Wikipedia:Reference desk/Archives/Mathematics/2013 May 25

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May 25

Parameters of a mechanical system

I have been told that some constants of motion, such as energy, are not external parameters of a system. What does this mean? Is it because energy is not a Casimir function, or am I missing something more fundamental?--Leon (talk) 08:36, 25 May 2013 (UTC)

Find the value of 'x' in the triangle of linked image

This is not my homework. I tried my best to solve the problem I have given, but I failed. My question is - Find the value of 'x' (= angle DEB) in the triangle of linked image. http://upload.wikimedia.org/wikipedia/commons/d/d3/Find_the_value_of_%27X%27_in_this_triangle.jpg — Preceding unsigned comment added by LearnNext111 (talk • contribs) 16:21, 25 May 2013 (UTC)

I think you might need to use something like the sine rule. Set the length of the base BC to be 1, calculate all the easy angles, BEC, BDC, BFC, DFB, CFE, DFE. Using the sine rule allows you to calculate lengths BF, CF, CE, BD, then DF, FE. We can now calculate length DE. The three length of triangle DEF are now known as well as one angle DFE so the other angles can then be found.--Salix (talk): 17:08, 25 May 2013 (UTC)

Have a look at the very nice site [1] which has a number of solutions to this and related problems, so you can have a look at one solution and then try a related problem. Dmcq (talk) 17:49, 25 May 2013 (UTC)

I do not know what the solution is, but it might be important to realize that the image has false proportions. Sum of given angles is 30°+50°+60°+20° = 160° so the angle A is 20°, and the triangle should be drawn approx. 3 times that high on the same base.
And triangle CDA is isosceles, CD = DA. --CiaPan (talk) 21:43, 27 May 2013 (UTC)

There's another isosceles triangle: BC=BE. – b_jonas 18:41, 28 May 2013 (UTC)

height of arc from length of arc and length of chord

I am trying to solve the problem in on this page

Suppose a steel beam, one mile long, is fastened securely to the ground at each end. As the day heats up, the metal expands. Let us assume that at the hottest part of the day, the metal is actually one mile and one foot long. Let us further assume that the beam is fastened in such a way that it can only buckle upward, and not side to side. Your problem is to estimate how high the beam will rise above the ground.

I am assuming a uniform curvature across the length of the beam so that would make it a arc of a circle and the un-expanded beam would be a mile long chord between the points of the arc. So I need to figure out the radius of the arc and the angle subtended using lenght of the chord and the arc.

working in radiens

a = arc length (5281 feet)

r = radius of curvature

θ = angle subtended

c = length of cord(5280 feet)

a = r * θ

(c/2) = r * sin(θ/2)

2640 = (5281/θ) * sin(θ/2)

θ/(sin(θ/2) = (5281/2640)

I am not sure how to solve this. Does anybody know how to solve this or simplify the problem to an easier equation? Diwakark86 (talk) 18:48, 25 May 2013 (UTC)

Approximate sin and cos assuming the angle is small in radians by

$\sin \alpha \approx 1-{\frac {\alpha ^{2}}{2}}$

$\cos \alpha \approx \alpha -{\frac {\alpha ^{3}}{6}}$

and the angle in radians of the entire sector is 5281/r. Dmcq (talk) 21:39, 25 May 2013 (UTC)

These expressions for sin and cos are the wrong way round. Using the approximation

$\sin \alpha \approx \alpha -{\frac {\alpha ^{3}}{6}}$, I find that the radius is about 78340 feet and the required height is about 44.5 feet. 86.139.121.71 (talk) 23:27, 25 May 2013 (UTC)

Eek, that was very silly of me! Dmcq (talk) 23:38, 25 May 2013 (UTC)

• The way the computer would solve this equation would be to use Newton's method - I'm quite sure that the solution to this equation is transcendental.--Jasper Deng (talk) 23:29, 25 May 2013 (UTC)

Note that to get an order of magnitude solution to verify that a more accurate answer is using the right formula, just do it as a two linear section pair of right triangles. 2640 along the ground each, 2640.5 along the expanded beam. Pythag gives a height of 51.4 feet, so 44.5 is quite plausible. -- SGBailey (talk) 23:36, 25 May 2013 (UTC)

One is just ${\sqrt {3}} \over 2$ times the other, and yes I'd guess estimating covers both. Dmcq (talk) 00:00, 26 May 2013 (UTC)

$\scriptstyle {\tfrac {sin(\operatorname {HalfAngle} )}{\operatorname {HalfAngle} }}\ =\ {\tfrac {\operatorname {Chord} }{\operatorname {Arc} }}\ \approx \ 1\ ,\qquad \qquad {\tfrac {\sin x}{x}}\ \approx \ 1\ -\ {\tfrac {x^{2}}{6}}\quad {\color {white}.}$ (see Taylor series)

therefore

$\scriptstyle \operatorname {HalfAngle} \ \approx \ {\sqrt {6\ \left(1\ -\ {\tfrac {\operatorname {Chord} }{\operatorname {Arc} }}\right)}}\ =\ {\sqrt {\tfrac {6}{5281}}}\ \approx \ {\sqrt {\tfrac {6}{5280}}}\ =\ {\sqrt {\tfrac {1}{880}}}\ =\ {\tfrac {1}{\sqrt {880}}}\ \approx \ {\tfrac {1}{29{\tfrac {2}{3}}}}\ =\ {\tfrac {3}{89}}$ radians.

Which should come as no surprise given that ${\color {white}.}{\tfrac {\sin x}{x}}$ ≈ 1 ⇔ x ≈ 0 (see sinc function and l'Hopital's rule)

Furthermore

$\scriptstyle \operatorname {Radius} \ =\ {\tfrac {\operatorname {Arc} }{2\operatorname {HalfAngle} }}\ ,\qquad \qquad \qquad \sin x\ \to \ x\ \operatorname {when} \ x\ \to \ 0$

and

$\scriptstyle \operatorname {Height} \ =\ \operatorname {Radius} \ \cdot \ [\ 1\ -\ cos(\operatorname {HalfAngle} )\ ]\ =\ 2\operatorname {Radius} \ \cdot \ \sin ^{2}\left({\tfrac {\operatorname {HalfAngle} }{2}}\right)\ \approx \ 2\operatorname {Radius} \ \cdot \ \left({\tfrac {\operatorname {HalfAngle} }{2}}\right)^{2}\ =$

$\scriptstyle \ =\ 2\ \cdot \ {\tfrac {\operatorname {Arc} }{2\operatorname {HalfAngle} }}\ \cdot \ {\tfrac {\operatorname {HalfAngle\ } ^{2}}{4}}\ =\ {\tfrac {\operatorname {Arc} \ \cdot \ \operatorname {HalfAngle} }{4}}\ \approx \ {\tfrac {5280}{4}}\ \cdot \ {\tfrac {3}{89}}\ =\ 1320\ \cdot \ {\tfrac {3}{89}}\ \approx \ 44{\tfrac {1}{2}}$ feet

since

$\scriptstyle {\color {white}.}1\ -\ \cos x\ =\ 2\ \cdot \ \sin ^{2}\left({\tfrac {x}{2}}\right){\color {white}.}$ → see half-angle formula — 79.113.220.51 (talk) 11:43, 26 May 2013 (UTC)

This approach is fundamentally wrong as it totally ignores the relevant physics, see here for a better approach. Count Iblis (talk) 16:51, 26 May 2013 (UTC)

I guess that would generally be the problem with asking a physics question on the math reference desk, instead of on the science reference desk. :-) — 79.113.215.108 (talk) 18:19, 26 May 2013 (UTC)

Well, possibly for such a long beam you should also take the Earth ellipsoid curvature into account, and the geoid undulation as well? --CiaPan (talk) 11:58, 27 May 2013 (UTC)

The shape of the beam approximates a sine curve rather than a circular arc. See buckling. Bo Jacoby (talk) 19:03, 26 May 2013 (UTC).

I agree it should be straighter near the ends of the arc and more bent in the middle, but I didn't see where it described the shape in that article. In fact one of the diagrams implied a circular shape which as you say is wrong. Dmcq (talk) 21:42, 26 May 2013 (UTC)

The curvature is proportional to the bending moment, which is proportional to the height. The curvature is approximately equal to second derivative. So the curve y=f(x) satisfies the differential equation y"=-Ay the solution of which is

y = y_max sin(π x / 5280 feet)

Bo Jacoby (talk) 06:25, 27 May 2013 (UTC).

I have found the solution, albeit it in a very rough approximation. If you take the beam radius to be 30 centimeters, it will come off the ground for only about 84 meters from both ends. The beam height from one end at x = 0 till x = p = 84 meters is:

y(x) = c x^2 (x - p)^2

where c = 7.13*10^(-7) meters^(-3).

The maximum height is then reached at x = p/2 where it is about 2.2 meters. Count Iblis (talk) 23:42, 26 May 2013 (UTC)

Could you explain the physics behind it (for lay people) ? (Maybe even draw a picture, with the various forces, and all ?) You've made me curious.. — 79.113.215.108 (talk) 02:02, 27 May 2013 (UTC)

If we're talking about the real world with gravity etc then it would fall over sideways and the maximum height would be zero ;-) Dmcq (talk) 11:05, 27 May 2013 (UTC)

Yeah, but let's say we put it between two long sheets of material that prevents it from falling to the side... I mean, I get how it's supposed to bend starting from where pressure is exerted, by being fixed to the ground... but how on earth we derive the numbers 84 and 2 is beyond me... Of course, the Count gave a formula, which I believe it in my heart to be true by the power of faith... but I just don't get how he arrived at it, and I'm curious by nature... — 79.113.227.125 (talk) 16:01, 27 May 2013 (UTC)

The geometric notion you describe is Sagitta. --CiaPan (talk) 05:31, 27 May 2013 (UTC)

See also Versine#"Versines"_of_arbitrary_curves_and_chords. Bo Jacoby (talk) 09:47, 29 May 2013 (UTC).