CIRCULAR MOTION. b gb g CHAPTER 5 DYNAMICS OF UNIFORM PROBLEMS

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1 HAPTER 5 DYNAMIS O UNIORM IRULAR MOTION PROBLEMS 1. SSM REASONING The peed of the plne i ien by Eqution 5.1: π / T, whee T i the peiod o the time equied fo the plne to omplete one eolution. SOLUTION Solin Eqution 5.1 fo T we he π π( 850 m) T 110 m / 160. REASONING AND SOLUTION The peiod i T π / o the diu of the tk i T/(π) (17 m/)(118 )/(π) 30 m 3. REASONING AND SOLUTION The time fo one eolution, the peiod T, n be found fom T π / π (0.053 m)/(1 m/) REASONING AND SOLUTION Sine the peed of the objet on nd off the ile emin ontnt t the me lue, the objet lwy tel the me ditne in equl time intel, both on nd off the ile. uthemoe ine the objet tel the ditne OA in the me time it would he moed fom O to P on the ile, we know tht the ditne OA i equl to the ditne lon the of the ile fom O to P. The iumfeene of the ile i π π(3.6 m).6 m. The OP ubtend n nle of θ 5 ; theefoe, ine ny ile ontin 360, the OP i 5/360 o 6.9 pe ent of the iumfeene of the ile. Thu, b b OP.6 m m nd, fom the ument ien boe, we onlude tht the ditne OA i 1.6 m.

2 hpte 5 Poblem SSM REASONING AND SOLUTION In eh e, the mnitude of the entipetl eletion i ien by Eqution 5., /. Theefoe, A B A B / A / B Sine eh bot expeiene the me entipetl eletion, of the peed ie A A 10 m m B B A B 1. Solin fo the tio 6. REASONING The entipetl eletion i ien by Eqution 5. /. The lue of the diu i ien, o to detemine we need infomtion bout the peed. But the peed i elted to the peiod T by (π )/T, odin to Eqution 5.1. We n ubtitute thi expeion fo the peed into Eqution 5. nd ee tht b T π / 4π T SOLUTION To ue the expeion obtined in the eonin, we need lue fo the peiod T. The peiod i the time fo one eolution. Sine the ontine i tunin t.0 eolution pe eond, the peiod i T (1 )/(.0 eolution) Thu, we find tht the entipetl eletion i b b 4π 4π 0. 1 m T m /

3 180 DYNAMIS O UNIORM IRULAR MOTION 7. REASONING AND SOLUTION Let epeent the lenth of the pth of the pebble fte it i eleed. om oneptul Exmple, we know tht the pebble will fly off tnentilly. Theefoe, the pth i pe-pependiul to the diu of the ile. Thu, the ditne,, nd d fom iht tinle with hypotenue d hown in the fiue t the iht. om the fiue we ee tht Tet d α Pebble o 1 o α d α o 1 H G I K J θ uthemoe, fom the fiue, we ee tht α +θ Theefoe, θ 145 α REASONING AND SOLUTION. At the equto peon tel in ile whoe diu equl the diu of the eth, R e m, nd whoe peiod of ottion i T 1 d We he πr e /T 464 m/ The entipetl eletion i b 464 m / m m / b. At 30.0 ltitude peon tel in ile of diu, Thu, R e o m π/t 40 m/ nd /.9 10 m/

4 hpte 5 Poblem SSM REASONING The mnitude of the entipetl eletion of ny point on the heliopte blde i ien by Eqution 5., /, whee i the diu of the ile on whih tht point moe. om Eqution 5.1: π / T. ombinin thee two expeion, we obtin 4 π T All point on the blde moe with the me peiod T. SOLUTION The tio of the entipetl eletion t the end of the blde (point 1) to tht whih exit t point loted 3.0 m fom the ente of the ile (point ) i 1 4π 1 / T m 4π / T 3.0 m 10. REASONING AND SOLUTION The mple mke one eolution in time, T, ien by T π/. The peed i The peiod i ( m)( )(9.80 m/ ) o tht 55.3 m/ T π ( m)/(55.3 m/) min The numbe of eolution pe minute 1/T e/min. 11. SSM REASONING In Exmple 3, it w hown tht the mnitude of the entipetl eletion fo the two e e. Rdiu 33 m 35 m / Rdiu 4 m 48 m / Aodin to Newton' eond lw, the entipetl foe i m (ee Eqution 5.3). SOLUTION. Theefoe, when the led undeoe the tun of diu 33 m, m ( 350 k)(35 m / ) N b. Similly, when the diu of the tun i 4 m,

5 18 DYNAMIS O UNIORM IRULAR MOTION m ( 350 k)(48 m / ) N 1. REASONING The entipetl foe tht t on the kte i m /, odin to Eqution 5.3. Thi expeion n be oled dietly to detemine the m m. SOLUTION Solin Eqution 5.3 fo the m m ie b b b m 460 N 31 m 14 m / 73 k 13. REASONING AND SOLUTION The entipetl foe i poided by the mximum foe of tti fition, f mx m /. The new fitionl foe, f, i one-thid the oiinl lue o f m / f mx /3 m /3. Solin fo the new eloity,, we obtin () (1 m/) 1 m/ REASONING The peon feel the entipetl foe tin on hi bk. Thi foe i m /, odin to Eqution 5.3. Thi expeion n be oled dietly to detemine the diu of the hmbe. SOLUTION Solin Eqution 5.3 fo the diu ie b b m 83 k 3. m / 560 N 1. 5 m 15. REASONING AND SOLUTION. In tem of the peiod of the motion, the entipetl foe i witten 4π m/t 4π (0.010 k)(0.100 m)/(0.500 ) N

6 hpte 5 Poblem 183 b. The entipetl foe ie the que of the peed. Thu, doublin the peed would inee the entipetl foe by fto of REASONING AND SOLUTION Initilly, the tone exeute unifom iul motion in ile of diu whih i equl to the diu of the tie. At the intnt tht the tone flie out of the MAX tie, the foe of tti fition jut exeed it mximum lue f µ (ee Eqution 4.7). N The foe of tti fition tht t on the tone fom one ide of the ted hnnel i, theefoe, MAX f (1.8 N) 1.6 N nd the mnitude of the totl fitionl foe tht t on the tone jut befoe it flie out i 1.6 N 3. N. If we ume tht only tti fition upplie the entipetl foe, then, 3. N. Solin Eqution 5.3 ( m / ) fo the diu, we he 3 m k (13 m / ) 3. N 0.31 m 17. SSM WWW REASONING Let 0 be the initil peed of the bll it bein it pojetile motion. Then, the entipetl foe i ien by Eqution 5.3: m 0 /. We e ien the lue fo m nd ; howee, we mut detemine the lue of 0 fom the detil of the pojetile motion fte the bll i eleed. In the bene of i eitne, the x omponent of the pojetile motion h zeo eletion, while the y omponent of the motion i ubjet to the eletion due to ity. The hoizontl ditne teled by the bll i ien by Eqution 3.5 (with x 0 ): x t ( o θ) t 0 x 0 with t equl to the fliht time of the bll while it exhibit pojetile motion. The time t n be found by onidein the etil motion. om Eqution 3.3b, + t y 0 y y Afte time t, y 0 y. Aumin tht up nd to the iht e the poitie dietion, we he nd t y 0 inθ 0 y y

7 184 DYNAMIS O UNIORM IRULAR MOTION x ( o θ) 0 H G Uin the ft tht in θ o θ in θ, we he 0 y inθ I K J x 0 oθ inθ 0 inθ y y (1) Eqution (1) (with upwd nd to the iht hoen the poitie dietion) n be ued to detemine the peed 0 with whih the bll bein it pojetile motion. Then Eqution 5.3 n be ued to find the entipetl foe. SOLUTION Solin eqution (1) fo 0, we he 0 x y in θ (. m)( 9.80 m / ) in (41 ) 9. 3 m / Then, fom Eqution 5.3, m 0 (7.3 k)(9.3 m / ) 1.8 m 3500 N 18. REASONING AND SOLUTION. The etil omponent of the tenion in the ble uppot the weiht of the hi nd oupnt. Newton' eond lw pplied to the etil ie Then T o 65.0 m 0 T m/o N b. The hoizontl omponent of the tenion i the entipetl foe o T in 65.0 m / The diu i found fom the dwin: (1.0 m) in m. Now,

8 hpte 5 Poblem 185 T in 65.0 o m (10.9 m)(5.1 x 10 3 N ) in 65.0 o 0 k 15 m / 19. REASONING AND SOLUTION The entipetl eletion of the blok i / (8 m/) /(150 m) 5. m/ The nle θ n be obtined fom θ tn HG I K J tn 1 1 HG 5. m / 9.80 m / I 8 KJ 0. REASONING AND SOLUTION Aumin tht thee i no fition between the tie nd the ie, Eqution 5.4 pplie. o tn θ / (5 m/) /(150 m)(9.80 m/ ) 0.43 θ 3 1. SSM REASONING AND SOLUTION Eqution 5.4 ie the eltionhip between the peed the nle of bnkin, nd the diu of utue. Solin fo, we obtin tn θ (10 m)(9.80 m / ) tn m /. REASONING o n unbnked ue tti fition upplie the entipetl foe. Theefoe, we n obtin n expeion fo the peed by equtin the foe of tti fition to the entipetl foe, whih i ien by Eqution 5.3. o bnked ue without fition, Eqution 5.4 elte the bnkin nle, the peed, nd the diu of the ue. By equtin the peed fo the unbnked nd bnked ue we will be ble to obtin lue fo the bnkin nle. SOLUTION The foe of tti fition i ien by Eqution 4.7 f MAX µ N, whee µ i the oeffiient of tti fition nd N i the noml foe. Sine the doe not elete in the etil dietion, the noml foe nd the weiht of the blne, o tht N m. Thu, the

9 186 DYNAMIS O UNIORM IRULAR MOTION foe of tti fition i f MAX µ m. Subtitutin thi expeion fo the entipetl foe in Eqution 5.3, we find fo n unbnked ue tht m µ m o µ o bnked ue, we he fom Eqution 5.4 tht tn θ o tn θ Equtin the two expeion fo the peed, we find tht tnθ µ o tnθ µ h b 1 θ tn µ tn REASONING The nle θ t whih fition-fee ue i bnked depend on the diu of the ue nd the peed with whih the ue i to be neotited, odin to Eqution 5.4: tn θ /( ). o known lue of θ nd, the fe peed i tnθ Befoe we n ue thi eult, we mut detemine tn θ fo the bnkin of the tk. SOLUTION The dwin t the iht how oetion of the tk. om the dwin we he 18 m tn θ. 53 m 0 34 θ 165 m 11 m 53 m 18 m. Theefoe, the mllet peed t whih n moe on thi tk without elyin on fition i b. Similly, the let peed i b ) min 11 m (9.80 m / m / b ) mx 165 m (9.80 m / m /

10 hpte 5 Poblem REASONING AND SOLUTION. The hoizontl omponent of the lift, L, i the entipetl foe whih hold the plne in the ile, L in θ m / The etil omponent of the lift uppot the weiht of the plne, L o θ m Diidin the fit eqution by the eond yield the equied eult. b. Subtitutin lue yield tn θ L b O P NM b hq θ tn 195 m / m m / P SSM WWW REASONING Refe to iue 5.10 in the text. The hoizontl omponent of the lift L i the entipetl foe tht hold the plne in the ile. Thu, L inθ m (1) The etil omponent of the lift uppot the weiht of the plne; theefoe, Loθ m () Diidin the fit eqution by the eond ie tnθ (3) Eqution (3) n be ued to detemine the nle θ of bnkin. One θ i known, then the mnitude of L n be found fom eithe eqution (1) o eqution (). SOLUTION Solin eqution (3) fo θ ie

11 188 DYNAMIS O UNIORM IRULAR MOTION θ tn 1 L N M The liftin foe i, fom eqution (), (13 m / ) (3810 m)(9.80 m / m ( k)(9.80 m / L oθ o.1 5 O Q P. 1 ) ) REASONING The entipetl foe equied to keep n objet of m m tht moe with peed on ile of diu i m / (Eqution 5.3). om Eqution 5.1, we know tht π / T, whee T i the peiod o the time fo the uite to o ound one. Theefoe, the entipetl foe n be witten 6 N m T m ( π / ) 4 π T (1) Thi expeion n be oled fo T. Howee, we mut fit find the entipetl foe tht t on the uite. SOLUTION Thee foe t on the uite. They e the weiht m of the uite, the foe of tti fition f MAX, nd the noml foe N exeted on the uite by the ufe of the ouel. The followin fiue how the fee body dim fo the uite. In thi dim, the y xi i oiented lon the etil + y dietion. The foe of ity t, then, in the y MAX N f dietion. The entipetl foe tht ue the uite θ to moe on it iul pth i poided by the net foe θ + x in the +x dietion in the dim. om the dim, MAX we n ee tht only the foe N nd f he hoizontl omponent. Thu, we he θ MAX f o θ inθ, whee the minu in N m indite tht the x omponent of N point to the left in the dim. Uin Eqution 4.7 fo the mximum tti fitionl foe, we n wite thi eult in eqution (). µ o θ in θ ( µ o θ in θ) () N N N If we pply Newton' eond lw in the y dietion, we ee fom the dim tht MAX o θ + f in θ m m 0 o o θ + µ in θ m 0 N y N N

12 hpte 5 Poblem 189 whee we in he ued Eqution 4.7 fo the mximum tti fitionl foe. Solin fo the noml foe, we find N m oθ + µ inθ Uin thi eult in eqution (), we obtin the mnitude of the entipetl foe tht t on the uite: m ( µ o θ in θ) ( µ o θ in θ) oθ + µ inθ N With thi expeion fo the entipetl foe, eqution (1) beome Solin fo the peiod T, we find T m( µ o θ in θ) mπ 4 oθ + µ inθ T h b hb 4π oθ + µ inθ 4 π ( m) o in ( µ o θ in θ) 9.80 m / o 36.0 in SSM WWW REASONING Eqution 5.5 ie the obitl peed fo tellite in iul obit ound the eth. It n be modified to detemine the obitl peed ound ny plnet P by eplin the m of the eth M E by the m of the plnet M P : GM / P. SOLUTION The tio of the obitl peed i, theefoe, 45 Solin fo ie 1 GM / P GM / P ( m / ) m m 4 m /

13 190 DYNAMIS O UNIORM IRULAR MOTION 8. REASONING AND SOLUTION The noml foe exeted by the wll on eh tonut i the entipetl foe needed to keep him in the iul pth, i.e., m /. Renin nd lettin (1/)m yield / (35.8 m/) /(9.80 m/ ) 6 m 9. REASONING AND SOLUTION We he fo Jupite GM J / whee Thu, m m m (6.67 x N m /k )(1.90 x 10 7 k) 4.0 x 10 4 m / 7.0 x 10 7 m 30. REASONING AND SOLUTION The peiod of tellite i ien by T 4π 3 G M E 4π [()(6.38 x 10 6 m)] 3 (6.67 x N m /k )(5.98 x 10 4 k) 1.43 x REASONING AND SOLUTION. We he π π T m h m / b. o the eth-un ytem 4 11 M S G m / h mh N m / k 3. REASONING AND SOLUTION The peiod of ottion i ien by T 4π 3 /GM. ompin the Eth nd Venu yield 30 k

14 hpte 5 Poblem 191 The eth' obitl peiod i 365 dy o (T V /T E ) ( V / E ) 3 o tht T V /T E T V (0.611)(365 dy) 3 dy 33. SSM REASONING The tue weiht of the tellite when it i t et on the plnet' ufe n be found fom Eqution 4.4: W ( GM m) / whee M P P nd m e the me of the plnet nd the tellite, epetiely, nd i the diu of the plnet. Howee, befoe we n ue Eqution 4.4, we mut detemine the m M P of the plnet. The m of the plnet n be found by eplin M E by M P in Eqution 5.6 nd olin fo M P. When uin Eqution 5.6, we note tht oepond to the diu of the iul obit eltie to the ente of the plnet. SOLUTION The peiod of the tellite i T.00 h om Eqution 5.6, M P π 4 π ( m) +( m) 3 GT ( N m / k )( ) k Uin Eqution 4.4, we he W GM m 11 4 P ( N m / k )( k) ( 5850 k) N 6 ( m) 34. REASONING Eqution 5. fo the entipetl eletion pplie to both the plne nd the tellite, nd the entipetl eletion i the me fo eh. Thu, we he plne tellite o plne HG plne tellite plne tellite The peed of the tellite n be obtined dietly fom Eqution 5.5. I KJ tellite SOLUTION Uin Eqution 5.5, we n expe the peed of the tellite

15 19 DYNAMIS O UNIORM IRULAR MOTION tellite Gm E tellite Subtitutin thi expeion into the expeion obtined in the eonin fo the peed of the plne ie plne plne H G plne tellite I KJ tellite H G plne tellite I K GmE J tellite plne Gm tellite 11 4 b15 m N m / k h kh m E 1 m / 35. REASONING AND SOLUTION. The entipetl eletion of point on the im of hmbe A i the tifiil eletion due to ity, A A /A 10.0 m/. A point on the im of hmbe A moe with peed A π A /T whee T i the peiod of eolution, Subtitutin the eond eqution into the fit nd enin yield A A T /(4π ) 91 m b. Now B A / m. A point on the im of hmbe B h entipetl eletion B B /B. The point moe with peed B π B /T. Subtitutin the eond eqution into the fit yield B 4π B T 4π (8 m) (60.0 ).50 m / 36. REASONING A the pilot oe ound the etil ile, the mnitude of the noml foe tht i exeted on the pilot hne. The mnitude of the noml foe i let t the bottom of the ile whee the noml point upwd nd the weiht m point downwd (ee iue 5.1). The eultnt of the noml foe nd the weiht poide the entipetl foe tht keep the pilot moin on the etil ile. If we tke upwd the poitie dietion in iue 5.1, then Eqution 5.3 ( m / ) n be witten

16 hpte 5 Poblem 193 m N m When the noml foe exeted on the pilot by hi et i equl to thee time hi weiht, 3 N m, nd the boe expeion beome SOLUTION Solin fo we obtin m 3m m o (30 m / ).7 10 (9.80 m / ) 37. SSM REASONING Thi itution i imil to the loop-the-loop tik diued in Setion 5.7 of the text. When the plne pe oe the top of etil ile of diu uh tht the pene expeiene ppent weihtlene, the entipetl foe i poided entiely by the tue weiht m. Thu, m m /. SOLUTION Solin the boe expeion fo ie 15 ( m / ) m / 38. REASONING The entipetl foe i the nme ien to the net foe pointin towd the ente of the iul pth. At point 3 t the top the net foe pointin towd the ente of the ile onit of the noml foe nd the weiht, both pointin towd the ente. At point 1 t the bottom the net foe onit of the noml foe pointin upwd towd the ente nd the weiht pointin downwd o wy fom the ente. In eithe e the entipetl foe i ien by Eqution 5.3 m /. SOLUTION At point 3 we he m N + m At point 1 we he m N m m 3 m

17 194 DYNAMIS O UNIORM IRULAR MOTION Subttin the eond eqution fom the fit ie Renin ie Thu, we find tht m3 m m m / 3 0 m + 15 m / 17 m / 1 1 hb b 39. SSM REASONING A the motoyle pe oe the top of the hill, it will expeiene entipetl foe, the mnitude of whih i ien by Eqution 5.3: m /. The entipetl foe i poided by the net foe on the yle + die ytem. At tht intnt, the net foe on the ytem i ompoed of the noml foe, whih point upwd, nd the weiht, whih point downwd. Tkin the dietion towd the ente of the ile (downwd) the poitie dietion, we he m. Thi expeion n be oled fo N, the noml foe. SOLUTION. The mnitude of the entipetl foe i N m (34 k)(5.0 m / ) N 16 m b. The mnitude of the noml foe i 3 3 m (343 k)(9.80 m / ) N N N 40. REASONING AND SOLUTION When the tone i whiled in hoizontl ile, the entipetl foe i poided by the tenion in the tin nd i ien by T h m When the tone i whiled in etil ile, the mximum tenion ou when the tone i t the lowet point in it pth.

18 hpte 5 Poblem 195 The dim t the iht how tht foe tht t on the tone in thi e. The entipetl foe i poided by the eultnt of T nd the weiht of the tone: T m m The tenion in the tin fo thi e i T m T m + m Sine the mximum tenion in the tin fo the e of etil motion i 10.0% le thn tht in the e of hoizontl motion, o Solin fo yield T ( ) T h m + m (1.100 ) m (9.80 m/ ) (0.950 m) 9.65 m/ REASONING AND SOLUTION In enel, when the olle-blde i t the top of the et, thee e two foe tht t on he. They e he weiht m, whih t downwd, nd the noml foe N exeted on the olle-blde by the ound. If we tke upwd the poitie dietion, then, odin to Eqution 5.3 ( m / ) the ondition fo the olle-blde to follow the iul et of the hill i, m N m If the olle-blde oe ft enouh to loe ontt with the ound, N 0, nd the boe expeion beome m m Solin fo ie (9.80 m / )(0.0 m) 14.0 m /

19 196 DYNAMIS O UNIORM IRULAR MOTION 4. REASONING The dwin t the iht how the two foe tht t on piee of lothin jut befoe it loe ontt with the wll of the ylinde. At tht intnt the entipetl foe i poided by the noml foe N nd the dil om-ponent of the weiht. om the dwin, the dil omponent of the weiht i ien by m o φ m o (90 θ) m inθ N θ φ lothe m Theefoe, with inwd tken the poitie dietion, Eqution 5.3 ( N + m in θ m m / ) ie At the intnt tht piee of lothin loe ontt with the ufe of the dum, N 0, nd the boe expeion beome m m in θ Aodin to Eqution 5.1, π / T, nd with thi ubtitution we obtin ( π / T) 4π in θ T Thi expeion n be oled fo the peiod T. Sine the peiod i the equied time fo one eolution, the numbe of eolution pe eond n be found by lultin 1/T. SOLUTION Solin fo the peiod, we obtin 4π 0.3 m T π π 117. in θ inθ 9.80 m / in 70.0 h Theefoe, the numbe of eolution pe eond tht the ylinde hould mke i 1 1 T e/ 43. SSM REASONING AND SOLUTION The mnitude of the entipetl foe on the bll i ien by Eqution 5.3: m /. Solin fo, we he

20 hpte 5 Poblem 197 m (0.08 N)(0.5 m) k 0.68 m / 44. REASONING AND SOLUTION The diu of ynhonou tellite i lulted in Exmple m. The peed i theefoe, G M E (6.67 x N m /k )(5.98 x 10 4 k) 4.3 x 10 7 m 3070 m / 45. REASONING AND SOLUTION. We know / (1.4 m/) /(0.039 m) m/ b. Between poket, the hin i tiht nd tel t ontnt peed, o it eletion i zeo.. o the font poket / (1.4 m/) /(0.10 m) m/ 46. REASONING AND SOLUTION The foe P upplied by the mn will be let when the ptne i t the lowet point in the win. The dim t the iht how the foe tin on the ptne in thi itution. The entipetl foe neey to keep the ptne winin lon the of ile i poided by the eultnt of the foe upplied by the mn nd the weiht of the ptne. om the fiue m P m Theefoe, P m + m P m Sine the weiht of the ptne, W, i equl to m, it follow tht m (W/) nd W P ( / ) + W [( N) / (. m / )] (. m / ) ( 475 N) 594 N ( m)

21 198 DYNAMIS O UNIORM IRULAR MOTION 47. SSM REASONING AND SOLUTION Sine the mnitude of the entipetl eletion i ien by Eqution 5., /, we n ole fo nd find tht ( 98.8 m / ) 33 m 3.00(9.80 m / ) 48. REASONING The entipetl foe i the nme ien to the net foe pointin towd the ente of the iul pth. At the lowet point the net foe onit of the tenion in the m pointin upwd towd the ente nd the weiht pointin downwd o wy fom the ente. In eithe e the entipetl foe i ien by Eqution 5.3 m /. SOLUTION () The entipetl foe i b b m 9. 5 k.8 m / m 88 N (b) Uin T to denote the tenion in the m, t the bottom of the ile we he T m m b h b b T m m 9. 5 k.8 m / k m / m 181 N 49. SSM REASONING AND SOLUTION Sine the tenion ee the me pupoe the noml foe t point 1 in iue 5.1, we he, uin the eqution fo the itution t point 1 with N1 epled by T, Solin fo T ie m T m m + + m T m HG I K J L(7.6 m / ) ) + N M 15 m O Q (100 k ( m / ) P N

22 hpte 5 Poblem REASONING AND SOLUTION The peiod of the moon' motion (ppoximtely the lenth of month) i ien by T 4π 3 G M E 4π (3.85 x 10 8 m) 3 (6.67 x N m /k )(5.98 x 10 4 k).38 x d y 51. REASONING The entipetl eletion fo ny point tht i ditne fom the ente of the di i, odin to Eqution 5., /. om Eqution 5.1, we know tht π / T whee T i the peiod of the motion. ombinin thee two eqution, we obtin T ( π / ) 4π T SOLUTION Uin the boe expeion fo, the tio of the entipetl eletion of the two point in quetion i T T 4π / 4 T / π / / T Sine the di i iid, ll point on the di mut moe with the me peiod, o T 1 T. Mkin thi nelltion nd olin fo, we obtin h m H G I 10 m / K J.0 10 m / m Note tht een thouh T 1 T, it i not tue tht 1. Thu, the implet wy to ppoh thi poblem i to expe the entipetl eletion in tem of the peiod T whih nel in the finl tep. 5. REASONING AND SOLUTION The entipetl eletion fo ny point on the blde ditne fom ente of the ile, odin to Eqution 5., i /. om Eqution 5.1, we know tht π / T whee T i the peiod of the motion. ombinin thee two eqution, we obtin T ( π / ) 4π T

23 00 DYNAMIS O UNIORM IRULAR MOTION. Sine the tubine blde otte t 617 e/, ll point on the blde otte with peiod of T (1/617) Theefoe, fo point with 0.00 m, the mnitude of the entipetl eletion i 4π ( m) 3 ( ) m / b. Expeed multiple of, thi entipetl eletion i m / h H G IKJ m / 53. SSM WWW REASONING If the effet of ity e not inoed in Exmple 5, the plne will mke n nle θ with the etil hown in fiue A below. The fiue B how the foe tht t on the plne, nd fiue how the hoizontl nd etil omponent of thee foe. T o θ T θ L θ L T in θ m m A B om fiue we ee tht the eultnt foe in the hoizontl dietion i the hoizontl omponent of the tenion in the uideline nd poide the entipetl foe. Theefoe, T in θ m om fiue A, the diu i elted to the lenth L of the uideline by L in θ; theefoe, m T in θ (1) L in θ The eultnt foe in the etil dietion i zeo: T o θ m 0, o tht om eqution () we he T o θ m ()

24 hpte 5 Poblem 01 T m o θ (3) Eqution (3) ontin two unknown, T nd θ. it we will ole eqution (1) nd (3) imultneouly to detemine the lue() of the nle θ. One θ i known, we n lulte the tenion uin eqution (3). SOLUTION Subtitutin eqution (3) into eqution (1): Thu, HG m o θ I K J in o in θ θ θ m L in θ L (4) Uin the ft tht o θ + in θ 1, eqution (4) n be witten o 1 o o θ 1 o θ θ o θ L L Thi n be put in the fom of n eqution tht i qudti in o θ. Multiplyin both ide by o θ nd enin yield: Eqution (5) i of the fom o θ + o θ 1 0 L (5) x + bx + 0 (6) with x o θ, 1, b /(L), nd 1. The olution to eqution (6) i found fom the qudti fomul: x b b ± 4 When 19.0 m/, b.17. The poitie oot fom the qudti fomul ie x o θ Subtitution into eqution (3) yield m (0.900 k)(9.80 m / ) T o θ N

25 0 DYNAMIS O UNIORM IRULAR MOTION When 38.0 m/, b The poitie oot fom the qudti fomul ie x o θ Subtitution into eqution (3) yield m (0.900 k)(9.80 m / ) T o θ N 54. REASONING AND SOLUTION. The entipetl foe i poided by the noml foe exeted on the ide by the wll. b. Newton' eond lw pplied in the hoizontl dietion ie N m / (55.0 k)(10.0 m/) /(3.30 m) 1670 N. Newton' eond lw pplied in the etil dietion ie µ N m 0 o µ (m)/ N REASONING AND SOLUTION If i the foe on m, then i the foe on m 1, nd we he o m 1: m 1 1 /1 o m : m / Diidin the eqution nd enin ie m /m 1 (1/)( / 1 )( 1 / ) ( 1 / ) ine 1 The peiod of eolution i the me fo both me o 1 π 1 /T nd π /T. Diidin thee ie 1 / 1 / 1/. Now m /m 1 (1/) ONEPT QUESTIONS. The peiod fo the eond hnd i the time it tke it to o one ound the ile o T eond 60. Similly, fo the minute hnd the peiod i T minute 1 h b. The eltionhip between the entipetl eletion nd the peiod n be obtined by uin Eqution 5. nd 5.1: 1

26 hpte 5 Poblem 03 b π T π / T 4π T (5.) (5.1) SOLUTION Uin the expeion fo the entipetl eletion obtined in the nwe to onept quetion b, we he, eond, minute 4π / Teond T 4π / T T b b minute minute eond 57. ONEPT QUESTIONS. Exmple h the mllet entipetl eletion. Unifom iul motion on ile with n infinitely le diu i like motion lon tiht line t ontnt eloity. In uh e thee i no entipetl eletion. b. Exmple 1 h the etet entipetl eletion. Aodin to Eqution 5., the eletion i /. We note tht the peed i in the numeto nd the diu i in the denominto of thi expeion. Theefoe, the etet peed nd the mllet diu, i the e fo Exmple 1, podue the etet entipetl eletion. SOLUTION With Eqution 5., we find the followin lue of the entipetl eletion. Exmple 1 Exmple Exmple 3 1 m / m b b 35 m / b.3 m / 1.8 m 90 m / 0 m /.9 m /

27 04 DYNAMIS O UNIORM IRULAR MOTION 58. ONEPT QUESTIONS. Stti, the thn kineti, fition poide the entipetl foe, beue the penny i ttiony nd not lidin eltie to the eod. b. The peed n be detemined fom the peiod T of the motion nd the diu, odin to Eqution 5.1 ( π /T). The peiod n be found fom the ft tht the eod mke eolution in eh minute o 60. Theefoe, 1 min e 1 min T H G I K J SOLUTION Uin Eqution 5.3 fo the entipetl foe ( m /) nd Eqution 5.1 fo the peed in tem of the peiod ( π /T), we he b m m T m π / 4π T mx Aodin to Eqution 4.7, the mximum foe of tti fition i f µ N, whee N i the noml foe. Sine the penny doe not elete in the etil dietion, the upwd noml mx foe mut be blned by the downwd-pointin weiht, o tht N m nd f µ m. Uin eqution (1), we find µ m π µ m 13 4 T entipetl foe b hb 4π 4π m T m / ONEPT QUESTIONS. The entipetl eletion depend only on the peed nd the diu of the ue, odin to Eqution 5. ( /). The peed of the e the me, nd ine they e neotitin the me ue, the diu i the me. Theefoe, the he the me entipetl eletion. b. The entipetl foe depend on the m m, well the peed nd the diu of the ue, odin to Eqution 5.3 ( m /). Sine the peed nd the diu e the me fo eh, the with the ete m, whih i B, expeiene the ete entipetl eletion. SOLUTION Uin Eqution 5. nd 5.3, we find the followin lue fo the entipetl eletion nd foe: (1)

28 hpte 5 Poblem 05 A B b 7 m / 10 m m A 1100 k 7 m / 10 m b 7 m / 10 m.1 m / (5.) 6700 N (5.3).1 m / (5.) m B 1600 k 7 m / 9700 N (5.3) 10 m 6 b b 6 b b 60. ONEPT QUESTIONS. The tellite with the lowe obit h the ete peed, odin to Eqution 5.5 ( GM / ), whee i the diu of the obit. Thu, tellite A h the ete peed. E b. In Eqution 5.5 ( GM / ) the tem i the obitl diu, meued fom the E ente of the eth, not the ufe of the eth. Theefoe, you do not ubtitute the heiht of m nd m fo the tem. Inted, thee heiht mut be dded to the diu of the eth ( m) in ode to et the dii. SOLUTION it we dd the obitl heiht to the diu of the eth to obtin the obitl dii. Then we ue Eqution 5.5 to lulte the peed. Stellite A A m m m GM E A N m / k h kh m 7690 m / Stellite B A m m m GM E A N m / k h kh m 7500 m/ 61. ONEPT QUESTIONS. Sine the peed nd m e ontnt nd the diu i fixed, the entipetl foe i the me t eh point on the ile. b. When the bll i t the thee o lok poition, the foe of ity, tin downwd, i pependiul to the tin nd nnot ontibute to the entipetl foe. (See iue 5.1, point

29 06 DYNAMIS O UNIORM IRULAR MOTION fo imil itution.) At thi point, only the tenion of T 16 N ontibute to the entipetl foe. onidein tht the entipetl foe i the me eeywhee, we n onlude tht it h lue of 16 N eeywhee.. At the twele o lok poition the tenion T nd the foe of ity m both t downwd (the netie dietion) towd the ente of the ile, with the eult tht the entipetl foe t thi point i T m. (See iue 5.1, point 3.) The mnitude of the entipetl foe hee, then, i T + m. At the ix o lok poition the tenion point upwd towd the ente of the ile, while the foe of ity point downwd, with the eult tht the entipetl foe t thi point i T m. (See iue 5.1, point 1.) The only wy fo entipetl foe to he the me mnitude of 16 N t both of thee ple i fo the tenion t the ix o lok poition to be ete. The ete tenion ompente fo the ft tht the foe of ity point wy fom the ente of the ile. SOLUTION Aumin tht upwd i the poitie dietion, we find t the twele nd ix o lok poition tht Twele o' lok Six o' lok T m 16 N entipetl foe T 16 N 0. 0 k m / 14 N T m 16 N 13 entipetl foe b h b h T 16 N k m / 18 N