2 / r. Since the speed of the car is constant,

Transcription

1 CHAPER 5 DYAMICS OF UIFORM CIRCULAR MOIO COCEPUAL QUESIOS 1. REASOIG AD SOLUIO he will elete if it eloity hnge in mgnitude, in dietion, o both. If i teling t ontnt peed of 35 m/, it n be eleting if it dietion of motion i hnging.. REASOIG AD SOLUIO Conide two people, one on the eth' ufe t the equto, nd the othe t the noth pole. If we ombine Eqution 5.1 nd 5., we ee tht the entipetl eletion of n objet moing in ile of diu with peiod n be witten ( 4 π )/ C. he eth otte bout n xi tht pe ppoximtely though the noth pole nd i pependiul to the plne of the equto. Sine both people e moing on the eth' ufe, they he the me peiod. he peon t the equto moe in lge ile o tht i lge fo the peon t the equto. heefoe, the peon t the equto h lge entipetl eletion thn the peon t the noth pole. 3. REASOIG AD SOLUIO he eqution of kinemti (Eqution ) nnot be pplied to unifom iul motion beue n objet in unifom iul motion doe not he ontnt eletion. While the eletion eto i ontnt in mgnitude /, it dietion hnge ontntly -- it lwy point towd the ente of the h ile. A the objet moe ound the ile the dietion of the eletion mut ontntly hnge. Beue of thi hnging dietion, the ondition of ontnt eletion tht i equied by Eqution i iolted. 4. REASOIG AD SOLUIO Aeletion i the te of hnge of eloity. In ode to he n eletion, the eloity eto mut hnge eithe in mgnitude o dietion, o both. heefoe, if the eloity of the objet i ontnt, the eletion mut be zeo. On the othe hnd, if the peed of the objet i ontnt, the objet ould be eleting if the dietion of the eloity i hnging. 5. REASOIG AD SOLUIO When the i moing t ontnt peed long the tight egment (i.e., AB nd DE), the eletion i zeo. Along the ued egment, the mgnitude of the eletion i gien by /. Sine the peed of the i ontnt, the mgnitude of the eletion i lget whee the diu i mllet. Rnked fom mllet to lget the mgnitude of the eletion in eh of the fou etion e: AB o DE, CD, BC.

2 Chpte 5 Coneptul Quetion REASOIG AD SOLUIO Fom Exmple 7, the mximum fe peed with whih n ound n unbnked hoizontl ue of diu i gien by µ g. Sine the eletion due to gity on the moon i oughly one ixth tht on eth, the fe peed fo the me ue on the moon would be le thn tht on eth. In othe wod, othe thing being equl, it would be moe diffiult to die t high peed ound n unbnked ue on the moon omped to diing ound the me ue on the eth. 7. SSM REASOIG AD SOLUIO A bug lnd on windhield wipe. he wipe e tuned on. Sine the wipe moe long the of ile, the bug will expeiene entipetl eletion, nd hene, entipetl foe mut be peent. he mgnitude of the entipetl foe i gien by F m /. In ode fo the bug to emin t et on the wipe blde, the foe of tti fition between the bug nd the wipe blde mut ontibute in mjo wy to the entipetl foe. Without the entipetl foe, the bug will be dilodged. When the wipe e tuned on t highe etting, i lge, nd the entipetl foe equied to keep the bug moing long the of the ile i lge thn if the wipe e tuned on the low etting. Sine the high etting equie lge entipetl foe to keep the bug on the wipe, the bug i moe likely to be dilodged t tht etting thn t the low etting. 8. REASOIG AD SOLUIO Fom Exmple 7, the mximum fe peed with whih n ound n unbnked ue of diu i gien by µ g. hi expeion i independent of the m (nd theefoe the weight) of the. hu, the hne of light fely ounding n unbnked ue on n iy od i the me tht fo heie (uming tht ll othe fto e the me). 9. SSM REASOIG AD SOLUIO Sine the peed nd diu of the ile e ontnt, the entipetl eletion i ontnt. A the wte lek out, howee, the m of the objet undegoing the unifom iul motion deee. Centipetl foe i m time the entipetl eletion, o tht the entipetl foe pplied to the ontine mut be deeing. It i the tenion in the ope tht poide the entipetl foe. You e holding the fee end of the ope nd pulling on it in ode to ete the tenion. heefoe, you mut be eduing you pull the wte lek out. In tun, oding to ewton thid lw, the ope mut be pulling bk on you hnd with foe of deeing mgnitude, nd you feel thi pull wekening time pe. 10. REASOIG AD SOLUIO A the popelle otte fte, the entipetl eletion of the pt of the popelle inee. A the entipetl eletion inee, the entipetl foe equied to ue the iou pt of the popelle to otte in the ile lo inee. When the neey entipetl foe exeed the mehnil foe tht hold the popelle togethe, the popelle will ome pt.

3 4 DYAMICS OF UIFORM CIRCULAR MOIO 11. REASOIG AD SOLUIO he entipetl foe on the penny i gien by F m /, whee π / nd i the ontnt peiod of the tuntble. heefoe, the entipetl foe on the penny i gien by F 4 π m/. Clely, the penny will equie the lget entipetl foe to emin in ple when loted t the lget lue of ; tht i, t the edge of the tuntble. 1. REASOIG AD SOLUIO A model iplne on guideline n fly in ile beue the tenion in the guideline poide the hoizontl entipetl foe neey to pull the plne into hoizontl ile. A el iplne h no uh hoizontl foe. he i on the wing on el plne exet n upwd lifting foe tht i pependiul to the wing. he plne mut bnk o tht omponent of the lifting foe n be oiented hoizontlly, theeby poiding the equied entipetl foe to ue the plne to fly in ile. 13. SSM REASOIG AD SOLUIO. Refeing to Figue 5.10 in the text, we n ee tht the entipetl foe on the plne i Lin θ m /, whee L i the mgnitude of the lifting foe. In ddition, the etil omponent of the lifting foe mut blne the weight of the plne, o tht Lo θ mg. Diiding thee two eqution eel tht tn θ /( g). b. he bnking ondition fo teling t peed ound ue of diu, bnked t ngle θ i tn θ / ( g), oding to Eqution 5.4 in the text.. he peed of tellite in iul obit of diu bout the eth i gien by GM E /, oding to Eqution 5.5 in the text. d. he minimum peed equied fo loop-the-loop tik ound loop of diu i g, oding to the diuion in Setion 5.7 of the text. Aoding to Eqution 4.4 nd 4.5, g GM /. hu, ny expeion tht depend on g lo depend on M E nd E would be ffeted by hnge in the eth' m. Suh i the e fo eh of the fou itution diued boe. 14. REASOIG AD SOLUIO When the ting i whiled in hoizontl ile, the tenion in the ting, F, poide the entipetl foe whih ue the tone to moe in ile. Sine the peed of the tone i ontnt, m / F nd the tenion in the ting i ontnt. When the ting i whiled in etil ile, the tenion in the ting nd the weight of the tone both ontibute to the entipetl foe, depending on whee the tone i on the ile. ow, howee, the tenion inee nd deee the tone tee the etil ile. When the tone i t the lowet point in it wing, the tenion in the ting pull the tone upwd, while the weight of the tone t downwd. heefoe, the entipetl foe

4 Chpte 5 Coneptul Quetion 43 i m / F mg. Soling fo the tenion how tht F m / + mg. hi tenion i lge thn in the hoizontl e. heefoe, the ting h gete hne of beking when the tone i whiled in etil ile. 15. REASOIG AD SOLUIO A fighte pilot pull out of die on etil ile nd begin to limb upwd. A the pilot moe long the ile, ll pt of hi body, inluding the blood in hi hed, mut expeiene entipetl foe in ode to emin on the ile. he blood, howee, i not igidly tthed to the body nd doe not expeiene the equiite entipetl foe until it flow out of the hed, wy fom the ile' ente, nd ollet in the lowe body pt, whih ultimtely puh on it enough to keep it on the iul pth. 16. REASOIG AD SOLUIO When moe ound n unbnked hoizontl ue, the entipetl foe tht keep the on the od o tht it n negotite the ue ome fom the foe of tti fition. If A nnot negotite the ue, then the foe of tti fition between the ted of A' tie nd the od i not get enough to poide the entipetl foe. he oeffiient of tti fition between the tie nd the od e le fo A thn fo B, ine B n negotite the tun.

5 44 DYAMICS OF UIFORM CIRCULAR MOIO CHAPER 5 DYAMICS OF UIFORM CIRCULAR MOIO PROBLEMS 1. SSM REASOIG he peed of the plne i gien by Eqution 5.1: π /, whee i the peiod o the time equied fo the plne to omplete one eolution. SOLUIO Soling Eqution 5.1 fo we he π π ( 850 m) m /. REASOIG AD SOLUIO 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. Futhemoe 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 long the of the ile fom O to P. he iumfeene of the ile i π π (3.6 m).6 m. he OP ubtend n ngle 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. hu, b gb g OP.6 m m nd, fom the gument gien boe, we onlude tht the ditne OA i 1.6 m. 3. REASOIG Sine the tip of the blde moe on iul pth, it expeiene entipetl eletion whoe mgnitude i gien by Eqution 5.,, whee / i the peed of blde tip nd i the diu of the iul pth. he diu i known, nd the peed n be obtined by diiding the ditne tht the tip tel by the time t of tel. Sine n ngle of 90 oepond to one fouth of the iumfeene of ile, the 1 ditne i 4 ( π ). SOLUIO Sine / nd 1 ( π 4 ) / t π/ ( t ), the mgnitude of the entipetl eletion of the blde tip i π t π π 6.9 m/ 4t ( 0.45 m) ( )

6 Chpte 5 Poblem REASOIG he mgnitude of the entipetl eletion i gien by Eqution 5., whee i the peed of the nd i the diu of the tk. he diu / i m. he peed n be obtined fom Eqution 5.1 the iumfeene (π ) of the tk diided by the peiod of the motion. he peiod i the time fo the to go one ound the tk ( 360 ). SOLUIO Sine eletion i nd ( π ) /, the mgnitude of the entipetl / π 3 4π 4π (.6 10 m) 0.79 m/ ( 360 ) 5. SSM REASOIG AD SOLUIO In eh e, the mgnitude of the entipetl eletion i gien by Eqution 5., /. heefoe, A B Sine eh bot expeiene the me entipetl eletion, tio of the peed gie A A 10 m m B B A B / / A B A B 1. Soling fo the 6. REASOIG he tonut in the hmbe i ubjeted to entipetl eletion tht i gien by / (Eqution 5.). In thi expeion i the peed t whih the tonut in the hmbe moe on the iul pth of diu. We n ole thi eltion fo the peed. SOLUIO Uing Eqution 5., we he ( ) ( ) o m/ 15 m 33 m/ 7. REASOIG he entipetl eletion i gien by Eqution 5. /. he lue of the diu i gien, o to detemine we need infomtion bout the peed. But the peed i elted to the peiod by (π )/, oding to Eqution 5.1. We n ubtitute thi expeion fo the peed into Eqution 5. nd ee tht

7 46 DYAMICS OF UIFORM CIRCULAR MOIO b g π / 4π SOLUIO o ue the expeion obtined in the eoning, we need lue fo the peiod. he peiod i the time fo one eolution. Sine the ontine i tuning t.0 eolution pe eond, the peiod i (1 )/(.0 eolution) hu, we find tht the entipetl eletion i 4π 4π 01. m 050. b b g g 19 m/ 8. REASOIG AD SOLUIO he entipetl eletion fo ny point on the blde ditne fom ente of the ile, oding to Eqution 5., i /. Fom Eqution 5.1, we know tht π / whee i the peiod of the motion. Combining thee two eqution, we obtin ( π / ) 4π. Sine the tubine blde otte t 617 e/, ll point on the blde otte with peiod of (1/617) heefoe, fo point with 0.00 m, the mgnitude of the entipetl eletion i 4π ( m) 3 ( ) m/ 5 b. Expeed multiple of g, thi entipetl eletion i h F 1.00 g 3.0 g H G I m / K J 5 m/ SSM REASOIG he mgnitude of the entipetl eletion of ny point on the heliopte blde i gien by Eqution 5.,, whee i the diu of the ile on C / whih tht point moe. Fom Eqution 5.1: π /. Combining thee two expeion, we obtin 4 π C All point on the blde moe with the me peiod.

8 Chpte 5 Poblem 47 SOLUIO he 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 C1 C 4π / m 4π / 3.0 m 10. REASOIG he entipetl eletion fo ny point tht i ditne fom the ente of the di i, oding to Eqution 5., /. Fom Eqution 5.1, we know tht π / whee i the peiod of the motion. Combining thee two eqution, we obtin ( π / ) 4π SOLUIO Uing the boe expeion fo, the tio of the entipetl eletion of the two point in quetion i 4π / / 4π / / Sine the di i igid, ll point on the di mut moe with the me peiod, o 1. Mking thi nelltion nd oling fo, we obtin h m 1 1. F 1 1 H G I 10 m / K J.0 10 m / m ote tht een though 1, it i not tue tht 1. hu, the implet wy to ppoh thi poblem i to expe the entipetl eletion in tem of the peiod whih nel in the finl tep. 11. SSM REASOIG In Exmple 3, it w hown tht the mgnitude of the entipetl eletion fo the two e e Rdiu 33 m 35 m / Rdiu 4 m 48 m / C C Aoding to ewton' eond lw, the entipetl foe i F C m (ee Eqution 5.3). C SOLUIO. heefoe, when the led undegoe the tun of diu 33 m, F C m ( 350 kg)(35 m / ) C 4

9 48 DYAMICS OF UIFORM CIRCULAR MOIO b. Similly, when the diu of the tun i 4 m, F C m ( 350 kg)(48 m / ) C 1. REASOIG he mgnitude F of the entipetl foe tht t on the kte i gien by Eqution 5.3 / 4 F m, whee m nd e the m nd peed of the kte, nd i the ditne of the kte fom the piot. Sine ll of thee ible e known, we n find the mgnitude of the entipetl foe. SOLUIO he mgnitude of the entipetl foe i ( 80.0 kg)( 6.80 m/) m F m 13. REASOIG AD SOLUIO. In tem of the peiod of the motion, the entipetl foe i witten F 4π m/ 4π (0.010 kg)(0.100 m)/(0.500 ) b. he entipetl foe ie the que of the peed. hu, doubling the peed would inee the entipetl foe by fto of REASOIG At the mximum peed, the mximum entipetl foe t on the tie, nd tti fition upplie it. he mgnitude of the mximum foe of tti fition i peified MAX by Eqution 4.7 f µ F, whee µ i the oeffiient of tti fition nd F i the mgnitude of the noml foe. Ou ttegy, then, i to find the noml foe, ubtitute it into the expeion fo the mximum fitionl foe, nd then equte the eult to the entipetl foe, whih i F m, oding to Eqution 5.3. hi will led u to n / expeion fo the mximum peed tht we n pply to eh. SOLUIO Sine neithe elete in the etil dietion, we n onlude tht the weight mg i blned by the noml foe, o F mg. Fom Eqution 4.7 nd 5.3 it follow tht MAX m f µ F µ mg F hu, we find tht m µ mg o µ g

10 Chpte 5 Poblem 49 Applying thi eult to A nd B gie µ g nd µ g A, A B, B In thee two eqution, the diu doe not he ubipt, ine the diu i the me fo eithe. Diiding the two eqution nd noting tht the tem g nd e eliminted lgebilly, we ee tht B A µ g µ µ 0.85 o ( 5 m/) m/ µ µ 1.1, B, B, B B A, Ag µ, A, A 15. REASOIG he peon feel the entipetl foe ting on hi bk. hi foe i F m /, oding to Eqution 5.3. hi expeion n be oled dietly to detemine the diu of the hmbe. SOLUIO Soling Eqution 5.3 fo the diu gie b gb g. m 83 kg 3. m / F 560 C 15m 16. REASOIG AD SOLUIO 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 MAX out of the tie, the foe of tti fition jut exeed it mximum lue f µ F (ee Eqution 4.7). he foe of tti fition tht t on the tone fom one ide of the ted hnnel i, theefoe, MAX f 0.90(1.8 ) 1.6 nd the mgnitude of the totl fitionl foe tht t on the tone jut befoe it flie out i If we ume tht only tti fition upplie the entipetl foe, then, F 3.. Soling Eqution 5.3 ( F m / ) fo the diu, we he ( ) 3 m kg (13 m/) F m 17. SSM WWW REASOIG Let be the initil peed of the bll it begin it pojetile motion. hen, the entipetl foe i gien by Eqution 5.3: 0 F C m 0 /. We e gien 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.

11 50 DYAMICS OF UIFORM CIRCULAR MOIO 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 gity. he hoizontl ditne teled by the bll i gien by Eqution 3.5 (with x 0): x t ( o θ ) t 0x 0 with t equl to the flight time of the bll while it exhibit pojetile motion. he time t n be found by onideing the etil motion. Fom Eqution 3.3b, Afte time t, he nd y 0y + t y 0 y y. Auming tht up nd to the ight e the poitie dietion, we y t inθ 0 0 x ( o θ ) 0 y F HG Uing the ft tht inθ oθ in θ, we he y 0 y inθ I KJ x 0 oθ inθ inθ 0 y Eqution (1) (with upwd nd to the ight hoen the poitie dietion) n be ued to detemine the peed 0 with whih the bll begin it pojetile motion. hen Eqution 5.3 n be ued to find the entipetl foe. SOLUIO Soling eqution (1) fo, we he 0 y (1) 0 x y (. m)( 9.80 m / ) in θ in (41 ) 9. 3 m/ hen, fom Eqution 5.3, m0 (7.3 kg)(9.3 m / ) F 3500 C 1.8 m 18. REASOIG AD SOLUIO he entipetl eletion of the blok i / (8 m/) /(150 m) 5. m/

12 Chpte 5 Poblem 51 he ngle θ n be obtined fom F H G I KJ 1 C 1 5. m / θ tn tn 8 g 9.80 m / KJ 19. REASOIG AD SOLUIO If F i the net foe on m, then F i the net foe on m 1, nd we he F H G Fo m 1: F m 1 1 /1 Fo m : F m / Diiding the eqution nd enging gie I m /m 1 (1/)( / 1 )( 1 / ) ( 1 / ) ine 1 he peiod of eolution i the me fo both me o 1 π 1 / nd π /. Diiding thee gie 1 / 1 / 1/. ow m /m 1 (1/) REASOIG he eltion tnθ (Eqution 5.4) detemine the bnking ngle θ tht g bnked ue of diu mut he if i to tel ound it t peed without elying on fition. In thi expeion g i the mgnitude of the eletion due to gity. We will ole fo nd pply the eult to eh ue. he ft tht the diu of eh ue i the me will llow u to detemine the unknown peed. SOLUIO Aoding to Eqution 5.4, we he tn θ o gtnθ g Applying thi eult fo the peed to eh ue gie θ A g tn A nd B g tn B ote tht the tem nd g e the me fo eh ue. heefoe, thee tem e eliminted lgebilly when we diide the two eqution. We find, then, tht gtnθ tnθ tnθ tn19 o ( 18 m/) m/ tn tn tn13 B B B B B A A gtnθ θ A A θa θ

13 5 DYAMICS OF UIFORM CIRCULAR MOIO 1. SSM REASOIG AD SOLUIO Eqution 5.4 gie the eltionhip between the peed the ngle of bnking, nd the diu of utue. Soling fo, we obtin g tn θ (10 m)(9.80 m / ) tn m/. REASOIG he ngle 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 negotited, oding to Eqution 5.4: tn θ /(g). Fo known lue of θ nd, the fe peed i g tnθ Befoe we n ue thi eult, we mut detemine tn θ fo the bnking of the tk. SOLUIO he dwing t the ight how o-etion of the tk. Fom the dwing we he 18 m tn θ. 53 m 034 θ 165 m 11 m 53 m 18 m. heefoe, the mllet peed t whih n moe on thi tk without elying on fition i b. Similly, the lget peed i ( )( )( ) min 11 m 9.80 m/ m/ ( )( )( ) mx 165 m 9.80 m/ m/ 3. REASOIG Fom the diuion on bnked ue in Setion 5.4, we know tht n fely ound bnked ue without the id of tti fition if the ngle θ of the bnked ue i gien by tn θ / ( g, whee 0 ) o i the peed of the nd i the diu of the ue (ee Eqution 5.4). he mximum peed tht n he when ounding n unbnked ue i 0 µ g (ee Exmple 7). By ombining thee two eltion, we n find the ngle θ. SOLUIO he ngle of the bnked ue i tn 1 / ( g) expeion µ g into thi eqution gie 0 θ. Subtituting the 0 µ g θ tn tn tn ( µ ) tn ( 0.81) 39 g g

14 Chpte 5 Poblem REASOIG he ditne d i elted to the diu of the ile on whih the tel by d /in 50.0 (ee the dwing). F +y 50.0 C +x 50.0 d m g We n obtin the diu by noting tht the expeiene entipetl foe tht i dieted towd the ente of the iul pth. hi foe i poided by the omponent, F o 50.0, of the noml foe tht i pllel to the diu. Setting thi foe equl to the m m of the time the entipetl eletion ( / gie ) F o50.0 m m /. Soling fo the diu nd ubtituting it into the eltion d /in 50.0 gie m F o 50.0 m (1) d in 50.0 in 50.0 F o 50.0 in ( )( ) he mgnitude F of the noml foe n be obtined by obeing tht the h no etil eletion, o the net foe in the etil dietion mut be zeo, F 0. he y net foe onit of the upwd etil omponent of the noml foe nd the downwd weight of the. he etil omponent of the noml foe i +F in 50.0, nd the weight i mg, whee we he hoen the up dietion the + dietion. hu, we he tht + F in 50.0 mg 0 () F y Soling thi eqution fo F nd ubtituting it into the eqution boe will yield the ditne d. SOLUIO Soling Eqution () fo F nd ubtituting the eult into Eqution (1) gie

15 54 DYAMICS OF UIFORM CIRCULAR MOIO m m d ( F o 50.0 )( in 50.0 ) mg in 50.0 ( 34.0 m/) 184 m g o m/ o 50.0 ( ) ( o 50.0 )( in 50.0 ) 5. SSM WWW REASOIG Refe to Figue 5.10 in the text. he hoizontl omponent of the lift L i the entipetl foe tht hold the plne in the ile. hu, L inθ m (1) he etil omponent of the lift uppot the weight of the plne; theefoe, Loθ mg () Diiding the fit eqution by the eond gie tnθ g (3) Eqution (3) n be ued to detemine the ngle θ of bnking. One θ i known, then the mgnitude of L n be found fom eithe eqution (1) o eqution (). SOLUIO Soling eqution (3) fo θ gie θ tn 1 he lifting foe i, fom eqution (), L M (13 m / ) (3810 m)(9.80 m / mg ( kg)(9.80 m / L oθ o.1 5 O QP. 1 ) ) REASOIG he entipetl foe F equied to keep n objet of m m tht moe with peed on ile of diu i F m / (Eqution 5.3). Fom Eqution 5.1, we know tht π /, whee i the peiod o the time fo the uite to go ound one. heefoe, the entipetl foe n be witten 6

16 Chpte 5 Poblem 55 m m F ( π / ) 4 π (1) hi expeion n be oled fo. Howee, we mut fit find the entipetl foe tht t on the uite. SOLUIO hee foe t on the uite. hey e the weight mg of the uite, the MAX foe of tti fition f, nd the noml foe F exeted on the uite by the ufe of the ouel. he following figue how the fee body digm fo the uite. In thi digm, the y xi i long the etil dietion. +y he foe of gity t, then, in the y dietion. F MAX f he entipetl foe tht ue the uite to θ moe on it iul pth i poided by the net θ foe in the +x dietion in the digm. Fom the +x digm, we n ee tht only the foe F nd f MAX he hoizontl omponent. hu, we he MAX F f o θ F inθ, whee the minu ign mg indite tht the x omponent of F point to the left in the digm. Uing Eqution 4.7 fo the mximum tti fitionl foe, we n wite thi eult in eqution (). θ F µ F o θ F in θ F ( µ o θ in θ) If we pply ewton' eond lw in the y dietion, we ee fom the digm tht () MAX F o θ + f inθ mg m 0 o F o θ + µ F inθ mg 0 y whee we gin he ued Eqution 4.7 fo the mximum tti fitionl foe. Soling fo the noml foe, we find mg F oθ + µ inθ Uing thi eult in eqution (), we obtin the mgnitude of the entipetl foe tht t on the uite: mg( µ o θ inθ) F F ( µ o θ in θ) oθ + µ inθ With thi expeion fo the entipetl foe, eqution (1) beome mg( µ o θ in θ) mπ 4 oθ + µ inθ

17 56 DYAMICS OF UIFORM CIRCULAR MOIO Soling fo the peiod, we find ( + ) ( + ) 4π oθ µ inθ 4 π (11.0 m) o in 36.0 g( µ o θ in θ) 9.80 m/ o 36.0 in 36.0 ( )( ) 7. SSM WWW REASOIG Eqution 5.5 gie 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 epling the m of the eth by the m of the plnet : GM /. P M E SOLUIO he tio of the obitl peed i, theefoe, 45 M P Soling fo gie 1 GM / P GM / P ( m/) m m REASOIG wo piee of infomtion e poided. One i the ft tht the mgnitude of the entipetl eletion i 9.80 m/. he othe i tht the pe ttion hould not otte fte thn two eolution pe minute. hi te of twie pe minute oepond to thity eond pe eolution, whih i the minimum lue fo the peiod of the motion. With thee dt in mind, we will be ou olution on Eqution 5., whih gie the entipetl eletion, nd on Eqution 5.1, whih peifie tht the peed on iul pth of diu i / π /. SOLUIO Fom Eqution 5., we he o Subtituting π / into thi eult nd oling fo the diu gie 4 m/ ( π / ) ( 9.80 m/ )( 30.0 ) o 4π 4π 3 m 9. REASOIG AD SOLUIO We he fo Jupite GM J /, whee

18 Chpte 5 Poblem m m m hu, ( m / kg )( kg) m/ m 30. REASOIG he peed of the tellite i gien by Eqution 5.1 π /. Sine we e gien tht the peiod i , it will be poible to detemine the peed fom Eqution 5.1 if we n detemine the diu of the obit. o find the diu, we will ue 3/ Eqution 5.6, whih elte the peiod to the diu oding to π / GM, whee G i the uniel gittionl ontnt nd M E i the m of the eth. SOLUIO Aoding to Eqution 5.1, the obitl peed i π o find lue fo the diu, we begin with Eqution 5.6: E 3/ π 3/ o GM E GM π E ext, we que both ide of the eult fo 3/ : GM 3/ E 3 GM o E ( ) π 4π We n now tke the ube oot of both ide of the expeion fo 3 in ode to detemine : ( ) ( m /kg )( kg) GM 3 E 3 7 4π 4π With thi lue fo the diu, we n ue Eqution 5.1 to obtin the peed: 7 ( ) π π m m/ m 31. REASOIG In Setion 5.5 it i hown tht the peiod of tellite in iul obit bout the eth i gien by (ee Eqution 5.6)

19 58 DYAMICS OF UIFORM CIRCULAR MOIO π GM 3/ E whee i the diu of the obit, G i the uniel gittionl ontnt, nd M E i the m of the eth. he tio of the peiod of tellite A nd B i, then, A B π 3/ A 3/ E A 3/ 3/ B B GM π (1) GM E We do not know the dii A nd B. Howee we do know tht the peed of tellite i equl to the iumfeene (π ) of it obit diided by the peiod, o π /. SOLUIO Soling the eltion π / fo gie / π. Subtituting thi lue fo into Eqution (1) yield ( π ) ( π ) ( ) ( ) 3/ 3/ 3/ / A A A A A A 3/ 3/ 3/ B B / B B B B Squing both ide of thi eqution, lgebilly oling fo the tio A / B, nd uing the ft tht A 3 B gie A B B 3 3 B A ( 3 B ) 3. REASOIG AD SOLUIO he peiod of ottion i gien by 4π 3 /GM. Comping the obitl peiod fo Eth nd Venu yield he eth' obitl peiod i 365 dy o ( V / E ) ( V / E ) 3 o tht V / E V (0.611)(365 dy) 3 dy 33. SSM REASOIG he tue weight of the tellite when it i t et on the plnet' ufe n be found fom Eqution 4.4: W ( GM m)/ whee nd m e the me P M P

20 Chpte 5 Poblem 59 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 of the plnet. he m of the plnet n be found by epling M E by M P in Eqution 5.6 nd oling fo M P. When uing Eqution 5.6, we note tht oepond to the diu of the iul obit eltie to the ente of the plnet. SOLUIO he peiod of the tellite i.00 h Fom Eqution 5.6, M P M P π 4 π ( m) + ( m) 3 G ( m / kg )( ) kg Uing Eqution 4.4, we he W GM m 11 4 P ( m / kg )( kg) ( 5850 kg) ( m) 34. REASOIG Eqution 5. fo the entipetl eletion pplie to both the plne nd the tellite, nd the entipetl eletion i the me fo eh. hu, we he plne tellite o plne plne tellite F HG plne tellite he peed of the tellite n be obtined dietly fom Eqution 5.5. I KJ tellite SOLUIO Uing Eqution 5.5, we n expe the peed of the tellite tellite Gm E tellite Subtituting thi expeion into the expeion obtined in the eoning fo the peed of the plne gie plne plne F HG plne I K J tellite F HG plne GmE J plne Gm tellite tellite tellite tellite 11 4 b15 m g m / kg h kgh I K m E 1 m/

21 60 DYAMICS OF UIFORM CIRCULAR MOIO 35. REASOIG AD SOLUIO. he entipetl eletion of point on the im of hmbe A i the tifiil eletion due to gity, A A /A 10.0 m/ A point on the im of hmbe A moe with peed A π A / whee i the peiod of eolution, Subtituting the eond eqution into the fit nd enging yield A A /(4π ) 91 m b. ow B A / m. A point on the im of hmbe B h entipetl eletion B B /B. he point moe with peed B π B /. Subtituting the eond eqution into the fit yield 4π ( ) B 4π 8 m.50 m/ B ( 60.0 ) 36. REASOIG Aoding to Eqution 5.3, the mgnitude F of the entipetl foe tht t on eh penge i F m, whee m nd e the m nd peed of / penge nd i the diu of the tun. Fom thi eltion we ee tht the peed i gien by F / m. he entipetl foe i the net foe equied to keep eh penge moing on the iul pth nd point towd the ente of the ile. With the id of fee-body digm, we will elute the net foe nd, hene, detemine the peed. Penge mg mg SOLUIO he fee-body digm how penge t the bottom of the iul dip. hee e two foe ting: he downwd-ting weight mg nd the upwd-ting foe mg tht the et exet on he. he net foe i +mg mg +mg, whee we he tken up the poitie dietion. hu, F mg. he peed of the penge n be found by uing thi eult in the eqution boe. Subtituting F mg into the eltion F / m yield ( ) F mg g ( 9.80 m/ )( 0.0 m) 14.0 m/ m m

22 Chpte 5 Poblem SSM REASOIG AD SOLUIO Sine the tenion ee the me pupoe the noml foe t point 1 in Figue 5.1, we he, uing the eqution fo the itution t point 1 with F 1 epled by, m mg Soling fo gie F HG I L M m mg m (7.6 m / ) + +g (100 kg) + ( m/ ) KJ QP m O 38. REASOIG he noml foe (mgnitude F ) tht the pilot et exet on him i pt of the entipetl foe tht keep him on the etil iul pth. Howee, thee i nothe ontibution to the entipetl foe, the dwing t the ight how. hi dditionl ontibution i the pilot weight (mgnitude W). o obtin the tio F /W, we will pply Eqution 5.3, / whih peifie the entipetl foe F m. F W + SOLUIO oting tht the dietion upwd (towd the ente of the iul pth) i poitie in the dwing, we ee tht the entipetl foe i F F W. hu, fom Eqution 5.3 we he m F F W he weight i gien by W mg (Eqution 4.5), o we n diide the expeion fo the entipetl foe by the expeion fo the weight nd obtin tht F W m F o F 1 W mg W g Soling fo the tio F /W, we find tht F W ( 30 m/) ( 9.80 m/ )( 690 m) g 39. SSM REASOIG he entipetl foe i the nme gien to the net foe pointing towd the ente of the iul pth. At point 3 t the top the net foe pointing towd the ente of the ile onit of the noml foe nd the weight, both pointing towd the

23 6 DYAMICS OF UIFORM CIRCULAR MOIO ente. At point 1 t the bottom the net foe onit of the noml foe pointing upwd towd the ente nd the weight pointing downwd o wy fom the ente. In eithe e the entipetl foe i gien by Eqution 5.3 F m /. SOLUIO At point 3 we he m F F + mg At point 1 we he m F F mg Subtting the eond eqution fom the fit gie 3 1 Renging gie hu, we find tht m m 3 mg 3 g+ 1 1 hb g b g m/ 3 0 m 15 m/ 17 m/ 40. REASOIG A the motoyle pe oe the top of the hill, it expeiene entipetl foe, the mgnitude of whih i gien by Eqution 5.3 F m, whee m nd e / the m nd peed of the motoyle, nd i the diu of the iul et in the od. he peed of the motoyle i then F m. he entipetl foe i the net foe / ting on the motoyle nd i dieted towd the ente of the ile. When the motoyle et the hill, thee e two foe tht t long the dil dietion, the noml foe F (upwd) tht the od exet on the motoyle nd the weight mg (downwd) of the motoyle nd ide. king the dietion towd the ente of the ile (downwd) the poitie dietion, we he tht F + mg F. When the motoyle jut loe ontt with the od, the noml foe beome zeo. With thi infomtion, we n find the mximum peed tht the yle n he. SOLUIO Subtituting F + mg F into the eltion F / m gie ( ) mg F m

24 Chpte 5 Poblem 63 he mximum peed mx ou when the motoyle jut loe ontt with the od. At thi intnt the noml foe beome zeo. Setting F 0, we he ( )( ) g 9.80 m/ 45.0 m 1.0 m/ mx 41. REASOIG When the tone i whiled in hoizontl ile, the entipetl foe i poided by the tenion h in the ting nd i gien by Eqution 5.3 h Centipetl foe m (1 whee m nd e the m nd peed of the tone, nd i the diu of the ile. When the tone i whiled in etil ile, the mximum tenion ou when the tone i t the lowet point in it pth. he fee-body digm how the foe tht t on the tone in thi itution: the tenion in the ting nd the weight mg of the tone. he entipetl foe i the net foe tht point towd the ente of the ile. Setting the entipetl foe equl to m /, pe Eqution 5.3, we he + m mg ( Centipetl foe Stone Hee, we he umed upwd to be the poitie dietion. We e gien tht the mximum tenion in the ting in the e of etil motion i 15.0% lge thn tht in the e of hoizontl motion. We n ue thi ft, long with Eqution 1 nd, to find the peed of the tone. Solution Sine the mximum tenion in the ting in the e of etil motion i 15.0% lge thn tht in the hoizontl motion, ( ). Subtituting the lue of h nd fom Eqution (1) nd () into thi eltion gie h mg ( ) + h m m + mg ( ) Soling thi eqution fo the peed of the tone yield

25 64 DYAMICS OF UIFORM CIRCULAR MOIO g (9.80 m/ ) (1.10 m) 8.48 m/ REASOIG he dwing t the ight how the two foe tht t on piee of lothing jut befoe it loe ontt with the wll of the ylinde. At tht intnt the entipetl foe i poided by the noml foe F nd the dil omponent of the weight. Fom the dwing, the dil omponent of the weight i gien by mg o φ mg o (90 θ ) mg inθ F θ φ Clothe mg heefoe, with inwd tken the poitie dietion, Eqution 5.3 ( F m / ) gie F + mg in θ At the intnt tht piee of lothing loe ontt with the ufe of the dum, F 0, nd the boe expeion beome m mg in θ m Aoding to Eqution 5.1, π /, nd with thi ubtitution we obtin g in θ ( π / ) 4π hi expeion n be oled fo the peiod. Sine the peiod i the equied time fo one eolution, the numbe of eolution pe eond n be found by lulting 1/. SOLUIO Soling fo the peiod, we obtin π m π π g g 1.17 in θ inθ 9.80 m / in 70.0 h heefoe, the numbe of eolution pe eond tht the ylinde hould mke i e /

26 Chpte 5 Poblem SSM REASOIG AD SOLUIO he mgnitude of the entipetl foe on the bll i gien by Eqution 5.3: F C m /. Soling fo, we he F C (0.08 )(0.5 m) m kg 0.68 m / 44. REASOIG AD SOLUIO he noml foe exeted by the wll on eh tonut i the entipetl foe needed to keep him in the iul pth, i.e., F m /. Renging nd letting F (1/)mg yield /g (35.8 m/) /(9.80 m/ ) 6 m 45. REASOIG AD SOLUIO Let epeent the length of the pth of the pebble fte it i eleed. Fom Coneptul Exmple, we know tht the pebble will fly off tngentilly. heefoe, the pth i pepependiul to the diu of the ile. hu, the ditne,, nd d fom ight tingle with hypotenue d hown in the figue t the ight. Fom the figue we ee tht o 1 o α d α o 1F H G I K J get d α 35 θ C Pebble Futhemoe, fom the figue, we ee tht α +θ heefoe, θ 145 α

27 66 DYAMICS OF UIFORM CIRCULAR MOIO 46. REASOIG AD SOLUIO he foe P upplied by the mn will be lget when the ptne i t the lowet point in the wing. he digm t the ight how the foe ting on the ptne in thi itution. he entipetl foe neey to keep the ptne winging long the of ile i poided by the eultnt of the foe upplied by the mn nd the weight of the ptne. Fom the figue heefoe, P mg P m m +mg P mg Sine the weight of the ptne, W, i equl to mg, it follow tht m (W/g) nd ( Wg / ) [(475 )/(9.80 m/ )] (4.00 m/) P + W + (475 ) 594 (6.50 m) 47. SSM REASOIG AD SOLUIO Sine the mgnitude of the entipetl eletion i gien by Eqution 5.,, we n ole fo nd find tht C / ( 98.8 m / ) 3.00(9.80 m / ) C 33 m 48. REASOIG he entipetl foe i the nme gien to the net foe pointing towd the ente of the iul pth. At the lowet point the net foe onit of the tenion in the m pointing upwd towd the ente nd the weight pointing downwd o wy fom the ente. In eithe e the entipetl foe i gien by Eqution 5.3 F m /. SOLUIO () he entipetl foe i F b gb g m 95. kg.8 m / 085. m 88 (b) Uing to denote the tenion in the m, t the bottom of the ile we he m F mg ( )( ) ( )( ) m 9.5 kg.8 m/ mg+ 9.5kg 9.80 m/ m

28 Chpte 5 Poblem SSM REASOIG A the motoyle pe oe the top of the hill, it will expeiene entipetl foe, the mgnitude of whih i gien by Eqution 5.3: F m /. he C 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 weight, whih point downwd. king the dietion towd the ente of the ile (downwd) the poitie dietion, we he F mg F C. hi expeion n be oled fo F, the noml foe. SOLUIO. he mgnitude of the entipetl foe i F C b. he mgnitude of the noml foe i m (34 kg)(5.0 m / ) m 3 3 F mg FC (34 kg)(9.80 m/ ) REASOIG AD SOLUIO he peiod of the moon' motion (ppoximtely the length of month) i gien by π ( ) 3 8 4π m GM E 11 4 ( m / kg )( kg) dy 51. REASOIG AD SOLUIO he mple mke one eolution in time gien by π /. he peed i ( m)( )(9.80 m/ ) o tht 55.3 m/ he peiod i π ( m)/(55.3 m/) min he numbe of eolution pe minute 1/ e/min. 5. REASOIG AD SOLUIO. At the equto peon tel in ile whoe diu equl the diu of the eth, R e m, nd whoe peiod of ottion i 1 dy We he

29 68 DYAMICS OF UIFORM CIRCULAR MOIO he entipetl eletion i πr e / 464 m/ ( 464 m/) m/ m 6 b. At 30.0 ltitude peon tel in ile of diu, hu, R e o m π / 40 m/ nd /.9 10 m/ 53. REASOIG. he fee body digm how the wing ide nd the two foe tht t on hi: the tenion in the ble, nd the weight mg of the hi nd it oupnt. We note tht the hi doe not elete etilly, o the net foe F in the etil dietion mut be y zeo, F 0. he net foe onit of the upwd etil omponent of the tenion nd y the downwd weight of the hi. he ft tht the net foe i zeo will llow u to detemine the mgnitude of the tenion m +y +x m g b. Aoding to ewton eond lw, the net foe F in the hoizontl dietion i x equl to the m m of the hi nd it oupnt time the entipetl eletion ), o tht F m m. hee i only one foe in the hoizontl ( / x / dietion, the hoizontl omponent of the tenion, o it i the net foe. We will ue ewton eond lw to find the peed of the hi. SOLUIO. he etil omponent of the tenion i + o 60.0, nd the weight i mg, whee we he hoen up the + dietion. Sine the hi nd it oupnt he no etil eletion, we he tht F 0, o y

30 Chpte 5 Poblem 69 + o 60.0 mg 0 F y (1 ) Soling fo the mgnitude of the tenion gie ( 179 kg)( 9.80 m/ ) mg o 60.0 o b. he hoizontl omponent of the tenion i + in 60.0, whee we he hoen the dietion to the left in the digm the + dietion. Sine the hi nd it oupnt he entipetl eletion in thi dietion, we he in 60.0 m m ( F ) x Fom the dwing we ee tht the diu of the iul pth i (15.0 m) in m. Soling Eqution () fo the peed gie ( )( ) in m 3510 in m/ m 179 kg 54. REASOIG AD SOLUIO. he entipetl foe i poided by the noml foe exeted on the ide by the wll. b. ewton' eond lw pplied in the hoizontl dietion gie F m / (55.0 kg)(10.0 m/) /(3.30 m) ewton' eond lw pplied in the etil dietion gie µ F mg 0 o µ (mg)/f SSM WWW REASOIG If the effet of gity e not ignoed in Exmple 5, the plne will mke n ngle θ with the etil hown in figue A below. he figue B how the foe tht t on the plne, nd figue C how the hoizontl nd etil omponent of thee foe.

31 70 DYAMICS OF UIFORM CIRCULAR MOIO L θ L θ in θ o θ mg mg A B C Fom figue C we ee tht the eultnt foe in the hoizontl dietion i the hoizontl omponent of the tenion in the guideline nd poide the entipetl foe. heefoe, in θ m Fom figue A, the diu i elted to the length L of the guideline by L inθ ; theefoe, m in θ (1) L in θ he eultnt foe in the etil dietion i zeo: oθ mg 0, o tht Fom eqution () we he oθ mg () mg o θ (3) Eqution (3) ontin two unknown, nd θ. Fit we will ole eqution (1) nd (3) imultneouly to detemine the lue() of the ngle θ. One θ i known, we n lulte the tenion uing eqution (3). SOLUIO Subtituting eqution (3) into eqution (1): hu, F mg I in θ HG o θ K J in o θ θ gl m L in θ (4) Uing the ft tht o θ + in θ 1, eqution (4) n be witten

32 Chpte 5 Poblem 71 o 1 o o θ θ 1 o θ o θ gl gl hi n be put in the fom of n eqution tht i qudti in o θ. Multiplying both ide by o θ nd enging yield: Eqution (5) i of the fom o θ + o θ 1 0 gl (5) x + bx + 0 (6) with x o θ, 1, b /(gl), nd 1. he olution to eqution (6) i found fom the qudti fomul: x b b ± 4 When 19.0 m/, b.17. he poitie oot fom the qudti fomul gie x o θ Subtitution into eqution (3) yield mg (0.900 kg)(9.80 m / ) o θ When 38.0 m/, b he poitie oot fom the qudti fomul gie x o θ Subtitution into eqution (3) yield mg (0.900 kg)(9.80 m / ) 77 o θ COCEP QUESIOS. he peiod fo the eond hnd i the time it tke fo it to go one ound the ile o eond 60. b. he peiod fo the minute hnd i the time it tke fo it to go one ound the ile o minute 1 h he eltionhip between the entipetl eletion nd the peiod n be obtined by uing Eqution 5. nd 5.1 in the following wy:

33 7 DYAMICS OF UIFORM CIRCULAR MOIO π (5.) (5.1) ( π ) / 4π SOLUIO Uing the expeion fo the entipetl eletion obtined in the nwe to onept quetion, we he, eond, minute 4π / eond 4π / b b g g minute minute eond 57. COCEP QUESIOS. Exmple h the mllet entipetl eletion. Unifom iul motion on ile with n infinitely lge diu i like motion long tight line t ontnt eloity. In uh e thee i no entipetl eletion. b. Exmple 1 h the getet entipetl eletion. Aoding 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. heefoe, the getet peed nd the mllet diu, i the e fo Exmple 1, podue the getet entipetl eletion. SOLUIO With Eqution 5., we find the following lue of the entipetl eletion. Exmple 1 Exmple 1 m/ 050. m b b b 35 m / 90 m/ 0 m/.3 m / Exmple 3.9 m / 1.8 m 58. COCEP QUESIOS. he entipetl eletion depend only on the peed nd g g the diu of the ue, oding to Eqution 5. ( /). he peed of the e the me, nd ine they e negotiting the me ue, the diu i the me. heefoe, the he the me entipetl eletion. g

34 Chpte 5 Poblem 73 b. he entipetl foe depend on the m m, well the peed nd the diu of the ue, oding to Eqution 5.3 (F m /). Sine the peed nd the diu e the me fo eh, the with the gete m, whih i B, expeiene the gete entipetl eletion. SOLUIO Uing Eqution 5. nd 5.3, we find the following lue fo the entipetl eletion nd foe: C A C B F b 7 m/ 10 m m 1100 kg 7 m / A 10 m 7 m/ 10 m g b.1 m / (5.) 6700 ( 5.3).1 m / (5.) m 1600 kg 7 m / B F 9700 ( 10 m 6 b gb g g 6 b gb g 5.3) 59. COCEP QUESIOS. Stti, the thn kineti, fition poide the entipetl foe, beue the penny i ttiony nd not liding eltie to the dik. b. he peed n be detemined fom the peiod of the motion nd the diu, oding to Eqution 5.1 ( π /). SOLUIO Uing Eqution 5.3 fo the entipetl foe (F m /) nd Eqution 5.1 fo the peed in tem of the peiod ( π /), we he b g (1) m m F m π / 4π mx µ F Aoding to Eqution 4.7, the mximum foe of tti fition i f, whee F i the noml foe. Sine the penny doe not elete in the etil dietion, the upwd noml foe mut be blned by the downwd-pointing weight, o tht F mg nd mx f µ mg. Uing eqution (1), we find

35 74 DYAMICS OF UIFORM CIRCULAR MOIO m π µ mg 4 Centipetl foe b g hb g 4π 4π m µ g 980. m/ COCEP QUESIOS. he tellite with the lowe obit h the gete peed, oding to Eqution 5.5 ( GM E / ), whee i the diu of the obit. hu, tellite A h the gete peed. b. In Eqution 5.5 ( GM E / ) the tem i the obitl diu, meued fom the ente of the eth, not the ufe of the eth. heefoe, you do not ubtitute the height of m nd m fo the tem. Inted, thee height mut be dded to the diu of the eth ( m) in ode to get the dii. SOLUIO Fit we dd the obitl height to the diu of the eth to obtin the obitl dii. hen we ue Eqution 5.5 to lulte the peed. Stellite A A m m m GM E A m / kg h kgh m 7690 m/ Stellite B A m m m GM E A m / kg h kgh m 7500 m / 61. COCEP QUESIOS. 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 gity, ting downwd, i pependiul to the ting nd nnot ontibute to the entipetl foe. (See Figue 5.1, point fo imil itution.) At thi point, only the tenion of 16 ontibute to the entipetl foe. Conideing tht the entipetl foe i the me eeywhee, we n onlude tht it h lue of 16 eeywhee.. At the twele o lok poition the tenion nd the foe of gity mg both t downwd (the negtie dietion) towd the ente of the ile, with the eult tht the entipetl foe t thi point i mg. (See Figue 5.1, point 3.) he mgnitude of the

36 Chpte 5 Poblem 75 entipetl foe hee, then, i + mg. At the ix o lok poition the tenion point upwd towd the ente of the ile, while the foe of gity point downwd, with the eult tht the entipetl foe t thi point i mg. (See Figue 5.1, point 1.) he only wy fo entipetl foe to he the me mgnitude of 16 t both of thee ple i fo the tenion t the ix o lok poition to be gete. he gete tenion ompente fo the ft tht the foe of gity point wy fom the ente of the ile. SOLUIO Auming tht upwd i the poitie dietion, we find t the twele nd ix o lok poition tht wele o' lok Six o' lok mg 16 Centipetl foe kg m / 14 mg 16 Centipetl foe b g h b g h kg m / 18