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24-Ruby I

## Works of my students

42 REPLIES 42
4-Participant
(To:ValeryOchkov)
I like it.
🙂

Philip Oakley
Very nice problem and solution. Since all forces are conservative, you might be able to plot the gravitational potential energy, the instantaneous pendulum energy and the harmonic motion energy. It would seem to me that this problem will demonstrate the equipotential theorem wherein all coupled degrees of freedom come to the same average value over time.

Best regards for your entire body of work on this forum, and, as an educator, for this excellent numerical problem.
---------------------
Physics: Common Sense made Obscure by Mathematics Don Sparlin
24-Ruby I
(To:ptc-1295782)
24-Ruby I
(To:ValeryOchkov)
4-Participant
(To:ValeryOchkov)
On 12/21/2007 9:02:28 AM, VFO wrote:
>3-d
>work:
http://twt.mpei.ac.ru/MA
>S/Worksheets/3-circles.mcd
Val
>
http://twt.mpei.ac.ru/ochkov/
>v_ochkov.htm

I managed to get the projected line to be off page (i.e. not shown).

Interesting though.

Philip Oakley
19-Tanzanite
(To:PhilipOakley)
On 12/21/2007 3:02:10 PM, philipoakley wrote:

>I managed to get the projected line to
>be off page (i.e. not shown).

Yes. For example, change the radii to be 12, 10, 15. The green lines are then not correctly drawn either.

Interesting that the blue (as opposed to blut 😉 ) line must be straight. I assume you have a geometrical proof of that?

Richard
19-Tanzanite
(To:RichardJ)
Or try 4, 16, 7. One of the green lines is not even tangential to the circle.

Richard
19-Tanzanite
(To:RichardJ)
Lastly, try 2, 16, 7.

Richard
24-Ruby I
(To:RichardJ)
Thank to ALL!
It was (is?) attempt to demonstrate with Mathcad one old theorem about 3 circles!
Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
It's actually all in a plane, hence 2D, not 3D.

It would be interesting to see the actual sheet, to see how much code has to be written, and how comprehensible that code is. Here is a solution using my old plotting routines. These have been copied into the sheet with no modifications.

The random centers and radii of the circles can easily be replaced with some form of user input. This code should fail only in the case that the common tangents do not exist (i.e., one circle is nested inside another).

It is interesting to note that the colinearity continues to hold even if one of the circles is oriented opposite to the other two circles, resulting on two pairs of internal tangents and one pair of external tangents. Changing the radii from rnd(2) to rnd(4)-2 results in circles with random orientations as well as random positions and radii. But this also results in more failing configurations, as circles which have opposite orientation and intersect will not have common tangents.

� � � � Tom Gutman
1-Newbie
(To:ptc-1295782)
>>Since all forces are conservative,<<

Isn't the frictional force dissipative? Other than that it could be nicely set up using the Lagrangian. You have two potential energies, gravity and tension, and one kinetic energy. Set that up in polar coordinates, and Mathcad will take it from there (I have a couple of examples in the collaboratory). But I don't know if it is possible to modify the Lagrangian method to include frictional forces.

� � � � Tom Gutman
24-Ruby I
(To:TomGutman)
On 12/21/2007 3:15:04 AM, Tom_Gutman wrote:
>>>Since all forces are conservative,<<

Isn't the frictional force dissipative? Other than that it could be nicely set up using the Lagrangian. You have two potential energies, gravity and tension, and one kinetic energy. Set that up in polar coordinates, and Mathcad will take it from there (I have a couple of examples in the collaboratory). But I don't know if it is possible to modify the Lagrangian method to include frictional forces.

� � � � Tom Gutman
http://twt.mpei.ac.ru/MAS/Worksheets/3-circles.mcd

Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
24-Ruby I
(To:TomGutman)
I don't see a Lagrangian in there, in either sheet.

� � � � Tom Gutman
24-Ruby I
(To:TomGutman)
On 1/14/2008 4:53:10 PM, Tom_Gutman wrote:
>I don't see a Lagrangian in
>there, in either sheet.
>
>� � � � Tom Gutman
http://en.wikipedia.org/wiki/Lagrangian
For new students works!

Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
24-Ruby I
(To:TomGutman)
On 1/14/2008 4:53:10 PM, Tom_Gutman wrote:
>I don't see a Lagrangian in
>there, in either sheet.
>
>� � � � Tom Gutman

See with Lagrangian http://twt.mpei.ac.ru/MAS/Worksheets/Pendulum_Spring_Pole.mcd

Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
>>See with Lagrangian <<

I see a graph with a line that purports to be the Lagrangian. But I don't see where the Lagrangian is being used to generate the differential equation. Also, I don't think the potential energy is correct. The equation given is clearly wrong (whatever coordinate system is being used -- what is x?) and the curve is much too regular, compared to the kinetic energy curve. A useful check is to plot the Hamiltonian, which should be smooth and monotonically decreasing.

� � � � Tom Gutman
24-Ruby I
(To:TomGutman)
>whatever coordinate system is being used -- what is x?
We have x as vertical axis in Russia! And the zero point for Potential Energy is at the Russian President born place! Now in S-Peterburg! And for future 4 years in S-Peterburg tooo! It is goot - stability!

Joke! Thanks for the remark - see the correct plots:
http://twt.mpei.ac.ru/MAS/Worksheets/Pendulum_Spring_Pole.mcd
About Lagrangian and etc in classic mechanic.
Now we have powerful tools for ODE solution and can not use "good old methods". We must study its but after not before "??? new methods"!
Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
>>Now we have powerful tools for ODE solution and can not use "good old methods". We must study its but after not before "??? new methods"!<<

I heartily disagree. The pwerful tools for ODE solutions make the Lagrangian more, not less, useful. It allows setting up the problem in terms of simple first principles, with minimal chance for errors, and letting the computer work out the details of calculating the derivatives that are needed for the ODE solver. The Lagrangian does not replace the ODE solver, it makes the ODE solver usable.

Have you looked at my sheets using the Lagrangian? That would be the segmented pendulum and the gyroscope. I think the gyroscope one is particularly instructive. It is set up without any use of advanced concepts like angular momentum or moment of inertia. The physics used is just the very basic physics of linear motion, kinetic energy as ½mv² and potential energy of mgh. No need to figure out the moment of inertia of particular shapes, or to figure out how multiple rotations interact to produce net rotations, net angular kinetic energy, or forces or torques. There is not even a torque in the setup.

>>see the correct plots:<<

Well, the potential energy is a bit better, but still not correct. I believe that a plot of the Hamiltonian would clearly show that it is in error. Also, has there been any validation of the purported solution? Given that this is the second attempt at a proper potential energy calculation that is wrong, how can we have any confidence that the differential equations used (but not shown or described) are, in fact, correct?

� � � � Tom Gutman
24-Ruby I
(To:TomGutman)
One BIG plus of the MAS - I can correct student's work without sending its tou you with the asc delete uncorrect variants!
See again
http://twt.mpei.ac.ru/MAS/Worksheets/Pendulum_Spring_Pole.mcd
Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
I'd rather get the actual sheet. For one thing, it takes less time to download and run the sheet than to view the page on the MAS. That goes double when the failure of the pictures to show (happens about half the time) requires downloading the page again. And, in general, I am much more interested in how the result obtained than in the result itself. I don't get that on the MAS. My opinion on the MAS remains unchanged -- it's a nice toy for those that can afford \$15,000 toys, and also the platform to run them on.

Now, what would really be useful is the ability to compile the worksheet into a Java applet, and then have that run on the user's machine. With some improvements to the animation support so that animation can be done directly in the sheet, hence be included in the Java applet, it could be really neat and useful. It would still not be a replacement for the actual worksheet, as one could not study the mechanism, nor play with changing parts of it. But it could do a good job of delivering computational results.

In the meantime the potential energy is still wrong. I don't see the plot of the Hamiltonian, which I expect would make that obvious, but it's still obviously wrong on other grounds. However it's nice to see that you are not working in some obscure unit system where g is a dimensionless unity.

Try calculating and plotting the Hamiltonian. Is it fairly smooth and monotonically decreasing? Set the friction to zero. Is the Hamiltonian now constant, as it must be? Try deriving the differential equation from the Lagrangian. Do you get the same, or equivalent, ODE?

� � � � Tom Gutman
24-Ruby I
(To:TomGutman)
On 1/18/2008 3:44:45 AM, Tom_Gutman wrote:
>I'd rather get the actual
>sheet. For one thing, it
>and run the sheet than to view
>the page on the MAS.

If you have Mathcad requisite version!
And if not?
If you have only PDA and WiFi!
See one example with "a toy"
http://twt.mpei.ac.ru/ochkov/WSPHB/NREE-1-2008/Pic-11-Point.png

>Now, what would really be
>useful is the ability to
>compile the worksheet into a
>Java applet, and then have
>that run on the user's
>machine.

In Mathcad we can create one complex calculation in one day. But in Java???

Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
>>If you have Mathcad requisite version!<<

I do. And I was stating my preference.

>>If you have only PDA and WiFi!<<

The sheets are barely tolerable on a real computer with a reasonably large screen and a DSL connection. I can't imagine using it on a minature screen and WiFi.

Even with the large screen and high speed connection, it's something I look at only for the sake of discussion here. If I actually want to see a pendulum solution I'll look for one of the Java implementations, which download faster, compute nearly instaneously, and show the animation directly.

>>In Mathcad we can create one complex calculation in one day. But in Java???<<

Unless the compiler is extremely slow, about the same. The point is that you create the application in Mathcad and Mathcad then compiles it into Java, which can then run on the user's machine. Ideally you could generate either a stand-alone Java application, which can be distributed by any means and run like any other program by a user, or a Java applet which is provided as part of a web page and can be run as part of the browser.

Tom Gutman
24-Ruby I
(To:TomGutman)
>The sheets are barely tolerable on a real >computer with a reasonably large screen
I have created 3 types of WebMathcadSheets - for "real computer", for PDA and for SmartPhone!
Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
24-Ruby I
(To:TomGutman)
>Unless the compiler is extremely slow
No. I can use very powerfull computer as MAS. And net as not Internet but quickly intranet!

>Ideally you could generate either a stand-alone Java application

Ideally we all must be happy and healthy. But realy...
Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
24-Ruby I
(To:ValeryOchkov)
In polar coordinates...
http://twt.mpei.ac.ru/MAS/Worksheets/Pendulum/2.mcd
How to show only lower half of the Polar Plot? Or any sector?
Let think together!
Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
24-Ruby I
(To:ValeryOchkov)
On 1/15/2008 8:37:51 AM, VFO wrote:
>How to show only lower half of
>the Polar Plot? Or any sector?
>Let think together!

We can for example use upper half of the Polar Plot for... X-Y Plot for example.
See http://twt.mpei.ac.ru/MAS/Worksheets/Pendulum_Spring_Pole.mcd
Tom! I hope now all is ok! Thanks for help!
One more preference WebMathcadSheets (MAS) over one's WorkMathcadSheets - nobody has Sheets with errors! We can use only final Sheets!

Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
>>nobody has Sheets with errors!<<

Not so. Just that everybody has the same sheet, so everybody has the same errors. With no way to fix them. Nor even to run checks to detect them.

The graph seems much better. You did finally find the missing potential energy. Although it took me a while to notice that you had eliminated the Lagrangian and that you had instead plotted the two potential energies separately. But you did add the Hamiltonian, it it looks reasonable.

I do find some problems with the input specifications. No units for kH? And what is ξ, that it comes out dimensionless? The initial conditions specify only an angle and a speed (although you label it velocity). That would be sufficient for a normal pendulum with a rigid rod. But when you make the rod a spring you add degrees of freedom. The initial position can be anywhere within the range of the spring, not just on a circle, and the velocity need not be perpendicular to the rod, as the rod's length can change. It is also unclear what t is supposed to be. Also, the graph should run through tend, not stop at t.

� � � � Tom Gutman
24-Ruby I
(To:TomGutman)
On 1/19/2008 7:09:59 PM, Tom_Gutman wrote:
>>>nobody has Sheets with errors!<<
>
>Not so. Just that everybody
>has the same sheet, so
>everybody has the same errors.
>With no way to fix them. Nor
>even to run checks to detect
>them.
>
>The graph seems much better.
>You did finally find the
>missing potential energy.
>Although it took me a while to
>the Lagrangian and that you
>potential energies separately.
>Hamiltonian, it it looks
>reasonable.
>
>I do find some problems with
>the input specifications. No
>units for kH? And what is
>�, that it comes out
>dimensionless? The initial
>conditions specify only an
>angle and a speed (although
>you label it velocity). That
>would be sufficient for a
>normal pendulum with a rigid
>rod. But when you make the
>rod a spring you add degrees
>of freedom. The initial
>position can be anywhere
>within the range of the
>spring, not just on a circle,
>and the velocity need not be
>perpendicular to the rod, as
>the rod's length can change.
>It is also unclear what t is
>supposed to be. Also, the
>graph should run through tend,
>not stop at t.
>
>� � � � Tom Gutman

Thanks!
http://twt.mpei.ac.ru/MAS/Worksheets/Pendulum_Spring_Pole.mcd

Val
http://twt.mpei.ac.ru/ochkov/v_ochkov.htm
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