33-331 Physical Mechanics I. Fall Semester, 2009
Assignment No. 1 (Revised)
Due Friday, Sept. 4
READING
Thornton and Marion:
Ch. 1. Most of this you should have seen before, and you can review it when you need it.
Glance through it to see what is there.
C
Solution 1
1) Problem 2.1 of the text.
a)F (x, t) = f (x)g(t) = m = m dx
x
dt
No matter how we manipulate this, we are going to have the following variables in the equation: x, t, and vx = x.
Since we dont know the dependence of any of those on the others
Solution 2
1) Problem 2.2 of the text.
(Solution in back of the book is not completely correct.)
F = ma where F (, ) is a function of and but can (and must) have a component in the er direction.
Using the form of the acceleration in spherical coordinates,
Solution 3
1) Problem 2.43 of the text.
F = kx + kx3 /2 = dU
dx
x4
U = 1 kx2 k 2
2
4
This is maximized or minimized when
0 = dU = F = kx + kx3 /2
dx
x2
Either x = 0 or k = k 2 x =
U is clearly parabolic (upwards) for small x, so x = 0 is a local minimum
Solution 4
1) Problem 5.7 of the text.
Express your answer in terms of the ratio R/l (where possible).
If the mass per unit length, , is constant, what happens when L ? Explain.
Hint: The integral you need is in appendix E.
Let = M/L (and using L in place
Assignment 5
33.331
Due: Wednesday, Oct. 7
1) Problem 5.9 of the text.
Dont try to do the integral exactly (you cant) and dont try to beat it into the form of a standard elliptic integral.
But do expand the integrand in a/R (where R is the distance from t
Solution 6
1) Problem 7.4 of the text.
Of course energy is conserved. its a conservative force. To explicitly show conservation of energy,
they want you to evaluate d (T + U ). you may rst want to eliminate from T by making use of a
dt
conserved quantity.
Solution 8
1) Problem 7.22 of the text. (F(x,t) is the x-component of the force.)
Is H conserved? Is H = E? Is E conserved (for the system)?
k
i
F = Fx (x, t) = x2 et/ = U
i
U
k t/
So x = x2 e
k
1
U = x et/
(choosing U = 0 at x = ), while T = 2 mx2 so
1
k
Solution 10
1) For the two-body central force problem, we dened L as the angular momentum of the analogous body of mass
orbiting a xed center of force: L = r r. Explicitly nd the actual total angular momentum (about the center
of mass point) of the two b
Solution text.
9
1) Problem 7.33 of the
Be sure you understand how to nd the kinetic and potential energy. Its not trivial to write H in terms of the
momenta, solve for the generalized velocities (in terms of the momenta) explicitly and then substitute in
Solution 7
1) Problem 7.15 of the text.
This is a pendulum, swinging in the x-y plane, where y is vertical. But now the usual pendulum rod
or string has been replaced by a spring, with spring constant k, which can change length but remains
straight.
Let r
Solution 11
1) Problem 8.2 of the text.
Do a DEFINITE integral on both sides from the starting point (pericenter) to the present point (r and ) and
remember that you know the value of rmin .
Youll need an integral from Appendix E.
There is a typo in the b
33-331 Physical Mechanics I. Fall Semester, 2009
Assignment No. 2
Due Friday, Sept. 11
READING
Thornton and Marion:
Ch. 3, Secs. 3.1 through 3.6
Ch. 5, Secs. 5.1, 5.2
READING AHEAD:
Thornton and Marion
Ch. 5, Secs. 5.3 through 5.5
Tidal Forces (handout)
A
33-331 Physical Mechanics I. Fall Semester, 2009
Assignment No. 7
Due Friday, October 23
READING
Handout: Supplement on Lagrangian and Hamiltonian Mechanics
Thornton and Marion Secs. 7.5 to 7.12: Lagrangian with multipliers, conservation laws, Hamiltonian
33-331 Physical Mechanics I. Fall Semester, 2009
Assignment No. 8
Due Friday, October 30
READING
Thornton and Marion Ch. 8, Secs. 8.1 through 8.6
READING AHEAD:
Thornton and Marion Ch. 8, Secs. 8.7, 8.8
Handout: Hyperbolic orbits
EXERCISES
1. Turn in at m
Assignment 3
33.331
Due: Friday, Sept. 18
1) Problem 5.2 of the text.
2) Problem 5.3 of the text.
Two spacecraft have the same mass. One has just been launched at escape velocity, the other has just been put into
(very) low altitude orbit. Find the ratio
Assignment 4
33.331
Due: Friday, Oct. 2
1) Problem 5.9 of the text.
Dont try to do the integral exactly (you cant) and dont try to beat it into the form of a standard elliptic integral.
But do expand the integrand in a/R (where R is the distance from the
33-331 Physical Mechanics I. Fall Semester, 2008
Hour Examination No. 1, 17 September 2008
NAME
INSTRUCTIONS. This examination consists of one problem with ve parts, with each part
worth the same number of points. DO NOT attempt (e) unless you are condent
33-331 Physical Mechanics I. Fall Semester, 2008
Solutions to Third Hour Exam
a) The Hamiltonian is the sum of the kinetic plus potential energy, thus
H = p2 /2m + mgy.
Hamiltons equations are
y = H/p = p/m;
p = H/y = mg.
Dierentiating the rst and using t
33-331 Physical Mechanics I. Fall Semester, 2008
Hour Examination No. 3, 17 November 2008.
NAME
INSTRUCTIONS. This examination consists of two problems. The rst, with
3 parts, is worth 60 points, and the second, with 2 parts, is worth 40 points. It is
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a) The gravitational potential outside a spherically symmetrical mass distribution is ex
actly the same as if all t