.

Sep 5, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 5

Start reading Chapter 7 of Marion.

Consider the Lagrangian representing a mass m moving vertically in a uniform gravitational field:

L(y,

.
y

;t) º

.
y

- mgy

such that y(0) = h and y(T) = 0, where the fixed time T is defined to be T º Ö{[2h/ g]}.

Solve the Euler-Lagrange equations to find the particle trajectory y(t) and evaluate the action integral for that trajectory.
Consider the following alternative trajectories and evaluate the action integrals for them:
1. y₁(t) = h(1 - t/T)
2. y₂(t) = h(1 - (t/T)³)
Are your results consistent with Hamilton's principle?

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On 5 Sep 1999, 16:46.

Sep 7, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 6

Consider a stationary pulley (assumed to be massless and frictionless) with masses m₁ and m₂ at heights z₁(t) and z₂(t) held by a massless rope. Write the equations of motion for the heights z₁(t) and z₂(t) using the Lagrangian formalism and the constraint z₁(t) + z₂(t) - C = 0. Here C is a constant related to the length of the rope. Show that the Lagrange multiplier is related to the tension.

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On 7 Sep 1999, 22:53.

Sep 8, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 7

Continue reading Chapter 7 of Marion.

Consider the Lagrangian for the motion of a symmetric top under the acceleration of gravity:

L(q,f,y,

.
q

.
f

.
y

.
f

.
q

.
f

.
y

where A, B, M, g, and h are parameters related to the moments of inertia, the mass, the acceleration of gravity, and the location of the center of mass. The angles q, f, and y are called the ``Euler angles" and are the generalized coordinates for this system. Find the equations of motion and identify the constants of the motion.

File translated from T_EX by T_TH, version 2.20.
On 8 Sep 1999, 17:33.

Sep 15, 1999

PHY 337 -- Problem Set PHY 337 - Problem Set # 8

Continue reading Chapter 7 in Marion.

Work Problem #7-33 in Marion.

Sep 17, 1999

PHY 337 -- Problem Set PHY 337 - Problem Set # 9

Continue reading Chapter 7 in Marion.

Work Problem #7-24 in Marion.

Sep 20, 1999

PHY 337 -- Problem Set PHY 337 - Problem Set # 10

Continue reading Chapter 7 in Marion.

Work Problem #7-30 in Marion.

Sep 24, 1999

PHY 337-- Problem Set PHY 337- Problem Set # 11

Continue reading Chapter 12 of Marion.

A particle of mass m moves in one dimension near the equilibrium point of the potential:

V(r) = A
r⁸
- B
r
,

where A and B are positive constants and r > 0.
1. Find the equilibrium displacement r₀.
2. Find the frequency of small oscillations about the equilibrium displacement.
Express your answers in terms of A, B, and m.