{"id":4126,"date":"2022-09-14T12:40:39","date_gmt":"2022-09-14T10:40:39","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=4126"},"modified":"2023-01-26T14:10:58","modified_gmt":"2023-01-26T13:10:58","slug":"advanced-text-galileo-tank","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-galileo-tank\/","title":{"rendered":"Advanced text Galileo tank"},"content":{"rendered":"

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

Galilei’s Tank<\/h1>\n

As shown in Figure 1 below, the so-called Galilean trough<\/em> consists of a rectilinear channel inclined by an angle \"\alpha\" and having an approximately semicircular cross-section.[\/vc_column_text][vc_single_image image=”1524″ img_size=”large” alignment=”center”]Figure 1: Galileo tub[\/vc_single_image][vc_column_text]In EXPERIENCE LAND MATHEMATICS, the trough, i.e. the Galilean trough, is a blue tread with a length of \"3.77\mathrm m\", which is inclined at the angle \"\alpha=7^\circ\" with respect to the horizontal. The approximately semi-circular cross-section of the channel, which is constant over the entire length, is “bounded upwards<\/em>” by an “imaginary<\/em>” straight line. The distance it generates has length \"d= 430\mathrm{mm}\" as shown in Figure 2 below:[\/vc_column_text][vc_single_image image=”1530″ img_size=”large” alignment=”center”]Figure 2: Experimental set-up in the Adventure Land[\/vc_single_image][vc_column_text]There are small obstacles in the “bottom<\/em>” of the Galilei trough so that a ball starting at a right angle (to the trough) and at the top of the right edge (in the direction of travel) of the Galilei trough does not touch it as it travels “down<\/em>“, if a suitable start of the ball is made. For this, there are 10 different starting possibilities as “small<\/em>” rectilinear depressions attached perpendicular to the Galilei trough (see figure 2). The experimenter now has the task of finding the launch option that allows the ball to travel downwards “undisturbed<\/em>“, i.e. without touching the obstacles. The path traversed by the sphere then approximately represents a “distorted” cosine-shaped<\/em> movement.[\/vc_column_text][vc_column_text]<\/p>\n

And now … the mathematics:<\/h3>\n

[\/vc_column_text][vc_column_text]The motion of the sphere is approximately the motion of a point mass on a two-dimensional surface, represented by a Cartesian \"xy\" coordinate system. The (positive) \"x\"-direction describes the direction of the valley bottom running from top to bottom and the (positive) \"y\"-direction describes the direction running perpendicular to it from the valley bottom to the right (in running direction) upper edge of the Galilei trough (see following figure 3):[\/vc_column_text][vc_single_image image=”1534″ img_size=”large” alignment=”center”]Figure 3: Introduction of the coordinates \"x\" and \"y\"[\/vc_single_image][vc_column_text]The motion of the sphere in \"x\"-direction is then determined — neglecting friction — by the displacement-time law<\/em> <\/p>\n

  <\/span>   <\/span>\"\[x(t) =\frac{g}{2}t^2\sin(\alpha) \quad(1)\]\"<\/p>\n

for \"t>0\" (where \"t\" denotes the time<\/em> in seconds). Where \"g= 9.81\mathrm{m}/\mathrm{s}^2\" is the acceleration due to gravity<\/em> and \"\alpha=7^\circ\" is the angle of inclination<\/em> of the Galilean trough. In the \"y\"-direction there is approximately a harmonic oscillation<\/em> (“cosine oscillation<\/em>“) of the form <\/p>\n

  <\/span>   <\/span>\"\[y(t)=A\cos(\omega t)\quad (2)\]\"<\/p>\n

before. Here \"\omega\" denotes the angular frequency<\/em> of this oscillation and \"A\" the amplitude<\/em> with \"A=\pi d/4\".<\/p>\n

Rearranging the equation \"(1)\" according to time, we get: <\/p>\n

  <\/span>   <\/span>\"\[(1)\Rightarrow t^2=\frac{2x(t)}{g\sin(\alpha)}\Rightarrow t=\sqrt{\frac{2x(t)}{g\sin(\alpha)}}\quad (3).\]\"<\/p>\n

Consequently, for all time points \"t\" one obtains the path-time law \"y(t)\" in \"y\"-direction as a function of the path-time law \"x(t)\" in \"x\"-direction: <\/p>\n

  <\/span>   <\/span>\"\[(2),\,(3)\Rightarrow y(t)=A\cos\left(\omega\sqrt{\frac{2x(t)}{g\sin(\alpha)}}\right)\]\"<\/p>\n

for \"t>0\", i.e. <\/p>\n

  <\/span>   <\/span>\"\[y=A\cos\left(\omega\sqrt{\frac{2x}{g\sin(\alpha)}}\right)\]\"<\/p>\n

and thus <\/p>\n

  <\/span>   <\/span>\"\[y=A\cos(\omega'\sqrt{x})\]\"<\/p>\n

with \"\omega'=\omega\sqrt{\frac{2}{g\sin(\alpha)}}\" and \"x\geq 0\". So \"y\" with the concrete values for the angle of inclination \"\alpha=7^\circ\", the amplitude \"A=\pi d/4=337.7\mathrm{mm}\" and the angular frequency \"\omega=6.9\mathrm{Hz}\": <\/p>\n

  <\/span>   <\/span>\"\[y= 337.7\mathrm{mm}\cos(0.2822\sqrt{x/\mathrm{mm}}).\]\"<\/p>\n

Then the trajectory of the ball rolling “downwards<\/em>” is approximately described by the curve shown in figure 4:[\/vc_column_text][vc_single_image image=”1538″ img_size=”large” alignment=”center”]Figure 4: \"xy\" diagram of the path curve[\/vc_single_image][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text] Galilei’s Tank As shown in Figure 1 below, the so-called Galilean trough consists of a rectilinear channel inclined by an angle and having an approximately semicircular cross-section.[\/vc_column_text][vc_single_image image=”1524″ img_size=”large” alignment=”center”]Figure 1: Galileo tub[\/vc_single_image][vc_column_text]In EXPERIENCE LAND MATHEMATICS, the trough, i.e. the Galilean trough, is a blue tread with a length of , which … Advanced text Galileo tank<\/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-4126","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4126","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=4126"}],"version-history":[{"count":16,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4126\/revisions"}],"predecessor-version":[{"id":4390,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/pages\/4126\/revisions\/4390"}],"wp:attachment":[{"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/media?parent=4126"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/erlebnisland-mathematik.de\/en\/wp-json\/wp\/v2\/folder?post=4126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}