Dr. Ahmed Baroudy: New Applications
http://www.maplesoft.com/applications/author.aspx?mid=37315
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 23 Apr 2014 13:58:54 GMTWed, 23 Apr 2014 13:58:54 GMTNew applications published by Dr. Ahmed Baroudyhttp://www.mapleprimes.com/images/mapleapps.gifDr. Ahmed Baroudy: New Applications
http://www.maplesoft.com/applications/author.aspx?mid=37315
Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies
http://www.maplesoft.com/applications/view.aspx?SID=153515&ref=Feed
<p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p><img src="/view.aspx?si=153515/Elastic_Collision_image1.jpg" alt="Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies" align="left"/><p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p>153515Fri, 07 Mar 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyHohmann Elliptic Transfer Orbit with Animation
http://www.maplesoft.com/applications/view.aspx?SID=151351&ref=Feed
<p>Abstract<br /><br />The main purpose of this article is to show how to use Hohmann elliptic transfer in two situations:<br />a- When one manned spaceship is trying to catch up with an other one <br />on the same circular orbit around Earth.<br />b- When delivering a payload from Earth to a space station on a circular <br />orbit around Earth using 2-stage rocket .<br /><br />The way we set up the problem is as follows:<br />Consider two manned spaceships with astronauts Sally & Igor , the latter<br />lagging behind Sally by a given angle = 4.5 degrees while both are on the same<br />circular orbit C2 about Earth. A 2d lower circular orbit C1 is given. <br />Find the Hohmann elliptic orbit that is tangent to both orbits which allows<br />Sally to maneuver on C1 then to get back to the circular orbit C2 alongside Igor.<br /><br />Though the math was correct , however the final result we found was not !! <br />It was somehow tricky to find the culprit!<br />We have to restate the problem to get the correct answer. <br />The animation was then set up using the correct data. <br />The animation is a good teaching help for two reasons:<br />1- it gives a 'hand on' experience for anyone who wants to fully understand it,<br />2- it is a good lesson in Maple programming with many loops of the type 'if..then'.<br /><br />Warning<br /><br />This particular animation is a hog for the CPU memory since data accumulated <br />for plotting reached 20 MB! This is the size of this article when animation is <br />executed. For this reason and to be able to upload it I left the animation <br />procedure non executed which drops the size of the article to 300KB.<br /><br />Conclusion<br /><br />If I can get someone interested in the subject of this article in such away that he or <br />she would seek further information for learning from other sources, my efforts<br />would be well rewarded.</p><img src="/view.aspx?si=151351/Elliptic_image1.jpg" alt="Hohmann Elliptic Transfer Orbit with Animation" align="left"/><p>Abstract<br /><br />The main purpose of this article is to show how to use Hohmann elliptic transfer in two situations:<br />a- When one manned spaceship is trying to catch up with an other one <br />on the same circular orbit around Earth.<br />b- When delivering a payload from Earth to a space station on a circular <br />orbit around Earth using 2-stage rocket .<br /><br />The way we set up the problem is as follows:<br />Consider two manned spaceships with astronauts Sally & Igor , the latter<br />lagging behind Sally by a given angle = 4.5 degrees while both are on the same<br />circular orbit C2 about Earth. A 2d lower circular orbit C1 is given. <br />Find the Hohmann elliptic orbit that is tangent to both orbits which allows<br />Sally to maneuver on C1 then to get back to the circular orbit C2 alongside Igor.<br /><br />Though the math was correct , however the final result we found was not !! <br />It was somehow tricky to find the culprit!<br />We have to restate the problem to get the correct answer. <br />The animation was then set up using the correct data. <br />The animation is a good teaching help for two reasons:<br />1- it gives a 'hand on' experience for anyone who wants to fully understand it,<br />2- it is a good lesson in Maple programming with many loops of the type 'if..then'.<br /><br />Warning<br /><br />This particular animation is a hog for the CPU memory since data accumulated <br />for plotting reached 20 MB! This is the size of this article when animation is <br />executed. For this reason and to be able to upload it I left the animation <br />procedure non executed which drops the size of the article to 300KB.<br /><br />Conclusion<br /><br />If I can get someone interested in the subject of this article in such away that he or <br />she would seek further information for learning from other sources, my efforts<br />would be well rewarded.</p>151351Wed, 04 Sep 2013 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudySpherical Pendulum with Animation
http://www.maplesoft.com/applications/view.aspx?SID=132143&ref=Feed
<p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1- to get the differential equations of motion for the Spherical Pendulum (SP),<br />2- to solve them,<br />3- to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4- to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5- finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p><img src="/view.aspx?si=132143/433082\Spherical_Pendulum_p.jpg" alt="Spherical Pendulum with Animation" align="left"/><p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1- to get the differential equations of motion for the Spherical Pendulum (SP),<br />2- to solve them,<br />3- to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4- to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5- finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p>132143Mon, 26 Mar 2012 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyTerminator circle with animation
http://www.maplesoft.com/applications/view.aspx?SID=100509&ref=Feed
<p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p><img src="/view.aspx?si=100509/thumb.jpg" alt="Terminator circle with animation" align="left"/><p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p>100509Tue, 28 Dec 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyTwo Bodies Revolving Around Their Center of Mass with ANIMATION
http://www.maplesoft.com/applications/view.aspx?SID=99587&ref=Feed
<p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p><img src="/view.aspx?si=99587/thumb.jpg" alt="Two Bodies Revolving Around Their Center of Mass with ANIMATION" align="left"/><p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p>99587Mon, 29 Nov 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed Baroudyvan Roomen Problem
http://www.maplesoft.com/applications/view.aspx?SID=96978&ref=Feed
<p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p><img src="/view.aspx?si=96978/maple_icon.jpg" alt="van Roomen Problem" align="left"/><p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p>96978Sat, 18 Sep 2010 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyASTROLABE MATHEMATICS
http://www.maplesoft.com/applications/view.aspx?SID=35214&ref=Feed
<p>After a very short introduction about the origin & the evolution of the astrolabe through ages, we go into the basic geometry behind its design and construction. <br />
This involves: <br />
1- the therory of STEREOGRAPHIC PROJECTION with two simple properties that we prove, <br />
2- simple ANALYTIC GEOMETRY,<br />
3- OBLIQUE CONE THEORY according to LEGENDRE. <br />
We then explore the details of designing an astrolabe PLATE according to ancient constructors technics and today's technology using Maple powerful plotting engine. This resuls in a beautiful PLATE and on top of it the RETE. (See Figure below). <br />
As a whole the article is a good exercise in Maple programming and a way to stimulate further interest in Astronomy.</p><img src="/view.aspx?si=35214/Rete_on_top_of_plate.jpeg" alt="ASTROLABE MATHEMATICS" align="left"/><p>After a very short introduction about the origin & the evolution of the astrolabe through ages, we go into the basic geometry behind its design and construction. <br />
This involves: <br />
1- the therory of STEREOGRAPHIC PROJECTION with two simple properties that we prove, <br />
2- simple ANALYTIC GEOMETRY,<br />
3- OBLIQUE CONE THEORY according to LEGENDRE. <br />
We then explore the details of designing an astrolabe PLATE according to ancient constructors technics and today's technology using Maple powerful plotting engine. This resuls in a beautiful PLATE and on top of it the RETE. (See Figure below). <br />
As a whole the article is a good exercise in Maple programming and a way to stimulate further interest in Astronomy.</p>35214Sat, 27 Feb 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyElliptic Functions and Simple Pendulum Animation
http://www.maplesoft.com/applications/view.aspx?SID=7174&ref=Feed
The purpose of the present article is to built the equation of motion of a simple pendulum for large angles, then to use Maple commands: JacobiSN, to find the displacement angle for a given time.
A set of ordered pairs of coordinates for the bob is thus obtained and used for plotting.
The animation is somehow tricky since the set obtained doesn't allow for a straight forward animation and needs some careful manipulation of the data in the set.
The reward is double:
1- being able to put to good use Maple engine in finding Elliptic Functions (EF) with a simple command rather than resorting to tables for finding EF.
2- A good exercise in programming with Maple animation.<img src="/view.aspx?si=7174/thumb.gif" alt="Elliptic Functions and Simple Pendulum Animation" align="left"/>The purpose of the present article is to built the equation of motion of a simple pendulum for large angles, then to use Maple commands: JacobiSN, to find the displacement angle for a given time.
A set of ordered pairs of coordinates for the bob is thus obtained and used for plotting.
The animation is somehow tricky since the set obtained doesn't allow for a straight forward animation and needs some careful manipulation of the data in the set.
The reward is double:
1- being able to put to good use Maple engine in finding Elliptic Functions (EF) with a simple command rather than resorting to tables for finding EF.
2- A good exercise in programming with Maple animation.7174Wed, 28 Jan 2009 00:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyBilliard ball dynamics
http://www.maplesoft.com/applications/view.aspx?SID=5650&ref=Feed
The dynamics of billiard ball is analyzed and the behaviour of the ball deduced from this analysis according to the point of impact of the cue on the ball. 8 cases were considered and Maple was used to show animation of one of them.<img src="/view.aspx?si=5650/billiard ball dynamics.jpg" alt="Billiard ball dynamics" align="left"/>The dynamics of billiard ball is analyzed and the behaviour of the ball deduced from this analysis according to the point of impact of the cue on the ball. 8 cases were considered and Maple was used to show animation of one of them.5650Fri, 08 Feb 2008 00:00:00 ZDr. Ahmed BaroudyDr. Ahmed Baroudy