Finding the Centroid of Volume
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- Sharyl Gardner
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1 9 Fndng the Centod of olume ef: Hbbele 9., Bedfod & Fowle: Statcs 7.4 The centod of volume s the geometc cente of a bod. If the denst s unfom thoughout the bod, then the cente of mass and cente of gavt coespond to the centod of volume. The defnton of the centod of volume s wtten n tems of atos of ntegals ove the volume of the bod. d d d d d d Ethe analtcal o numecal ntegaton methods can be used to evaluate these ntegals and compute the centod of volume fo the bod. The ntegals ove volume take slghtl dffeent foms depng on the coodnate sstem ou use. Catesan Coodnates d Clndcal Coodnates d θ θ Sphecal Coodnates d φ φ θ θ d d d d dθ d sn φ d dθ dφ These ntegals can be evaluated usng analtcal o numecal ntegaton technques. Eample: Centod of a Hemsphee Fnd the centod of volume fo a hemsphee of adus 7 cm.
2 ote: Ths smple eample wll allow us to check ou esults aganst publshed values. Fo eample, Hbbele shows (nsde back cove) that the centod fo a hemsphee estng on the plane fomed b the and aes to be located at 0, 0, 3/8. Soluton: umecal Integaton The volume of the hemsphee can be calculated b usng the equaton fo the aea of a ccle, and ntegatng n onl one decton,. To use ths appoach, fst note that the aea of the base s easl calculated, as A π. A π The aea of the ccle fomed b an - plane though the hemsphee s calculated as a π. a π whee the value of deps on. The elatonshp between and s eadl detemned, snce and ae two sdes of a ght tangle n whch the hpotenuse s the adus of the hemsphee,. We then ntegate the aea of the ccle fom 0 to. d a d ( π ) d π ( ) To appomate the esult usng numecal ntegaton methods, ewte the ntegal as a sum and the d as a. Gaphcall (fo the volume calculaton n the denomnato) ths s equvalent to appomatng the volume of the hemsphee b a sees of stacked dsks of thckness. One such dsk s shown n the fgue below. d
3 π Hee, the value of at the cuent (pont ) value of has been used to calculate the volume of the dsk. Snce 7 cm n ths eample, we mght t a of cm, and calculate the volumes of seven dsks ( 7). Fo each dsk, the value of can be calculated usng The value of the coespondng adus s then The fomula fo volume can then be wtten n tems of, as π ( ) The sum of all of the volumes of all seven dsks s an appomaton of the volume of the hemsphee. π [ ( ) In MATLAB, ths calculaton looks lke ths:» 7; %adus» 7; %umbe of ntevals» deltaz /; %Inteval spacng» %Sum (p * (^ - (*deltaz)^) * deltaz) fo,,...,» _pedcted 0; %Intale pedcted volume to eo» fo : _pedcted _pedcted + p * (^ - (*deltaz)^) * deltaz;» _pedcted _pedcted We can see how good (o bad) the appomaton s b calculatng the actual volume of the hemsphee. _actual /3 * p * ^3 _actual The appomaton usng onl seven dsks s not too good. If we use moe dsks, sa 70, the appomaton of the volume s qute a bt bette.
4 » 7; %adus» 70; %umbe of ntevals» deltaz /; %Inteval spacng» %Sum (p * (^ - (*deltaz)^) * deltaz) fo,,...,» _pedcted 0; %Intale pedcted volume to eo» fo : _pedcted _pedcted + p * (^ - (*deltaz)^) * deltaz;» _pedcted _pedcted To calculate the centod usng numecal methods, smpl eplace the ato of ntegals b the numecal appomaton: d d π π [ ( ) [ ( ) The eta has been ncluded n the numeato as (I * deltaz). In MATLAB, the calculaton of the centod looks lke ths:» numeato 0; %Intale numeato to eo» denomnato 0; %Intale denomnato to eo» fo : %Sum both numeato and denomnato numeato numeato + (*deltaz) * p * (^ - (*deltaz)^) * deltaz; denomnato denomnato + p * (^ - (*deltaz)^) * deltaz;» _ba numeato / denomnato _ba.6530 We can use the analtcal esult to check ou answe.» _ba_act 3/8 * _ba_act.650
5 Annotated MATLAB Scpt Soluton %Defne the Sstem 7; %adus 70; %umbe of dsks deltaz /; %Calculate dsk thckness fpntf('system\n') fpntf('adus %3.f cm\t\t %3.0f\t\tdeltaZ %3.f\n\n',,,deltaZ) %Calculate the olume %Sum (p * (^ - (*deltaz)^) * deltaz) fo,,..., _pedcted 0; %Intale pedcted volume to eo fo : _pedcted _pedcted + p * (^ - (*deltaz)^) * deltaz; fpntf('calculate THE OLUME\n') fpntf('pedcted olume %3.3f cm^3\n',_pedcted) _actual /3 * p * ^3; fpntf('actual olume %3.3f cm^3\n\n',_actual) %Calculate the Centod numeato 0; %Intale numeato to eo denomnato 0; %Intale denomnato to eo fo : %Sum both numeato and denomnato numeato numeato + (*deltaz) * p * (^ - (*deltaz)^) * deltaz; denomnato denomnato + p * (^ - (*deltaz)^) * deltaz; _ba numeato / denomnato; fpntf('calculate THE CETOID\n') fpntf('calculated _ba %3.3f cm\n',_ba) _ba_act 3/8 *; fpntf('actual _ba %3.3f cm\n',_ba_act)
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