Physics Bonanza's New Writeups
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2005-03-24T18:07:49Z
weetos (thing)http://m.everything2.com/user/Physics+Bonanza/writeups/weetosPhysics Bonanzahttp://m.everything2.com/user/Physics+Bonanza2005-03-24T18:07:49Z2005-03-24T18:07:49Z
Everyone knows that Weetos are the choco-hoops of choice for today's distinguished <a href="/title/or+a+woman+if+you+are+one">gentleman</a>. However, everyone also knows that old Joe McShmoe down the street can go to town in his Joemobile and get himself some Weetos too! It is <a href="/title/my+face%2521">blindingly clear</a> that we need a way to differentiate between cool <a href="/title/or+a+woman+if+you+are+one">guys</a> like <a href="/title/Batman">you</a> and bums like Joe.
<p>
I have such a way.*
</p>
In this very writeup, I present to you the <a href="/title/I+don%2527t+know+what+I%2527m+doing">codiest C code</a> that will make you stand out from the rest when you are enjoying your morning dose of choco nutrients! In order to use this code effectively, you need to know the rules of the Game.
<p>
<b>The Game</b>
</p>
<p>
The <a href="/title/Game">Game</a> is very simple: get some Weetos, and <a href="/title/munch">munch</a> away. Once you are down to a countably <a href="/title/small">small</a> number of Weetos, the Game begins. First eat one Weeto. Then two at a time. Then three, and so on, until you have <a href="/title/every+man+for+himself">run out of Weetos</a>. The aim of the game…
eigenvector (idea)http://m.everything2.com/user/Physics+Bonanza/writeups/eigenvectorPhysics Bonanzahttp://m.everything2.com/user/Physics+Bonanza2004-12-16T00:44:50Z2004-12-16T00:44:50Z
The <a href="/title/equation">equation</a> A<b>v</b> = λ<b>v</b> can be interpreted so that <a href="/title/eigenvectors">eigenvectors</a> and <a href="/title/eigenvalues">eigenvalues</a> can be thought of in a somewhat more <a href="/title/intuitive">intuitive</a> fashion than simply their <a href="/title/definition">definition</a>s in this equation. This is good news for <a href="/title/mathematicians+are+monkeys">physicists</a>. We can express this equation in words by saying the following: take a <a href="/title/vector">vector</a>, and <a href="/title/transform">transform</a> it; if the new, transformed vector is simply a <a href="/title/multiple">multiple</a> of the old one, then that vector is an <a href="/title/eigenvector">eigenvector</a>. The multiplier is an <a href="/title/eigenvalue">eigenvalue</a>. (N.B. "Eigenvector" literally means "own vector" (correct me if I'm wrong) in German, and we can now see why they are so called: the transformation of a vector creates a multiple of <b>itself</b>).<p>
A nice way to visualise such a transformation is that of a <a href="/title/rubber">rubber</a> <a href="/title/square">square</a> with an <a href="/title/arrow">arrow</a> drawn on it (see <a href="http://www.physlink.com/Education/AskExperts/ae520.cfm">http://www.physlink.com/Education/AskExperts/ae520.cfm</a>). If you <a href="/title/stretch">stretch</a> the square along a particular <a href="/title/axis">axis</a>, only arrows in certain <a href="/title/direction">direction</a>s will keep their direction; this is true no…
Shortest distance from a point to a plane (idea)http://m.everything2.com/user/Physics+Bonanza/writeups/Shortest+distance+from+a+point+to+a+planePhysics Bonanzahttp://m.everything2.com/user/Physics+Bonanza2004-07-27T18:19:57Z2004-07-27T18:19:57Z
This writeup explains how to find the <a href="/title/shortest">shortest</a> <a href="/title/distance">distance</a> from a <a href="/title/point">point</a> to a <a href="/title/plane">plane</a>, when both the point and the plane are <a href="/title/define">define</a>d by <a href="/title/vector">vector</a>s, by an <a href="/title/example">example</a>.
<p>
<a href="/title/suppose">Suppose</a> that there is a plane with <a href="/title/scalar">scalar</a> <a href="/title/equation">equation</a>:
<p>
<b>r</b>.(2<b>i</b> - <b>j</b> - 3<b>k</b>) = 13
<p>
and we want to find the shortest distance from a point P with <a href="/title/position+vector">position vector</a> (3<b>i</b> + <b>j</b> + 16<b>k</b>) to this plane.
<p>
First, find the <a href="/title/perpendicular">perpendicular</a> distance of the plane from the <a href="/title/origin">origin</a>. To do this, <a href="/title/consider">consider</a> the position vector <b>r</b> of an <a href="/title/arbitrary">arbitrary</a> point on the plane. Now <a href="/title/imagine">imagine</a> a vector that is perpendicular to the plane and passes through the origin. (I would recommend <a href="/title/draw">draw</a>ing a <a href="/title/diagram">diagram</a>.) It is now evident that the vector <b>r</b>, that is the vector from the origin to the arbitrary point on the plane, is the <a href="/title/hypotenuse">hypotenuse</a> of a <a href="/title/triangle">triangle</a> with its <a href="/title/base">base</a> on the plane, and the remaining <a href="/title/side">side</a> <a href="/title/form">form</a>ed by the perpendicular vector. This perpendicular vector is called <b><a href="/title/n">n</a></b>…
tetrahedron (idea)http://m.everything2.com/user/Physics+Bonanza/writeups/tetrahedronPhysics Bonanzahttp://m.everything2.com/user/Physics+Bonanza2004-07-27T13:23:49Z2004-07-27T13:23:49Z
If a tetrahedron is <a href="/title/define">define</a>d by three <a href="/title/vector">vector</a>s, the <a href="/title/volume">volume</a> of that tetrahedron can be found by <a href="/title/evaluate">evaluating</a> the <a href="/title/value">value</a> of one-sixth of the <a href="/title/scalar+triple+product">scalar triple product</a>. So, if the tetrahedron has <a href="/title/corner">corner</a>s at <a href="/title/point">point</a>s A, B, C and D, the three vectors needed are those in the <a href="/title/direction">direction</a>s AB, AC and AD. Then the volume V is found by
<p>
V = (1/6)(<b>AD</b>.<b>AB</b>x<b>AC</b>).
<p>
As an example, let us consider a tetrahedron which is defined by the following vectors:
<p>
<b>AB</b> = 5<b>i</b> - <b>j</b> - <b>k</b>
<br>
<b>AC</b> = 2<b>i</b> - 8<b>j</b> + <b>k</b>
<br>
<b>AD</b> = -<b>i</b> + 2<b>k</b>
<p>
A quick way to evaluate the scalar triple product is to calculate the <a href="/title/modulus">modulus</a> of the <a href="/title/determinant">determinant</a> of a <a href="/title/matrix">matrix</a> consisting of these three vectors:
<p>
<pre>
| / 5 -1 -1 \ |
V = (1/6)| det | 2 -8 1 | |
| \-1 0 2 / |
</pre>
<p>
= (1/6)|5(-16 - 0) + 1(4 + 1) - 1(0 - 8)|
<br>
= (1/6)|-80 + 5 + 8|
<br>
= (1/6)|-67|
<br>
= (1/6)(67)
<br>
= 67/6
<p>
Therefore the tetrahedron has volume 67/6 units^3.
Benzene (idea)http://m.everything2.com/user/Physics+Bonanza/writeups/BenzenePhysics Bonanzahttp://m.everything2.com/user/Physics+Bonanza2004-07-26T21:32:40Z2004-07-26T21:32:40Z
Some of the <a href="/title/evidence">evidence</a> for benzene having the <a href="/title/structure">structure</a> detailed in other writeups (with a <a href="/title/delocalised">delocalised</a> <a href="/title/ring">ring</a> of <a href="/title/electron">electron</a>s at the centre) as opposed to the <a href="/title/Kekul%25C3%25A9">KekulÃ©</a> structure includes <a href="/title/thermodynamic">thermodynamic</a> considerations.
<p>
Let's look at the <a href="/title/enthalpy">enthalpy</a> change for the complete <a href="/title/hydrogenation">hydrogenation</a> of <a href="/title/cyclohexatriene">cyclohexatriene</a> (essentially the KekulÃ©e structure) to <a href="/title/cyclohexane">cyclohexane</a>. The problem is, cyclohexatriene doesn't exist due to its <a href="/title/instability">instability</a>. So we use our <a href="/title/imagination">imagination</a>s, and <a href="/title/Do+as+I+say%252C+not+as+I+do">think as follows</a>:
<br>
The enthalpy change for the hydrogenation of cyclohexene (like cyclohexane but with a double bond replacing one of the single bonds) is -120 k<a href="/title/joule">J</a>/<a href="/title/mole">mol</a>.
<br>
The enthalpy change for the hydrogenation of cyclohex-1,3-diene (like cyclohexane but with two double bonds) is -233 kJ/mol.
<br>
This is roughly twice the amount for the same reaction for cyclohexene; we could therefore say that the enthalpy of hydrogenation of cyclohexatriene will be roughly three times that for…
coefficient of restitution (idea)http://m.everything2.com/user/Physics+Bonanza/writeups/coefficient+of+restitutionPhysics Bonanzahttp://m.everything2.com/user/Physics+Bonanza2004-07-25T21:11:15Z2004-07-25T21:11:15Z
The <a href="/title/symbol">symbol</a> for the <a href="/title/coefficient">coefficient</a> of <a href="/title/restitution">restitution</a> is e. For two <a href="/title/collision">colliding</a> <a href="/title/body">bodies</a>:
<p>
e = (<a href="/title/speed">speed</a> of <a href="/title/separation">separation</a>) / (speed of <a href="/title/approach">approach</a>)
<p>
As a <a href="/title/consequence">consequence</a> of the principle of <a href="/title/conservation+of+energy">conservation of energy</a>, e lies between 0 (for a completely <a href="/title/inelastic">inelastic</a> collision, i.e. one where all <a href="/title/kinetic+energy">kinetic energy</a> is lost to the surroundings) and 1 (for a completely <a href="/title/elastic">elastic</a> collision).
<p>
To make the use of the coefficient of restitution clear, let us consider a typical <a href="/title/mechanics">mechanics</a> <a href="/title/problem">problem</a>.
<p>
<a href="/title/Immediately">Immediately</a> before a collision, a <a href="/title/particle">particle</a> A with <a href="/title/mass">mass</a> 2m is travelling with speed 2 m/s to the <a href="/title/right">right</a>, and a particle B with mass 4m is travelling with speed 1 m/s also to the right. Immediately after the collision, A travels with speed u, and B travels with speed v, both to the right. Given that the coefficient of restitution between A and B is 1/2 and that the particles do not <a href="/title/stick">stick</a> together, find the values of u and v.
<p>
In order to <a href="/title/solve">solve</a> this problem, two <a href="/title/law">law</a>s need to be used:…