6.6 Moments and Centers of Mass

Transcription

1 th oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder two sses d tht re ttched to rod of eglgle ss o opposte sdes of fulcru d t dstces d d d fro the fulcru. B the Lw of the Lever ( eperetl fct dscovered Archedes), the rod wll lce f d = d. () (Thk of lghter perso lcg hever oe o seesw sttg frther w fro the ceter.). Suppose the rod les log the -s wth t d t d the ceter of ss t. Equto () gves = + = + = + () + The uers d re clled the oets of sses d (wth respect to the org), + s clled the oet of the sste (wth respect to the org), d + s the totl ss. 3. sses Alog Le I geerl, f we hve sste of prtcles wth sses, =,, locted t the pots,, o the -s, t c e show slrl tht the ceter of ss of the sste s locted t (3) = The uer s clled the oet of the sste wth respect to the org, d = s the totl ss of the sste. The ceter of ss s tht pot where sgle prtcle of ss would hve the se oet s the sste.

2 4. sses Dstruted over Ple Rego Now we cosder sste of prtcles wth sses,, B log wth the oe-desol cse, we defe the oet of the sste out the -s to e = locted t the pots,,, the -ple. = d the oet of the sste out the -s to e =. esures the tedec of the sste to rotte out the -s d the -s. As the oe-desol cse, the coordtes (, ) foruls (4) = = esures the tedec to rotte out of the ceter of ss re gve ters of the oets the = =. The ceter of ss s tht pot where sgle prtcle of ss would hve the se oets s the sste. Eercse Fd the oets d ceter of ss of the sste of ojects tht hve sses 3, 4, d 8 t the pots 3,. (, ),, ( ), d

3 3 5. Pltes Bouded Oe Curve Cosder flt plte wth ufor dest δ tht occupes rego R the ple. We wsh to locte the ceter of ss of the plte, whch s clled the cetrod of R (whe the dest s costt). We ll use the followg phscl prcple: The setr prcple If rego R s setrc out le l, the the ceter of ss of R les o l. Therefore, the ceter of ss of rectgle s ts ceter. Suppose the rego R les etwee the les = d =, ove the -s, d eeth the grph of f, where f s cotuous fucto. Dvde the tervl [, ] to sutervls [ ],,,, of equl legth Choose the sple pots, the dpot of the th sutervl,, d costruct the rectgulr pproto of the rego. The ceter of ss of the th pprotg rectgle R s ts ceter, f. The ss of the rectgle R s Fd the oet of the rectgle ( ) δ f ( ss=dest tes re). R out the -s, δ R = f Addg these oets, we ot the oet of the polgol pproto to R, the tkg the lt s we ot the oet of R out the -s: Fd the oet of the rectgle = δ = l δf = f d R out the -s, R = δ f f = δ f Addg these oets, we ot the oet of the polgol pproto to R, the tkg the lt s we ot the oet of R out the -s: δ = lδ f f d = =

4 The ss of the plte s the product of ts dest d ts re: 4 δa δ f d = = The ceter of ss of R s gve the coordtes: δ f d = = ; f the dest s costt, the δ f d f d f d = = = δ f d f d A δ f d = = ; f the dest f costt, the δ f d f d f d = = = δ f d f d A Eercse Fd the ceter of ss of secrculr plte of ufor dest of rdus r.

5 5 Eercse 3 (Eple /6.6) The trgulr plte show hs costt dest of δ = 3 g/ c. () Fd the plte s oets out the es. () Fd the plte s ss. (c) Fd the coordtes of the ceter of ss. Eercse 4 (Eple /6.6) Fd the ceter of ss of th plte coverg the rego ouded ove the prol 4, s δ =. = d elow the -s. Assue the dest of the plte t the pot

6 6 6. Pltes Bouded Two Curves Suppose plte covers rego R tht les etwee two curves = f d = g, where f g d. The ceter of ss of R s gve the coordtes: δ [ f g ] d = = ; f the dest s costt, the δ f [ g ] d f [ g ] d = = δ [ f g ] d [ f g ] d δ f g d = = ; f the dest f costt, the δ f g d f g d = = δ [ f g ] d [ f g ] d Eercse 5 (Eercse 4/6.6) Fd the ceter of ss of th plte of costt dest coverg the rego eclosed the prols = 3 d =.

7 Eercse 6 (Eercse 7/6.6) The rego ouded the curves out the -s to geerte sold. ) Fd the volue of the sold. 7 4 =± d the les = d = 4 s revolved ) Fd the ceter of ss of th plte coverg the rego f the plte s dest t the pot (, ) s δ =. c) Sketch the plte d show the ceter of ss our sketch. Eercse 8 (Eercse 3/6.6) Fd the cetrod of the th plte ouded = π. g = 0, f = + s, = 0 d

8 The Theores of Pppus 8 Theore Pppu s Theore for Volues If ple rego s revolved oce out le the ple tht does ot cut through the rego s teror, the the volue of the sold t geertes s equl to the rego s re tes the dstce trvele d the rego s cetrod durg the revoluto. If d s the dstce fro the s of revoluto to the cetrod, the V = π da. Theore Pppu s Theore for Surfce Ares If rc of sooth ple curve s revolved oce out le tht does ot cut through the rc s teror, the the re of the surfce geerted the rc equls the legth L of the rc tes the dstce trveled the rc s cetrod durg the revoluto. If d s the dstce fro the s of revoluto to the cetrod, the S = π dl. Eercse 9 (Eerse 35/6.6) The squre rego wth vertces (0,), (,0), (4,) d (,4) s revolved out the -s to geerte sold. Fd the volue d surfce re of the sold.