Phyc 0 Len 9H Ranal Knemac In Len 1 9 we learned abu lnear mn knemac and he relanhp beween dplacemen, velcy, acceleran and me. In h len we wll learn abu ranal knemac. The man derence beween he w ype mn whle lnear knemac deal wh lnear dplacemen ( d ), ranal knemac deal wh angular dplacemen (). I. Ranal Mn and Angular Dplacemen When a rgd bdy rae hrugh a crcular pah, mu d arund a cener knwn a ax ran. A he rgd bdy rae weep u an angle abu he xed ax. Th angle knwn a he angular dplacemen. Angular dplacemen he angle () wep u by a lne pang hrugh any pn n he rang bjec and nerecng he ax ran perpendcularly. Angular dplacemen pve (+) cunerclckwe and negave (-) clckwe. O r P Reerence Lne In ranal mn all pn he rgd bdy mve n crcle abu me ax ran, O. Reerrng he dagram abve, he pn P mvng cuner clckwe arund he crcle, ha an angular dplacemen, relave he pve x-ax. Ne ha he lnear dplacemen ha he pn ha mved hrugh an arc () he crcle. Angle are cmmnly meaured n degree, bu he mahemac crcular mn much mpler we ue he an r angular meaure. One an () he angle wep u by an arc whe lengh equal he u. In ur crcle, pn P a dance r rm he ax ran. I ha mved a dance alng he arc he crcle and = r, hen = 1. In general, (1) r Anher way underand an relae hem ur prevu knwledge abu degree. Radan and degree are bh angular un. In prevu wrk yu learned ha here are 360 n a crcle. Fr an, he angle relaed he crcumerence a crcle C = r In an, here are an n a crcle. In her wrd = 360 1 = 57.3 Dr. Rn Lch 9H - 1 www.rucuredndependenlearnng.cm
Example 1 A wheel rae hrugh 70, wha he ran n an? Se up a ra x 360 70 3 x r x 4.71 Example ynchrnu aelle are pu n an rb whe u 4.3 x 10 7 m. The rb n he plane he equar, and he w adjacen aelle have an angular eparan =.00. Fnd he arc lengh ha eparae he aelle. ** Once he angle () cnvered n an, he relan = r can be ued. 360.00 x = 0.0349 = r = (0.0349 )(4.3 x 10 7 m) = 1.48 x 10 6 m II. Angular Velcy T decrbe ranal mn, we al make ue angular quane uch a angular velcy ( prnunced mega) and angular acceleran ( prnunced alpha). Angular velcy dened n a mlar way wh lnear velcy, bu nead dance raveled we ue he angular dance. Thu, he average angular peed,, dened a () Angular peed he angle hrugh whch he bdy ha raed n me and un are /. T nd u hw angular peed relaed lnear, n h cae angenal, peed, cnder an bjec already undergng unrm crcular mn. The bjec manan a cnan peed a revlve arund a crcle u (r) n a perd me (). Clearly, he al dance raveled arund he crcle crcumerence, r, and he me r ne cmplee ran, hen he cnan average peed d v r v (3) Dr. Rn Lch 9H - www.rucuredndependenlearnng.cm
and he angular peed gven by (4) Subung equan (4) n equan (3) we ge: r v r r v r (5) Equan (5) draw ur aenn an mpran dea. Cnder w peple andng n a mvng merry-g-rund; ne pern near he cener and ne near he uer edge he merry-g-rund. Bh peple experence he ame angular peed nce hey bh weep u equal angle n equal me nerval, bu he pern n he uer edge ha a greaer angenal peed, nce he weepng u a larger arc lengh n he ame amun me. Example 3 A gymna n a hghbar wng hrugh w revlun n a me 1.9. Fnd he average angular velcy he gymna n /. =.00 rev x ( an / revlun) = 1.6 an 1.6 1.90 = 6.63 / III. Angular Acceleran Angular acceleran, mlar lnear acceleran, dened a he change n angular velcy dvded by he me requred make h change. The angular acceleran,, 1 r (6) The un /. Example 4 A je wang ake rev' h engne. A he engne dle, he an blade rae wh an angular velcy +110 /. A he plane ake, he angular velcy he blade reache +330 / n a me 14. Fnd he angular acceleran he blade. ung equan (5) 1 330 110 14 = 16 / Dr. Rn Lch 9H - 3 www.rucuredndependenlearnng.cm
IV. Ranal Knemac Our underandng ranal velcy and acceleran ake n he realm ranal knemac. Ung ur prevu knemac equan we are able derve equan explanng ranal mn. The mahemacal rm he lnear and ranal knemac equan are dencal wh he excepn he replacemen he lnear varable wh ranal varable. Quany Ranal Mn Lnear Mn Dplacemen d Inal velcy v Fnal velcy v Acceleran a Tme Ranal Mn Lnear Mn ( = cnan) (a = cnan) v v v v d d v a a 1 1 v v ad Example 5 The blade an elecrc blender are whrlng wh an angular velcy +375 / whle he puree bun depreed. When he blend bun preed, he blade accelerae and reach a greaer angular velcy n +44.0 (even revlun). The angular acceleran ha a cnan value +1740 /. Fnd he nal angular velcy he blade. 54 (375 ) (1740 )(44.0) Dr. Rn Lch 9H - 4 www.rucuredndependenlearnng.cm
V. Hand-n Agnmen 1. A bcycle dmeer (whch meaure dance raveled) aached near he wheel hub and degned r 70 cm wheel. Wha happen yu ue n a bcycle wh 60 cm wheel?. Wha are he llwng angle n an: (a) 30, (b) 90, and (c) 40? (0.54, 1.57, 7.33) 3. A laer beam dreced a he mn, 380,000 km rm earh. The beam dverge a an angle 1.8 x 10-5. Hw large a p wll make n he mn? (6.8 km) 4. A bcycle wh 68-cm-dameer re ravel.0 km. Hw many revlun d he wheel make? (936) 5. A 0-cm-dameer grndng wheel rae a 000 rpm. a. Calculae angular velcy n /. (.1 x 10 /) b. Wha he lnear peed a pn n he edge he grndng wheel? (41.9 m/) 6. A 70-cm-dameer wheel rang a 100 rpm brugh re n 15. Calculae angular acceleran. (-8.4 / ) 7. A 33-rpm phngraph recrd reache raed peed.8 aer urnng n. Wha wa he angular acceleran? (+1. / ) 8. Calculae he angular velcy he earh (a) n rb arund he un, and (b) abu ax. (1.99 x 10-7 /, 7.7 x 10-5 /) 9. Wha he lnear peed a pn (a) n he equar, (b) a a laude 50 N, due he earh' ran? (463 m/, 98 m/) 10. An aumble engne lw dwn rm 4500 rpm 1000 rpm n 6.5. Calculae (a) angular acceleran, (b) angular dplacemen, and (c) he al number revlun he engne make n h me. (-56 /, 187, 3.0 x 10 rev) 11. A cenruge accelerae a +10 / rm re 10,000 rpm. Wha wa he angular dplacemen? (4.6 x 10 3 ) 1. A phngraph urnable reache raed peed 33 rpm aer makng 1.5 revlun. Wha wa he angular acceleran? (0.63 / ) 13. A 40-cm-dameer wheel accelerae unrmly rm 80 rpm 300 rpm n 3.6. Hw ar wll a pn n he edge he wheel have raveled n h me? (14.3 m) Dr. Rn Lch 9H - 5 www.rucuredndependenlearnng.cm