{"id":3730,"date":"2022-09-27T13:03:52","date_gmt":"2022-09-27T11:03:52","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=3730"},"modified":"2023-06-09T14:47:13","modified_gmt":"2023-06-09T12:47:13","slug":"advanced-text-the-circular-surface","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-the-circular-surface\/","title":{"rendered":"Advanced Text The Circular Surface"},"content":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text]<\/p>\n

The Circular Surface<\/h1>\n

[\/vc_column_text][vc_single_image image=”2481″ img_size=”large” alignment=”center”]Figure 1: The exhibit in its original state[\/vc_single_image][vc_column_text] As the above Figure 1 shows, the experiment at the exhibit “The Circular Surface<\/a>“<\/em> consists of bending up a circular<\/em> spring by means of two (initially vertical and parallel) rotatable levers in such a way that its shape in the final position (see Figure 2 below) represents a distance from point \"A\" to point \"B\".[\/vc_column_text][vc_single_image image=”2486″ img_size=”large” alignment=”center”]Figure 2: The exhibit in its final state[\/vc_single_image][vc_column_text]Here <\/p>\n

  <\/span>   <\/span>\"\[\lvert\overline{AB} \rvert=2\lvert\overline{AC} \rvert=2\lvert\overline{BC} \rvert=2\pi r\]\"<\/p>\n

is the length<\/em> of the circumference<\/em> of the circle that the spring originally formed. Furthermore, the following applies: The area<\/em> \"F_\Delta\" of the right triangle \"\Delta\" with the vertices \"M\", \"B\" and \"C\" (see Figure 1) is equal to half the area of the circle, i.e. <\/p>\n

  <\/span>   <\/span>\"\[F_\Delta=\frac{1}{2}\pi r^2.\]\"<\/p>\n

Thus the area \"\pi r^2\" of a circle with radius \"r\" can be represented by the sum of the areas of two congruent<\/em> triangles.[\/vc_column_text][vc_column_text]<\/p>\n

And now … the mathematics:<\/h3>\n

[\/vc_column_text][vc_single_image image=”2490″ img_size=”large” alignment=”center”]Figure 3: Mechanism of the exhibit[\/vc_single_image][vc_column_text]If the two levers are each rotated by the angle \"\varphi\" against the vertical axis, the contact point<\/em> \"(x_0,y_0)\" between the right lever arm and the bent-up circle results on the right side. The latter now represents itself as a circular arc<\/em> with the radius<\/em> \"r^\ast\" and the opening angle<\/em> \"\varphi^\ast\" (in radians!). Thus <\/p>\n

  <\/span>   <\/span>\"\[\pi r=r^\ast\varphi^\ast\quad(1)\]\"<\/p>\n

and <\/p>\n

  <\/span>   <\/span>\"\[x_0^2+(r^\ast-y_0)^2=(r^\ast)^2\quad(2)\]\"<\/p>\n

apply.<\/p>\n

According to figure 3 above, the following applies to the straight lines \"y_1\" and \"y_2\": <\/p>\n

  <\/span>   <\/span>\"\[y_1\colon y_1(x)=r+\tan(\pi/2-\varphi)x\]\"<\/p>\n

and <\/p>\n

  <\/span>   <\/span>\"\[y_2\colon y_2(x)=r^\ast+\tan(\pi/2+\varphi^\ast)x.\]\"<\/p>\n

Their intersection \"(x_0,y_0)\" then results as the solution of the equation <\/p>\n

  <\/span>   <\/span>\"\[y_0\coloneqq y_1(x_0)=y_2(x_0)\]\"<\/p>\n

i.e. <\/p>\n

  <\/span>   <\/span>\"\[r+\tan(\pi/2-\varphi)x_0=r^\ast+\tan(\pi/2+\varphi^\ast)x_0\]\"<\/p>\n

and thus <\/p>\n

  <\/span>   <\/span>\"\[x_0(\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast))=r^\ast-r.\]\"<\/p>\n

This results in equation (1): <\/p>\n

  <\/span>   <\/span>\"\[x_0=\frac{r^\ast-r}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}=r\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}\]\"<\/p>\n

and thus <\/p>\n

  <\/span>   <\/span>\"\[y_0=r+\tan(\pi/2-\varphi)x_0=r+\tan(\pi/2-\varphi)r\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}.\]\"<\/p>\n

Equations (1) and (2) then give <\/p>\n

  <\/span>   <\/span>\"\[x_0^2+(r^\ast-y_0)^2=(r^\ast)^2=\frac{\pi^2 r^2}{(\varphi^\ast)^2}\]\"<\/p>\n

and thus <\/p>\n

  <\/span>   <\/span>\"\[x_0^2+\left(\frac{\pi r}{\varphi^\ast}-y_0\right)^2=\frac{\pi^2 r^2}{(\varphi^\ast)^2},\]\"<\/p>\n

which after multiplying out and reducing leads to <\/p>\n

  <\/span>   <\/span>\"\[x_0^2-\frac{2\pi r y_0}{\varphi^\ast}+y_0^2=0.\]\"<\/p>\n

That now means <\/p>\n

  <\/span>   <\/span>\"\begin{gather*} \left(r\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}\right)^2\\ -\frac{2\pi r}{\varphi^\ast}\left(r+\tan(\pi/2-\varphi)r\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}\right)\\+\left(r+\tan(\pi/2-\varphi)r\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}\right)^2=0.\end{gather*}\"<\/p>\n

Consequently, it can be assumed without restriction that \"r=1\", so that for a given opening angle \"\varphi\" (in radians) of the right lever (see figure 3), the opening angle \"\varphi^\ast\" of the corresponding circular arc (with radius \"r^\ast\") is obtained as the solution of the following equation: <\/p>\n

  <\/span>   <\/span>\"\begin{gather*} \left(\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}\right)^2\\ -\frac{2\pi}{\varphi^\ast}\left(1+\tan(\pi/2-\varphi)\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}\right)\\+\left(1+\tan(\pi/2-\varphi)\frac{\pi/\varphi^\ast-1}{\tan(\pi/2-\varphi)-\tan(\pi/2+\varphi^\ast)}\right)^2=0.\end{gather*}\"<\/p>\n

Finally, we give — determined numerically as approximate values — for <\/p>\n

  <\/span>   <\/span>\"\[\varphi_i=\frac{i}{10}(\pi/2+\varphi_0)=\frac{i}{10}(\pi/2+\arctan(1/\pi))\]\"<\/p>\n

(\"i =1,\ldots,10\") the corresponding angles \"\varphi^\ast\" and the radii \"r^\ast\" (using the above equation).[\/vc_column_text][vc_column_text]\n\n\n\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t
\"i\"<\/th>\"\varphi_i\"<\/th>\"\varphi_i^\ast\"<\/th>\"\varphi_i^\ast\" (in Radian)<\/th>\"r^\ast_i\"<\/th>\"x_0^i\"<\/th>\"y_0^i\"<\/th>\n<\/tr>\n<\/thead>\n
\"1\"<\/td>\"0,1879\"<\/td>\"2,942\"<\/td>\"168,5\"<\/td>\"1,068\"<\/td>\"0,2148\"<\/td>\"2,1298\"<\/td>\n<\/tr>\n
\"2\"<\/td>\"0,3760\"<\/td>\"2,716\"<\/td>\"155,6\"<\/td>\"1,157\"<\/td>\"0,4770\"<\/td>\"2,2089\"<\/td>\n<\/tr>\n
\"3\"<\/td>\"0,5637\"<\/td>\"2,462\"<\/td>\"141,1\"<\/td>\"1,276\"<\/td>\"0,8015\"<\/td>\"2,2680\"<\/td>\n<\/tr>\n
\"4\"<\/td>\"0,7516\"<\/td>\"2,179\"<\/td>\"124,8\"<\/td>\"1,442\"<\/td>\"1,1819\"<\/td>\"2,2647\"<\/td>\n<\/tr>\n
\"5\"<\/td>\"0,9395\"<\/td>\"1,869\"<\/td>\"107,1\"<\/td>\"1,681\"<\/td>\"1,6070\"<\/td>\"2,1748\"<\/td>\n<\/tr>\n
\"6\"<\/td>\"1,1274\"<\/td>\"1,534\"<\/td>\"87,9\"<\/td>\"2,048\"<\/td>\"2,0477\"<\/td>\"1,9726\"<\/td>\n<\/tr>\n
\"7\"<\/td>\"1,3153\"<\/td>\"1,176\"<\/td>\"67,4\"<\/td>\"2,671\"<\/td>\"2,4656\"<\/td>\"1,6441\"<\/td>\n<\/tr>\n
\"8\"<\/td>\"1,5031\"<\/td>\"0,800\"<\/td>\"45,8\"<\/td>\"3,927\"<\/td>\"2,8173\"<\/td>\"1,1908\"<\/td>\n<\/tr>\n
\"9\"<\/td>\"1,6911\"<\/td>\"0,4071\"<\/td>\"23,3\"<\/td>\"7,717\"<\/td>\"3,0555\"<\/td>\"0,6307\"<\/td>\n<\/tr>\n
\"10\"<\/td>\"1,8789\"<\/td>\"0\"<\/td>\"0\"<\/td>\"\infty\"<\/td>\"\pi\"<\/td>\"0\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/p>\n

Table 1: The values \"\varphi^\ast\" and \"r^\ast\" as a function of \"\varphi\".<\/p>\n

[\/vc_column_text][vc_column_text]Figure 4 below summarises this in a diagram.[\/vc_column_text][vc_single_image image=”2494″ img_size=”large” alignment=”center”]Figure 4: Diagram showing the points \"(x_0,y_0)\" for the different opening angles[\/vc_single_image][vc_column_text]Note<\/em>: This exhibit is closely related to the exhibits “What is Pi?”<\/a>, “What is the area of a circle?”<\/a> and “Twelve corners”<\/a>.[\/vc_column_text][vc_column_text]<\/p>\n

Literature<\/h3>\n

[\/vc_column_text][vc_column_text][1] https:\/\/en.wikipedia.org\/wiki\/Area_of_a_circle<\/a><\/p>\n

[2] https:\/\/www.youtube.com\/watch?v=2wL_dIDTc3w<\/a>[\/vc_column_text][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text] The Circular Surface [\/vc_column_text][vc_single_image image=”2481″ img_size=”large” alignment=”center”]Figure 1: The exhibit in its original state[\/vc_single_image][vc_column_text] As the above Figure 1 shows, the experiment at the exhibit “The Circular Surface“ consists of bending up a circular spring by means of two (initially vertical and parallel) rotatable levers in such a way that its shape … Advanced Text The Circular Surface<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"folder":[],"class_list":["post-3730","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3730","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/comments?post=3730"}],"version-history":[{"count":21,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3730\/revisions"}],"predecessor-version":[{"id":4664,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/3730\/revisions\/4664"}],"wp:attachment":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/media?parent=3730"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/folder?post=3730"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}