3344 Updates
Apr 28
Since the current set of problems is your last, you may hand them
in any time through Wednesday without penalty; we're not in any
rush. Also, if you missed any homeworks and would like to hand them
in for 25% credit, I'll accept them for the next week, until 6pm May 6.
If you're planning to do this, please let me know so I'll know to
expect them.
Also, for the wedge problem, depending on how you solve it, it
might be helpful to make use of the conservation of energy. Note
that in this problem, H = E = const by the usual conditions.
Apr 27
For 7-34, in order to solve part b, you'll need to use coordinates
with too much freedom, and then use a constraint to force the mass
to sit on the wedge. Furthermore, if you're interested in the normal
force, you want a constraint that fixes the coordinate normal to
the surface. A good choice to consider would be one which includes
the radial distance and angle with respect to the (moving) center of
the circle. That is, polar coordinates about the center, but with r
at first allowed to be anything. The constraint is then very simple
to impose, and directly involves the direction of the normal force.
For the last problem, the particle moves in three dimensions. The
potential can be any function of r, but won't depend on theta or phi.
Apr 19
I've posted the solution to Marion 7.11. It might be
useful to check your solution before writing up your discussion.
Note that it's the pivot point that's relevant for the pseudo force,
not the center of the circle from which you measure the mass's angle.
You'll be interested in the component of the pseudo force that acts
as a restoring force for the mass tangent to the circle.
For the 7.15, you will also oscillations in length about equilibrium,
but note that the equilibrium length won't be b because of gravity.
At rest, it will be stretched until the forces balance, but you should
see this directly in the E-L eqns.
As you think about your result from the 7.27 and added parts, it might
be useful to know that the total angular momentum about the origin
for a system is equal to the angular momentum of the center of mass
about the origin plus the angular momentum of the system about the
position of its center of mass. Look at section 9.4 if you'd like
to learn more.
Apr 11
I've decided to postpone the last part of 7.11 (the discussion,
as described below) for next week, since there's enough to do
already.
Apr 9
For problem 7.4, you're given a radial force F (one which only
depends on r), but you need the potential U in order to determine
the Lagrangian, so you'll first need to derive U from F. Recall
that F = -grad U. It might help to find the form of the gradient
in polar coordinates in the appendices, and apply it when F and
U only depend on r. If you're still unsure, look at sect 5.2.
To determine whether E and angular momentum L are conserved, you
can either follow what we did in lecture for the particle-on-a-cone,
or compute dE/dt and dL/dt directly and see what the Euler-Lag eqns
tell you. Note that for motion in a plane, there is only one component
to L, perpendicular to the plane.
For problem 7.11, use the angle whose vertex is at the center of
the circle, rather than at the pivot point, to specify the location
of the mass. Otherwise, the motion couldn't be similar to that of
a pendulum. (Why?)
To answer why the result is reasonable, I'd like you to consider
the system from the rest frame of the circle, as we did for the
pendulum on the train. It's not trivial to set up the picture
from this frame, but it's a great exercise in pseudo forces (as
well as a modest exercise in plane geometry).
You can determine the pseudo force by asking what actual force is
necessary to keep a particle at rest relative to this rotating frame.
Then in the absence of such a force, the particle would appear to
accelerate in the opposite direction with the same magnitude, as if
it were acted on by a force in the opposite direction. (Recall
centripetal vs centrifugal forces for rotating systems.)
Finally, you'll need to determine what the component of that pseudo
force is tangent to the circle, acting as a restoring force
for the mass. It should look just like the restoring force for a
pendulum at rest.
For problem 7.12, you'll actually get to solve the Euler-Lagr.
equation, not just write it down. But lucky for you, you've
already solved an almost identical differential equation in HW 7:
appx C prob 2(c). (Don't forget both the particular and homogenous
solutions.) We waste nothing in this course.
Apr 4
For the maximum enclosed area problem, part d, if
you choose to start with what you suspect is a solution (ie
a parametrized circle), I recommend looking at what I did
in lecture for the helix, and redefine x(t) and y(t)
in terms of an angle theta(t). On the other hand, it turns out
that the equations really aren't that difficult to solve
in general; maybe even easier than checking a specific solution.
The first step is to recognize that both sides are a derivative,
and so can be trivially integrated. The next step is to relate
x to y (which requires one more trivial integration), which gives
the equation for a circle. (Some students in the course the
last time I taught it figured out this solution.)
Incidently, I recommend the regular form of the Euler equations
(including constraints). The second form we derived in class
applied to the case where the integrand depended on only one
function. You can rederive it for the case with more than one
(I put this in my notes but didn't discuss it in lecture), but
it has a different form and in this problem doesn't help; it
just gives 0 = 0.
Mar 31
I've updated the lecture notes on Ch 6, so they should now
better match the last lecture or two, especially with respect
to Lagrange multipliers.
Mar 28
For problem 6-4 (the circular cylinder), I recommend using
a coordinate system that is natural for this surface; see
appx F (on different coordinate systems).
For problem 6.5, the equation 6.39 referred to is the
second form of the Euler equation (in case you're using
a different edition). You should be able to work the
equation into x = integral of some fn of y. When you get
to that point, think trig or hyperbolic substitution.
I've pretty much decided not to give any exams with this course,
since I think I'm getting a good idea from the homework how
everyone is doing. Let me know if you have any objections to
this.
Incidently, I tend not to worry about grades until the course
is over, but I vaguely recall being an undergrad, when grades
had something to do with getting into grad school or getting
a job. So I thought it might be helpful at this point to let
you know roughly how I'll compute grades for this course. Keep
in mind that this is just an estimate.
.85 - 1.0 A
.70 - .85 B
.55 - .70 C
.40 - .55 D
Here the number is the fraction of points given on hw out
of the total possible (not including points for reading),
with +'s or -'s added to grades depending on whether you're
near the top or bottom of each range, as you'd expect.
Mar 24
For the last problem, you'll need to refer to the Fourier Series
problem from last week's set; the solution is posted.
Mar 20
For the Green's function problem, the integral is fairly simple
if you convert the sin fns to exponentials; then it's just some
exponential. It's even easier if you give it to Mathematica:
Integrate[Exp[a tprime] Sin[w tprime] Sin[v (t-tprime)], {tprime, 0, t}]
(which integrates tprime from 0 to t) gives
(2 a v w Cos[v t] + w (a^2 - v^2 + w^2 ) Sin[v t] +
Exp[a t] v (-2 a w Cos[w t] + (a^2 + v^2 - w^2 ) Sin[w t]))
-------------------------------------------------------------------------
(a^2 + v^2 )^2 + 2 (a - v) (a + v) w^2 + w^4
(Someone check this for me.)
Mar 18
Please be sure to do problem 3.32 from the 4th edition. Apparently
the numbering for problems in that section is different in the 5th.
Problem 3-18: We probably won't discuss the parameter Q in
lecture, but from your reading you've found that one definition
is omega_R/2 beta. But more useful observations about Q are
that it gives both a measure of how narrow a resonance is in
omega, and also of the inverse of the fraction of energy lost
per oscillation for an underdamped system, which is what you'll
be showing in this problem. This makes a high Q desirable in
circuit design. Technically, giving the total E that the system
has during one particular cycle is not well-defined, because
the system is losing energy during that cycle. On the other
hand, for a lightly damped system, it doesn't lose a lot, so
it doesn't matter much whether you use E at the beginning,
middle or end of the cycle; that is, to compute the E in
the numerator of this expression, you can treat the system
as approximately conservative. (This is obviously not true
for computing the denominator, since you'll be explicitly
calculating the energy lost during the cycle.)
Mar 17
I'll be out of town today, but in email contact, if you have
questions about the current homework. I hope you enjoyed your
break.
Mar 10
Let me remind you again to keep units in your problems;
it's good form, and it helps you catch errors. No number
should appear in your homework without its associated unit
unless it truly is dimensionless, either in the answer or
in intermediate steps. (You shouldn't be inserting values
for quantities until the very last step, in any case.)
Has anyone seen the copy of Marion I left in the Physics UGrad
Lounge? I notice that it's been missing for the last week or
more. Thanks very much.
Mar 7
For 3.32, many of the integrals you'll need
to compute the Fourier coefficients are identically zero.
(Why?) Using this can save you some work. Also, you'll
need to generate some plots for this problem. You've got
lots of choices: Mathematica, Matlab, Maple, spreadsheets,
many hand calculators, and so on. One program that I use a
lot is gnuplot, which is available free on the web for
linux or windows. If you have some problems finding a way
to do plots, let me know.
Two further comments:
1. The expression for the coefficients a_n and b_n in class
used t from 0 to T, while the function F(t) in this problem is
given from -T/2 to T/2 (ie from -pi/omega to pi/omega).
Because all the functions discussed (including the extended
version of F(t)) are periodic, you can either
(a) rewrite the integral for the coeffs as an integral
from -pi/omega to pi/omega, as discussed in the text, or
(b) do the integral from 0 to T, and use the fact that
F is periodic to tell you what the value is from T/2 to T.
2. F changes abruptly at one point, so to do the integrals
discussed above (in whichever form you prefer), you should
break the integration into two regions and treat them separately.
Mar 1
For the pendulum problem, when deriving the force from
the potential, it might help to have a version of the gradient
in polar coordinates. You can find this in the appendices.
(You can extract it from the gradient either in spherical
or cylindrical coordinates restricted to 2d, the plane
of the pendulum.) Of course, you could also figure out
the component of F from a free-body diagram; you
should find it gives the same answer.
You shouldn't have to solve the diff eqn (Newton's law)
for the pendulum to figure out the angular frequency omega
or the function theta(t) (though you may if you like). The
eqn should look just like the mass on a spring (as promised),
but with a different constant. So you should be able to give
omega and theta(t) just by looking at the eqn. (Incidently,
you also know what solutions look like if you account for
resistance for a pendulum moving in a fluid, such as air.)
For the second derivative problem, you may find that
you end up with a version that depends on t, t + delta t,
and t + 2 delta t. That's fine, since it does reduce
to the second derivative as delta t goes to zero. But
it's actually a better approximation for the second deriv
at t + delta t than it is at t. You could adjust this
definition in an obvious way to make it better for t.
Feb 26
For 2.26, to keep things simple, take 2 meters as the
entire distance over which friction acts; that is, ignore
frictional losses while the spring is compressing.
For the conservation law checklist, we won't be trying to
keep track of energy in the form of heat or chemical structure,
for example; we won't try to construct a potential energy
associated with these as part of mechanics.
Feb 21
For the iterative problem, there are many algorithms you
could invent, some of which are faster than others. I'd
recommend trying something similar to the one I used for
square root in lecture.
For the curl problem (showing that you can find a potential),
I'd like you to use cartesian coordinates. Also, remember
that the big advantage of this test is that you don't need
to know what U is to find out whether it exists; you can
apply the test directly to F.
Feb 19
For the last problem, I wanted to emphasize that you'll need to
keep terms on both sides to order k^2 in order to get enough
information to figure out v0 and v1. That means that inside the logs,
you'll need to keep to k^3, because one order gets cancelled.
It doesn't seem fair, but that's the way this one works out.
When you're doing the series expansion, it's probably simplest
to break the expression into simple pieces, and expand them
separately. For example, you could expand the logs separately,
then expand vf inside of that expression, always discarding
terms that go beyond what you need. It's easier than trying
to apply the Taylor series expansion on the entire expression
at once, though that will also certainly work.
Feb 18
For 2.21, we already showed this should work in general. For this
problem, the idea is to test it out for a simple case where you
know the explicit solution. That is, take the solution for a projectile,
and compare the torque (from the force acting on it) to the angular
momentum for this particular trajectory.
For the numerical problem, I have in mind the method I discussed in
the lecture on Feb 5, page v.1.1 in my notes. (It was labelled method (e).)
Feb 16
For the last set of solutions, I've added a couple pages to the
solution from the manual for 2.12. I thought the way they handled the
second stage of the trajectory was unnecessarily confusing, so I wrote
up an alternative. Either is an ok way to solve it.
I've put a copy of the two chapters on vector calculus
(gradient, curl, divergence, etc) from the Feynman Lectures vol II
in with the course lectures. It's a very nice discussion on the
subject, and an excellent introduction if this is all new.
Feb 14
For the Taylor series discussion in the appx, h is
just x - x0, or delta x; that is, for a series in which
you're approximating a function near a point x0 other than
zero, x - x0 is how far away you are from that point.
The series approximation will be a polynomial in x - x0,
rather than in just x.
Just to make sure the numerical problem (problem 4) is clear:
even though you may be able to solve for v(t) exactly, that's not
what I'm asking for. Just as we discussed in lecture, I want
you to convert the differential equation into an approximate
equation that lets you evaluate v numerically at a discrete
number of times (10) between 0 an 10s. Ask me if this
isn't clear.
In the last problem (problem 5), v really is the component
of velocity, not speed, and so can be positive or negative.
Both choices correspond to possible solutions, but only one
is relevant. (The equation comes from asking for the
velocity when x=0. The obvious trivial solution is the velocity
when it just lifts off; ie, the initial velocity. The interesting
one is when it's back to x=0 after taking a trip.)
I should note that the log in the last problem is
the natural log (not log base 10). It's so rare
in physics to need log base 10 that both the symbols
ln and log mean natural log. (log base 10 is always
denoted with a "10" subscript.)
Feb 7
Please note that there was a typo in the last problem; a c was
missing from the final expression. It's corrected now.
For 2.2, by equation of motion, Marion means Newton's laws.
Since the object is confined to a sphere (by whatever force
is necessary to do that), we're interested in the behavior
in the angles theta and phi that determine where the particle
is on the sphere. That is, work out Newton's laws for the
theta and phi components.
Also, recall that you've already worked out the form of
acceleration a in spherical coordinates, so you've
got almost all of this problem done. (You might want to
check the solutions to make sure your form for a
is correct.)
For 2.3, you can assume that there's no resistive force.
For 2.12, remember that the sign of the resistive force
term changes once the particle changes direction; that is,
both forces act down on the way up, but on the way down gravity
still pulls down but the resistive term is up. (When you're
using only one power of v, the sign takes care of itself.) So
you'll need to solve a different equation for the two different
parts of this problem.
One thing you can do that will simplify this problem a lot is
to rewrite the differential equation in terms of v vs x,
rather than v vs t. (We did this in lecture for a somewhat
different case.) That is, use the chain rule to replace
dv/dt = dv/dx dx/dt = v dv/dx
and solve for v(x) (or x(v)). You can get away with this
because the info you're asked for only relates v and x;
you don't need to know what time. On the other hand, if
you have trouble with this and would like to solve first for
v(t) and x(t), that's ok; just give the answer in whatever form
you get. It might be too difficult to get it into the simple
form in the text.
For the last problem, if you decide to look up the integral you'll
need in Appx E, you may notice that you have a choice to make.
You should be able to decide on physical grounds.
Feb 4
For Marion 1.31, r is just the magnitude of a position vector,
with r = Sqrt(x12 + x22 + x32). As mentioned in the assignment,
I'd like you to handle this using Cartesian coordinates (instead
of converting the gradient into spherical coordinates, which is
another approach), so that means converting all the rs into
x1, x2 and x3.
Jan 31
Note that I gave a general expression for a in spherical
coordinates in lecture. You might want to check your answer to
problem 25 against it before going on to the last problem.
Text typo: To prevent some (probably minor) confusion in
your reading, I wanted to mention that the right side of Eq 2.45
in the text (4th ed) should have a T, not a t.
Although the solutions manual that I'm using for posted
solutions often gives answers in degrees, as much as possible
get into the habit of using radians. It's a much more rational
system, and allows the use of expressions such as s = R theta
and v = R omega for circular motion, which aren't correct if you
use degrees.
Jan 29
For the last problem, a proper rotation is one without
an axis inversion such as parity. Think about what det[lambda]
would be for a proper rotation of 1 (that is, all angles
are zero), then for angles infinitesimally close to zero,
then etc.
Jan 28
For prob 22, involving the identity for epsilon tensors,
sorry but yes, it's a bit tedious. You can make it a bit
less so by not explicitly enumerating (ie over 1, 2, 3)
each case, but giving results in sets, and showing the
results are the same using the right hand side. For example,
when i=l, the sum gives 1 when j=m and i != j or m, 0 otherwise.
This way you won't need to enumerate all 81 cases (though
you may if you really want to).
Jan 23
For prob 8 from the text: there are several ways to show
what they're asking. I can think of a pictorial proof by
recalling the geometrical picture of the dot product.
I can interpret A . r = A r cos(theta) to be
A x [component of r parallel to A] (or as r x [component
of A parallel to r]). Think of the equation as a condition
on the allowed rs.
For prob 11, you can look up the formula for the volume of a
parallelepiped if you like. (It's just a skewed brick.) If you
can't find it, let me know and I'll post it.
For the last problem, remember that the transpose of lambda
is also the inverse.
Jan 18
I've reposted both the back of the text solutions and the Ch 1 problems
from Marion 4th ed in higher resolution than was up yesterday, and in
proper orientation. I've found that if they're hard to read on the
screen, printing them out helps a lot; the printed versions are quite
readable. But let me know if you have problems reading them, or if any
other format would be more useful.
Jan 13
I'll post hints for current problems, answers to questions, and
other useful information here.
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