Re: [Phys-L] Math List Serves

[Brian Whatcott <betwys1@sbcglobal.net>]

 This...Consider a long uniform thin rigid rod of negligible thickness and length, L precariously balanced and standing vertically upright at rest on a frictionless horizontal floor. 
...is a contradiction in terms.
    On Monday, September 19, 2022 at 12:00:21 PM CDT, David Bowman <david_bowman@georgetowncollege.edu> wrote:  
 
 Regarding Don P's request:

> Can anyone help me find a math/physics list serve which would
> discuss math/physics problems at the college level (first two
> years preferred)?  I'm a senior citizen who would like to keep
> his mind sharper by solving undergraduate level math/physics
> problems.  I'm a subscriber to this list, but I find the
> discussions somewhat challenging at times.
> Thanks,
> Don Polvani

I don't know of any other actual old fashioned math/physics listservs at the appropriate level that are still active (present company excepted).  However, you might want to try lurking in some of the math or physics corners of Quora or some other question/answer social media site where students often ask questions about their homework problems.  You could read the questions and then attempt to answer them before looking at the published answers of those answering the questions.  BTW, I think students using Quora for homework problems or for problems on take-home exams is a form of cheating, and those that explain their answers--showing all the intermediate work steps in arriving at the answer, are complicit in that cheating.

I'm not sure about the level of the problems you want, but I can provide some of the challenge questions I've put up on the hallway whiteboard in the math/physics wing of the science building of the 4-year liberal arts college where I used to teach before retiring.  As a sample here are the first few problems I've put up there so far this semester:


1) Consider a long uniform thin rigid rod of negligible thickness and length, L precariously balanced and standing vertically upright at rest on a frictionless horizontal floor.  Since this situation is unstable it soon tips over and falls down.  Because the contact between the bottom of the rod and the floor is frictionless the rod's center of mass stays centered directly over the same spot on the floor as the rod tips and the rod's bottom kicks out sideways and slides across the floor during the falling process.  The tipping/falling process has negligible air resistance (or is conducted in an effective vacuum).
A) Find an expression for the upward normal force (in units of the rod's weight) the floor exerts on the rod's bottom as a function of the rod's tilt angle, θ.
B) Does the rod's bottom lift off of the floor before the top hits the floor?  If so, at what angle, θ_0 does this happen?  If, not what is the minimum upward normal force (in units of the rod's weight) on the rod's bottom during the tipping process, and at what angle, θ_m does this minimum force occur?
C) At what speed does the rod's top hit the floor (in units of √(g*L)), and compare this speed with the speed a small object would impact the floor if it was dropped from rest from a height L above the floor (with negligible air resistance).
D) How is the available (liberated potential) energy partitioned between the rod's center of mass translational degrees of freedom and the rotational degree of freedom about the center of mass at the moment just before impact?


2) Again consider an upright rod like the above problem, except this time assume it is standing up in a corner of and against 2 adjoining vertical walls which allows it to only fall over into a quadrant defined by the walls and the frictionless horizontal floor.  The walls prevent the rod's bottom from kicking out across the floor away from the falling top.  Assume any contact between the rod and any wall is also frictionless. Again this situation is unstable, and the rod soon tips over and falls down into the allowed quadrant.  Again there no significant air resistance.
A) Does the rod's bottom ever lift up off the floor & slide up the wall corner at some point of the tipping process?  If so, at what tipping angle, θ_l does this happen?
B) If the rod's bottom does not slide up the wall corner, does it instead begin to slide out across the open floor away from the corner?  If so, at what angle, θ_s does this sliding process begin?  And how far has the bottom slid across the floor from the corner at the moment when the top hits the floor?
C) Find an expression for the upward vertical normal force (in units of the rod's weight) the floor exerts on the rod's bottom as a function of the rod's tilt angle, θ.
D) Find an expression for the horizontal normal force (in units of the rod's weight) the wall corner exerts on the rod's bottom as a function of the rod's tilt angle, θ.
E) What is the velocity *vector* of the rod's top at the moment just before its impact with the floor?
F) How is the available (liberated potential) energy partitioned between the rod's center of mass translational degrees of freedom and the rotational degree of freedom about the center of mass at the moment just before impact?


3) A very large number, N of point particles of an ideal gas are situated at random in a large volume, V.  The probability density for the location of each particle is uniform over that volume V, and the location of each particle in V is statistically independent of the locations of all of the other particles in V.  In the limit of N → ∞, V → ∞  with λ ≡ N/V remaining fixed, find:
A) the average (mean) distance, <r_0> from any such particle to its nearest neighbor in V.  Express your answer in terms of λ.
B) Because the particles are an ideal gas in equilibrium they obey N/V = λ = P/(kT) where p is the pressure of the gas, k is Boltzmann's constant, and T is the absolute temperature of the gas particles.  Suppose T = 1 atm (= 101325 Pa) and T = 295 K (= 21.85 °C = 71.33 °F).  What is the numerical value of <r_0> in nanometers?


And here is the next problem I plan to put up on the board soon (maybe even later today).

4) Consider a point in a d-dimensional Cartesian vector space.  Suppose a particle is placed in this space such that each of the Cartesian components of the particle's location in the vector space has a Gaussian-Normal distribution, and assume the values of each of the components is statistically independent of those of all the other components.  We adjust things so the space's origin is located at the mean value for the particle (so the multivariable distribution has zero mean vector).  BTW, each component's value does not have to have the same variance as for the values of the other components.  Suppose r_vec is the vector location of the particle and <f(r_vec)> is the mean value of some given function, f() of that location.  Now consider a second particle in the same space and also distributed with the same distribution as the first particle, and statistically independent of it.  Let s_vec be the relative displacement in the space from the first particle to the 2nd one, i.e. s_vec ≡ (r2_vec - r1_vec).
A) Find <f(s_vec)> in terms of the corresponding mean, <f(r_vec)>, of just one (either) of these particles.
B) Use the result of part A) to find the mean relative speed between 2 identical classical gas particles in a common thermal equilibrium terms of the mean speed of each one by itself.  BTW, in this special case we have d = 3, the space is a Cartesian space of particle *velocities* not actual positions, and the variance of each velocity component distribution is kT/m.

Have fun with these.  If you want I can supply some more, on request.  (BTW, the problem I discussed a few months earlier about the cosine of multiples of π/5 was an earlier hallway board problem from last year.)

Also I'll bet JSD & multiple others on this list have a plethora of favorite problems they could probably share with you to keep you busy for a long time.


David Bowman
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