5.1 Angles and Their Measure

Transcription

1 5. Angles and Their Measure Secin 5. Nes Page This secin will cver hw angles are drawn and als arc lengh and rains. We will use (hea) represen an angle s measuremen. In he figure belw i describes hw yu knw if an angle is psiive r negaive. The verex f he angle is a he rigin f a recangular crdinae sysem. The psiive x axis is always where an angle is measure frm, and his is called he iniial side. An angle drawn his way is said be in sandard frm. An angle ha ges cunerclckwise is always psiive, and clckwise angles are negaive. Angles are measured a cuple f differen ways. The firs uni f measuremen is a degree in which 60 (degrees) is equal ne revluin. Ms likely he reasn why we use 60 is frm he Babylnians, whse year is based n 60 days. EXAMPLE: Draw 60 in sandard psiin. Firs we sar by drawing a hriznal line. Then ur angle is measured frm his line. Since he angle is psiive we need g in he cunerclckwise direcin. Our answer lks like: EXAMPLE: Draw 0 in sandard psiin. Firs we sar by drawing a hriznal line. Then ur angle is measured frm his line. Since he angle is negaive we need g in he clckwise direcin. Our answer lks like:

2 EXAMPLE: Draw 95 in sandard psiin. Secin 5. Nes Page Firs we sar by drawing a hriznal line. Then ur angle is measured frm his line. Since he angle is psiive we need g in he cunerclckwise direcin. If we ake 95 and divide i by 60 we will ge wih 5 lef ver. This means his angle ges arund wice and hen ges an exra 5 degrees. Ne he fllwing is hw yu draw and angle ha ges arund mre han nce: Alhugh he bk alks abu degrees, minues and secnds we are ging ignre his since hese unis are n really used excep fr in surveying. Anher uni f measuremen fr angles is radians. In radians, is equal ne revluin. S a cnversin beween radians and degrees is = 60, r =. When cnvering frm degrees radians: Muliply yur degrees by When cnvering frm radians degrees: Muliply yur radians by EXAMPLE: Cnver 60 radians. We will ake 60 and muliply i by and yu will ge: 60. This reduces. EXAMPLE: Cnver 05 radians. We will ake 05 and muliply i by and yu will ge: 50. This reduces 9. EXAMPLE: Cnver We will ake in degrees. and muliply i by and yu will ge:. This reduces 0. EXAMPLE: Cnver We will ake in degrees. and muliply i by and yu will ge:. This reduces 70.

3 Arc lengh his is he lengh f he arc beween he w lines shwn wih. Secin 5. Nes Page The equain is S = r, where S is he arc lengh, r is he radius, and MUST be measured in RADIANS! The is als called he cenral angle. EXAMPLE: Find he arc lengh f a secr whse radius is inches and whse cenral angle is. Using ur frmula S =, s S = inches. S = r we knw ha r = and =. Puing his in he frmula we will ge: EXAMPLE: Find he arc lengh f a secr whse radius is 7 inches and whse cenral angle is 5. Using ur frmula S = r we knw ha r = 7. We will n use 5 degrees as ur angle because his is n in degrees. We need muliply i by ge i in radians: 5. Reducing his we ge =. Puing his in he frmula we will ge: 7 7 S =, s S = inches. Area f a Secr his is he area f he piece f pie shwn belw. The equain is A = r, where A is he area f he secr, r is he radius, and again MUST be measured in RADIANS! EXAMPLE: Find he area f a secr wih a radius f inches wih a cenral angle f Using ur frmula A = r we knw ha r = and =. Puing his in he frmula will give us: A =. S we have 6 A = which simplifies A = square inches..

4 Secin 5. Nes Page EXAMPLE: Find he area f a secr wih a radius f 6 inches wih a cenral angle f 50. Using ur frmula A = r we knw ha r = 6. We will n use 50 degrees as ur angle because his is n in degrees. We need muliply i by ge i in radians: Reducing his we ge =. 6 5 Puing his in he frmula will give us: A =. S we have A = which simplifies 6 6 A = 5 square inches. Angular Speed (ω ) Angular speed is he angle ha can be swep u in ime. The frmula is: ω = where is in radians. Linear Speed (v) This is he speed a which a pin n he circle is mving. I is measured by wha arc lengh is raveled in ime. The frmula is: v = rω whereω is measured in radians per uni f ime. EXAMPLE: A circular gear raes a a rae f 75 rpm (revluins per minue). Wha is he angular speed (in radians per minue)? Wha is he linear speed f a pin n he gear mm frm he cener? In rder find he angular speed we need find he angle in radians. We knw ha revluin is radians, s we can muliply 75 by ge 50 radians. This is ur angle. Our uni f ime is minue, 50 s we can use he frmula ω =. We will ge ω = = 50 radians per minue. Fr he linear speed we will use millimeers per minue. v = rω wih r = and ω = 50. Yu will ge v = 50, s v = 50 EXAMPLE: A wheel raes a a rae f 60 per secnd. Wha is he angular speed (in radians per minue)? Wha is he linear speed f a pin n he gear 0 cm frm he cener? We need firs change 60 in radians by muliplying by. Yu will ge: 60 which reduces radians. Our uni f ime is secnd, s we can use he frmula ω =. We will ge ω = = radians per secnd. The quesin is asking us fr radians per minue, s we need muliply ur answer by 60. Yu will ge ω = 70 radians per minue. Nw we can find he linear speed. We will use v = rω wih r = 0 and ω = 70. Yu will ge v = 0 70

5 which is v = 600 cenimeers per secnd. Secin 5. Nes Page 5 Pulleys When we have pulleys, he bel ha drives he pulleys is always mving a he same speed, s acually bh pulleys will have he same linear speed. This is impran when slving hese kind f prblems. EXAMPLE: A -inch radius pulley is raing a rpm. Deermine he rpm f he larger pulley, which has a radius f 8 inches (see figure) Firs we need find he linear speed f he small pulley. Then we can use his infrmain find he rpm f he larger pulley. Fr he small pulley we will muliply rpm by ge 6 radians per minue. Then since r = we can pu hese in he frmula v = rω and we will have v = 6 s v = radians per minue. We knw ha he larger pulley will have he same linear speed as he smaller pulley. We will sill use v = rω. We knw v = frm he previus par and als r = 8. The nly hing lef slve fr is ω. Afer subsiuing we ge: = 8ω. Slving his we ge ω =, r ω =. The quesin is asking us express 8 his as rpm, s nw we mus divide his by. Yu will ge:, which is rpm. EXAMPLE: T apprximae he speed f he curren f a river, a circular paddle wheel wih a radius f fee is lwered in he waer. If he curren causes he wheel rae a a speed f 0 rpm, wha is he speed f he curren in miles per hur ( mile = 580 fee). We can cnver 0rpm radians per minue by muliplying i by ge 0 radians per minue. This is ur angular speed, ω. The waer mving is a linear speed, s we need use he frmula v = rω. We knw r = and ω = 0, s v = 0, r v = 80 fee per minue. We need nw change his in miles per hur, s we need use sme dimensinal analysis. Basically we ge he unis cancel by using he apprpriae cnversins: 80 f 60min min hr mile 580 f =.86 mph