Secion. Linear and Angular Speed 7. From exercise, =. A= r A = ( 00 ) (. ) = 7,00 in 7. Since 7 is in quadran IV, he reference 7 8 7 angle is = =. In quadran IV, he cosine is posiive. Thus, 7 cos = cos =. Convering 7 o degrees, we have 7 7 = ( 80 ) =. The reference angle is 80 =. Thus, 7 cos = cos = cos =. is coerminal wih 7 + = + =. Since 7 is in quadran III, he reference angle is 7 7 = =. In quadran III, he sine is negaive. Thus, 7 sin = sin = sin =. Convering 7 o degrees, we have 7 7 = ( 80 ) = 0. The reference angle is 0 80 = 0. Thus, 7 sin = sin = sin 0 = sin 0 =. is coerminal wih =. So an = an = 0 Convering o degrees, we have 80 = 0. The reference angle is 0º 0º = 80º. Thus an = an 0 = an80 = 0. 0., ; sins = Recall ha sin = and in quadran II, sin s is posiive. Therefore, sin = sin =, and hus, s =. Secion. Linear and Angular Speed. The circumference of he uni circle is. radian per, = = =. The circumference of he uni circle is. v = uni per, s = unis s v = = =. r = 0 cm, (a) radian per, = = = (b) s = r s = 0 = 0 cm r 0 0 (c) v = v = = = cm per. r = 0 cm,. (a) (b) (c) radian per, = 0 = = = 0 0 s = r s = 0 = cm r 0 v = v = = = cm per per, = = = Copyrigh 0 Pearson Educaion, Inc.
8 Chaper Radian Measure and he Uni Circle. 7.. 0. per min, = min = = =, = 8 = = = radian per 8 8 =, = 0 = = radian per 0 0 = radian, radian per min 7 = = 7 7 = = = min =, radian per min 8 8 = = 8 7 8 = 7 = = min 8. =.87, =.7.87 0.80 radian per.7. =., =...078 per... 0.07 radian per min, =.88 min 0.07 =.88 = 0.07.88 0.77. per min, =.7 min. =.7 =..7 0.. r = m, per v = = 8 m per. r = 8 m, per 7 v = 8 = cm per 7. v = m per, r = m = ω per v = 8 f per, r = f 8 = ω per. v = 07.7 m per, r = 7 m 07.7 = 7ω 07.7.8 per 7 0. r =. cm, 0.7 radian per v =. 0.7. cm per. r = cm, per, = s = ( ) = 8 cm. r = yd, per, = s = ( ) = yd. s = cm, r = cm, radian per = = = =. s = m, r = m, per = = = = Copyrigh 0 Pearson Educaion, Inc.
Secion. Linear and Angular Speed. s = km, r = km, = = ω = 8ω = radian per 8. s = 8 m, r = m, = 8 = ω 8 8 = ω = radian per 8 7. The hour hand of a clock moves hrough an angle of (one complee revoluion) in hours, so = = radian per hr. The ond hand makes one revoluion per minue. Each revoluion is, so we = per min. There have are 0 onds in min, so = 0 0 radian per.. The minue hand makes one revoluion per hour. Each revoluion is, so we = per hr. There have are 0 minues in hour, so = 0 0 radian per min. 0. The line makes 00 revoluions per minue. Each revoluion is, so we have 00 = 00 per min. The minue hand of a clock moves hrough an angle of in 0 min, and a he ip of he minue hand, r = 7 cm, so we have r 7 ( ) 7 v = v = = cm per min 0 0. The ond hand makes one revoluion per minue. Each revoluion is, and a he ip of he ond hand, r = 8 mm, so we have v = 8 = mm per min. There are 0 onds in min, so v = = mm per. 0. The flywheel making roaions per min = 8 urns hrough an angle in minue wih r = m. So, r 8 ( ) v = v = = 8 m per min. The poin on he read of he ire is roaing imes per min. Each roaion is. Thus, we have = 70 per min. Since, we have v = 8 70 = 0 cm per min.. A 00 roaions per min, he propeller urns hrough an angle of = 00( ) = 000 in min wih r = =. m, we have r.( 000 ) v = v = = 00 m per min.. The poin on he edge of he gyroscope is roaing 80 imes per min. Each roaion is. 80 = 0 per min, so v = 8 0 =,880 cm per min 7. A revoluions per minue, he bicycle ire is moving ( ) = 0 per min. This is he angular velociy ω. The linear velociy of he bicycle is =.0 0 = 0 in. per min. Conver his o miles per hour: 0 in. 0 min f mi v = min hr in. 80 f. mph Mars will make one full roaion (of ) during he course of one day. Thus, hr. hr 0. radian. (a) (b) = = radian radian per day = radian per hr = radian per hr 80 Copyrigh 0 Pearson Educaion, Inc.
0 Chaper Radian Measure and he Uni Circle (c) v = (,000,000) 7,000 mph 80 0. (a) The earh complees one revoluion per day, so i urns hrough = in ime = day = hr. So, we have: = per day = radian per hr = radian per hr (b) A he poles, r = 0, so = 0. (c) A he equaor, r = 00 km. So, v = 00 =,800 km per day,800 = km per hr km per hr (d) Salem roaes abou he axis in a circle of radius r a an angular velociy per day. r sin = 00 r = 00sin = 00 = 00 km v = 00 ( ) 00 km per day 8,000 km per day 00 v = km per hr 77 km per hr 00 km per hr. (a) Since s = cm of bel go around in = 8, he linear velociy is s 8 v = v = =. cm per 8 (b) Since he -cm bel goes around in 8, we have = (.) ω 8 8 = (.) ω 8 0. radian per.. The larger pulley roaes imes in or imes per. Thus, is angular velociy is = per. The 8 linear velociy of he bel is v = = cm per 8 To find he angular velociy of he smaller pulley, use v = cm per and r = 8 cm. = 8ω = per 8 8. = 0 per min 0 = per 0 7 = per 7. = r r =..7 cm 7. Le s = he lengh of he rack on he arc. Firs, convering 0 o, we have 0 = 0 radian = radian. 80 The lengh of rack, s, is he arc lengh when = and r = 800 f. s = r s = 800 = 00 f (coninued on nex page) Copyrigh 0 Pearson Educaion, Inc.
Chaper Review Exercises (coninued) Expressing he velociy v = 0 mph in f per, we have 0 v = 0 mph = mi per 00 ( 0) 80 = f per = f per 00 s 00 v = = = 00 00 00 = =. In one minue, he propeller makes 000 revoluions. Each revoluion is, so we have 000( ) = 0,000 per min. There are 0 in a minue, so 0,000 00 =. per 0. r = f; per Chaper v = = f per Review Exercises. A cenral angle of a circle ha inerceps an arc of lengh imes he radius of he circle has a measure of.. (a) Since.7 and., we have < <. Thus, he erminal side is in quadran II. (b) Since. and.7, we have < <. Thus, he erminal side is in quadran III. (c) Since.7 and., we have > >. Thus, he erminal side is in quadran III. (d) Since.8 and 7. we have < 7 <. Thus, he erminal side is in quadran I.. To find a coerminal angle, add or subrac muliples of. Three of he many possible answers are +, +, and +.. To find a coerminal angle, add or subrac muliples of. Since n represens any ineger, he expression + n generaes all coerminal angles wih an angle of radian.. = radian = radian 80. 7.. 0....... 0 = 0 radian = 80 7 = 7 radian = 80 0 = 0 radian = 80 0 800 = 800 radian = 80 7 00 = 00 radian = 80 = = = = 0 0 8 8 = = 80 = = = = 0 8 8 = = 7 7. Since = roaion, we have 0 = ( ) =. Thus, s = r s = = in. Copyrigh 0 Pearson Educaion, Inc.