B.Sc. MATHEMATICS - III YEAR

Transcription

1 MANONMANIAM SUNDARANAR UNIVERSITY DIRECTORATE OF DISTANCE & CONTINUING EDUCATION TIRUNELVELI 670, TAMIL NADU B.Sc. MATHEMATICS - III YEAR DJMC - MECHANICS (Fom the cdemic ye 06-7) Most Stdet fiedly Uivesity - Stive to Stdy d Le to Excel Fo moe ifomtio visit:

2 DJMC - MECHANICS Uit I: Foces ctig t poit Pllelogm of foces tigle of foces Lmi s Theoem, Pllel foces d momets Coples Eqilibim of thee foces ctig o igid body. Uit II: Fictio Lws of fictio eqilibim of pticle (i) o ogh iclied ple, (ii) de foce pllel to the ple, (iii) de y foce eqilibim of stigs eqtio of the commo ctey tesio t y poit geometicl popeties of commo ctey ifom chi de the ctio of gvity sspesio bidge. Uit III: Dymics pojectiles eqtio of pth, ge etc ge o iclied ple motio o iclied ple. Implsive foces collisio of elstic bodies lws of impct diect d obliqe impct impct o fixed ple. Uit IV: Simple hmoic motio i stight lie geometicl epesettio compositio of SHMS of the sme peiod i the sme le d log two pepedicl diectios pticles sspeded by spig SHM o cve simple pedlm simple eqivlet pedlm secod s pedlm. Uit V: Motio de the ctio of cetl foces velocity d cceletio i pol co-odites diffeetil eqtio of cetl obit pedl eqtio of cetl obit pses psidl distces ivese sqe lw. Refeece Books:. Sttics d Dymics: S. Ny. Sttics d Dymics: M.K. Vektm. Sttics : Mickvchgompilli. Dymics : Dipdi

3 Itodctio Mthemtics is the Qee of the Scieces d Nmbe Theoy is the Qee of Mthemtics - Gss. Mechics is bch of Sciece which dels with the ctio of foces o bodies. Mechics hs two bches clled Sttics d Dymics. Sttics is the bch of Mechics which dels with bodies emi t est de the iflece of foces. Dymics is the bch of Mechics which dels with bodies i motio de the ctio of foces. Defiitios: Spce: The egio whee vios evets tke plce is clled spce. Body: A potio of mtte is clled body. Rigid body: A body cosists of imeble pticles i which the distce betwee y two pticles emis the sme i ll positios of the body is clled igid body. Pticle: A pticle is body which is vey smll whose positio t y time coicides with poit. Motio: If body chges its positio de the ctio of foces, the it is sid to be i motio. Pth of pticle: It is the cve joiig the diffeet positios of the pticle i spce while i motio. Speed: The te t which the body descibes its pth. It is scl qtity. Displcemet (vecto qtity): It is the chge i the positios of pticle i ceti itevl. Velocity (vecto qtity): It is the te of chge of displcemet. Acceletio (vecto qtity): It is the te of chge of velocity. Eqilibim: A body t est de the ctio of y mbe of foces o it is sid to be i eqilibim. Eqilibim of two foces Q P If two foces P, Q ct o body sch tht they hve eql mgitde, opposite diectios, sme lie of ctio the they e i eqilibim. Foce (vecto): Foce is y cse which podces o teds to podce chge i the existig stte of est of body o of its ifom motio i stight lie. Foce is epeseted by stight lie (thogh the poit of pplictio) which hs both mgitde d diectio. Types of foces: Weight, ttctio, eplsio, tesio, thst, fictio etc. By Newto s thid lw, ctio d ectio e lwys eql d opposite.

4 Diectios of Noml Rectio R t the poit of cotct.. Whe od AB is i cotct with smooth ple, R is pepedicl to the ple t the poit of cotct A. R od B Smooth A Ple. Whe od AB is estig o smooth peg P, R is pepedicl to the od t the poit of cotct P. A R B O P -Peg. Whe od AB is estig o smooth sphee, R is oml to the sphee t the poit of cotct C. A R C B. Whe od AB is estig o the im of hemisphee, with R oe ed A i cotct with the ie sfce d C i cotct with the im. The the oml O R C B ectios R t A is oml to A the spheicl sfce d psses thogh the cete O, R t C is pepedicl to the od. Regl polygo is polygo with eql sides. Its vetices lie o cicle.

5 Itodctio UNIT I Foces Actig t Poit Foces e epeseted by stight lies with mgitde d diectio. Foces ctig o igid body my be epeseted by stight lies with mgitde d diectio pssig thogh the sme poit d we sy the foces e ctig t poit. If P P, P.. e the foces ctig o igid body it is esy to fid sigle foce whose effect is sme s the combied effect of P, P, P.. The the sigle foce is clled the esltt. P, P, P.. e clled the compoets of the esltt. I this sectio we stdy some theoems d methods to fid the esltt of two o moe foces ctig t poit.. Pllelogm lw of foces (Fdmetl theoem i sttics) If two foces ctig t poit be epeseted i mgitde d diectio by the sides of pllelogm dw fom the poit, thei esltt is epeseted both i mgitde d diectio by the digol of the pllelogm dw thogh tht poit. Q D R C, AB AD AC A P B ie) P + Q R The esltt of two foces ctig t poit D C Q R A P B E Let the two foces P d Q ctig t A be epeseted by AB d AD. Let be the gle betwee them. i.e. BAD Complete the pllelogm ABCD. The the digol AC will epeset the esltt.

6 5 Let CAB Dw CE to AB. Now BC AD Q. Fom the ight gled CBE, si C B E CE i.e. si BC CE Q CE Q si (i) cos BE BC BE Q BE Q cos (ii) R AC AE + CE (AB + BE) +CE (P + Q cos ) + (Q si ) P + PQcos + Q R P PQcos Q t CE AE Qsi P Qcos Reslt If the foces P d Q e t ight gles to ech othe, the 90 o ; R P Q t Q P Reslt If the foces e eql (i.e.) Q P, the R P P cos P P cos t ie) P.cos P cos si cos Psi si P Pcos cos cos t

7 6 Ths the esltt of two eql foces P, P t gle is P cos i diectio bisectig the gle betwee them. Reslt Resltt R is getest whe cos is getest. i.e. whe cos o 0 o. ie) Getest vle of R is R P +Q. R is lest whe cos is lest. i.e. whe cos o 80 o. Lest vle of R is P~Q. Poblem The esltt of two foces P, Q ctig t ceti gle is X d tht of P, R ctig t the sme gle is lso X. The esltt of Q, R gi ctig t the sme gle is Y, Pove tht. P (X + QR ) QR Q R Q R Y Pove lso tht, if P + Q + R 0, Y X. Soltio: Let be the gle betwee P d Q Give X P + Q + PQ cos... () X P + R + PR cos... () Y Q + R + QR cos... () () () gives 0 Q R + P cos ( Q R) i.e. 0 (Q R) (Q+R+P cos ) Bt Q R d so Q R 0 Q + R + Pcos 0 Q R cos... () P Sbstitte () i (), X P + Q Q R + PQ P + Q Q QR P P X + QR. i.e. P (X + QR )

8 7 Sbstitte () i (), Y Q + R Q R + QR P Q + R QR Q R QR Q R Q + R Y P P QR Q R Q R Y If P + Q + R 0, the Q + R P, Fom (), cos (5) (6) gives X Y cos P Q R P P P X P + R + PR (5) Y Q + R + QR (6) X Y P Q + PR QR (P Q) (P + Q + R) (P Q).0 0 Poblem Two foces of give mgitde P d Q ct t poit t gle. Wht will be the mximm d miimm vle of the esltt? Soltio: i. Mximm vle of the esltt P + Q ii. Miimm vle of the esltt P~ Q.

9 8 Poblem The getest d lest mgitdes of the esltt of two foces of costt mgitdes e R d S espectively. Pove tht, whe the foces ct t gle, the esltt is of mgitde R cos S si Soltio: Resltt the foces is Soltio: Give, R P + Q, S P-Q, whee P d Q e two foces. Whe P d Q e ctig t gle P Q PQ.cos P Q PQcos si P Q si cos PQcos si P Q PQcos P Q PQsi R cos S si. Poblem The esltt of two foces P d Q is t ight gles to P. Show tht the gle betwee cos P Q Let be the gle betwee the two foces P d Q. Give 90 o. D C Q R A P B We kow, t i.e. t 90 o Qsi P Q cos Qsi P Q cos

10 9 0 P P Qcos 0 cos cos P Q P Q Qsi Q cos Poblem 5 The esltt of two foces P d Q is of mgitde P. Show tht, if P be dobled, the ew esltt is t ight gles to Q d its mgitde will be P Q. Soltio: Let be the gle betwee P d Q D C P P A Q B Give, P P + Q + PQ cos. Q (Q+Pcos ) 0 Q cos P If P is dobled, let R be the ew esltt, d be the gle betwee Q d R. R P Q PQ.cos Q P Q PQ P P Q Q P Q R P Q

11 0 t Q Psi Pcos i.e. t Psi Q Q P P P si 0 cos 0 φ 90 0 Q is t ight gles to R. Poblem 6 Two eql foces ct o pticle, fid the gle betwee them whe the sqe of thei esltt is eql to thee times thei podct. Soltio: D C P R A P B Let be the gle betwee the two eql foces P, P, d let R be thei esltt. R P P P. P.cos P cos P cos i.e. R P cos R Pcos Give, R P P P P P cos cos cos

12 α o Poblem 7 If the esltt of foces P, 5P is eql to 7P fid i. the gle betwee the foces ii. the gle which the esltt mkes with the fist foce. Soltio: Let be the gle betwee P, 5P i. Give (7P) (P) + (5P) + (P) (5P).cos 9P 9P + 5P + 0P cos 5P 0P cos cos α 60 0 ii. Let be the gle betwee the esltt d P. t Qsi P Q cos 5P.si P 5P.cos 5P.si 60 P 5P.cos60 5 5

13 t 5 5 t. Tigle of foces If thee foces ctig t poit c be epeseted i mgitde d diectio by the sides of tigle tke i ode, they will be i eqilibim. M D C Q R Q Q O R P L A P B N Let the foces, P,Q,R ct t poit O d be epeseted i mgitde d diectio by the sides AB,BC,CA of the tigle ABC. To pove : They will be i eqilibim. Complete the pllelogm BADC. P+Q AB + AD AB + BC AC ie) The esltt of the foces P, Q t O is epeseted i mgitde d diectio by AC. The thid foce R cts t O d it is epeseted i mgitde d diectio by CA. Hece P+Q+R AC + CA 0

14 Piciple If two foces ctig t poit e epeseted i mgitde d diectio by two sides of tigle tke i the sme ode, the esltt will be epeseted i mgitde d diectio by the thid side tke i the evese ode.. Lmi s Theoem If thee foces ctig t poit e i eqilibim, ech foce is popotiol to the sie of the gle betwee the othe two. M X Q B D R O A P L Y Z N Poof: By covese of the tigle of foces, the sides of the tigle OAD epeset the foces P,Q,R i mgitde d diectio. By sie le i OAD, we hve OA AD DO si ODA si DOA si OAD. () Bt OAD lt. BOD 80 si ODA si 80 Also DOA 80 si DOA si 80 0 MON 0 MON si MON 0 NOL 0 NOL si NOL.. (). ()

15 Ad OAD 80 si OAD si 80 Sbstitte (), (), () i (), 0 BOA 80 0 LOM 0 LOM si LOM OA AD DO si MON si NOL si LOM i.e. si P MON si Q NOL si P Q R si( Q. R) si( R, P) si( P, Q) R LOM. () Poblem 8 Two foces ct o pticle. If the sm d diffeece of the foces e t ight gles to ech othe, show tht the foces e of eql mgitde. Soltio: D C Q A P B Let the foces P d Q ctig t A be epeseted i mgitde d diectio by the lies AB d AD. Complete the pllelogm BAD. The P+Q P-Q AB AD AB DA AB AD AC DA AB DB

16 5 Give AC d DB e t ight gles. The digols AC d BD ct t ight gles. ABCD mst be hombs. AB AD. P Q. Poblem 9 Let A d B two fixed poits o hoizotl lie t distce c pt. Two fie light stigs AC d BC of legths b d espectively sppot mss t C. Show tht the tesios of b c b : b c the stigs e i the tio Soltio A c D B T T b C E W Foces T, T, W e ctig t C. By Lmi s theoem, T T...() si ECB si ECA 0 Now si ECB si 80 DCB si DCB si 90 0 ABC cosabc 0 si ECA si 80 ACD si ACD si 90 0 BAC cosbac

17 6 T T cosabc cos BAC T T T c cos B cos A b b c c bc T c b bc bc b c b c b c Poblem 0 ABC is give tigle. Foces P,Q,R ctig log the lies OA,OB,OC e i eqilibim. Pove tht (i)p : Q : R b c : b c b : c b c if O is the cicmcete of the tigle. A B C (ii) P : Q : R cos : cos : cos if O is the icete of the tigle. (iii) P : Q : R :b:c if O is the otho cete of the tigle. (iv) P : Q : ROA : OB : OC if O is the cetoid of the tigle, Soltio: A A P F B Q O R C B D O E C By Lmi s theoem, P Q R si BOC si COA si AOB () (i) O is the cicmcete of the ABC BOC BAC A; COA B d AOB C

18 7 C R B Q A P si si si ) ( i.e. C C R B B Q A A P cos si cos si cos si. () Bt bc c b A cos d bc A si whee is the e of the tigle ABC bc bc c b A A cos si c b c b Similly cos si c b c B B si CcosC b c b Sbstitte i ().. c b b R b c Q c c b c P b Divide by c b c b c R b c b Q c b P (ii) O is the i-cete of the tigle, OB d OC e the bisectos of B d C

19 8 BOC 80 B C 80 B 0 0 C 0 0 A 0 A Similly COA 90 B, 0 0 C AOB 90 P Q R () 0 A 0 B 0 C si 90 si 90 si 90 i.e. P A cos Q B cos R C cos (iii) O is the otho-cete of the tigle AD, BE, CF e the ltitdes of the tigle AFOE is cyclic qdiltel. 0 FOE A 80, FOE 80 A BOC 0 80 A 0 Similly, COA 80 B, AOB 80 C Hece () becomes P Q A si 80 B si 80 C si 80 i.e. i.e. P Q R si A si B si C P Q R b c b c si A si B si C 0 R 0

20 9 (iv) O is the cetoid of the tigle BOC COA AOB BOC OB. OC si BOC ABC si BOC OB. OC ABC ABC Similly, ABC si COA, OC. OA ABC si AOB OAOB. Hece () becomes P.OB. OC Q.OC. OA R.OAOB. ABC ABC ABC i.e. P.OB.OC Q.OC.OA R.OA.OB Dividig by OA.OB.OC, we get P OA Q R. OB OC. Pllel foces: Foces ctig log pllel lies e clled pllel foces. Thee e two types of pllel foces kow s like d like pllel foces. Sice the pllel foces do ot meet t poit, i this chpte we stdy methods to fid the esltt of two like pllel d like pllel foces. Pllel foces ctig o igid body hve tedecy to otte it bot fixed poit. Sch tedecy is kow s momet of the pllel foces. Hee we stdy the theoem o momets of foces bot poit. Defiitio: Two pllel foces e sid to be like if they ct i the sme diectio, they e sid to be like if they ct i opposite pllel diectios.

21 0 The esltt of two like pllel foces ctig o igid body Y F O F Y P G F A C B F N Q R P X E D Q R L M Poof: Let P d Q be two like pllel foces ctig t A d B log the lies AD d BL.At A d B, itodce two eql d opposite foces F log AG d BN. These two foces F blce ech othe d will ot ffect the system. Now, R is the esltt of P d F t A d R is the esltt of Q d F t B s i the digm. Podce EA d MB to meet t O. At O, dw YOY pllel to AB d dw OX pllel to the diectio of P. Resolve R d R t O ito thei oigil compoets. R t O is eql to F log OY d P log OX. R t O is eql to F log OY d Q log OX. The two foces F, F t O ccel ech othe. The emiig two foces P d Q ctig log OX hve the esltt P+Q (sm) log OX.

22 Fid the positio of the esltt Now, AB d OX meet t C. Tigles, OAC d AED e simil. OC AC OC AC ie) AD ED P F F. OC P. AC () Tigles OCB d BLM e simil. OC CB OC CB ie) BL LM Q F F. OC Q. CB.. () () & () P.AC Q.CB ie) AC CB Q P ie) C divides AB itelly i the ivese tio of the foces. The esltt of two like d eql pllel foces ctig o igid body: O Y F F Y P Q E D C P B F N X G F R A Q R L M

23 Poof: Let P d Q t A d B be two eql like pllel foces ctig log AD d BL. Let P > Q. At A d B itodce two eql d opposite foces F log AG d BN. These two blces ech othe d will ot ffect the system. Let R be the esltt of F d P t A d R be the esltt of F d Q t B. s i the digm. Podce EA d MB to meet t O. At O, dw Y OY pllel to AB d dw OX pllel to the diectio of P. Resolve R d R t O ito thei compoets. R t O is eql to F log XO. R t O is eql to F log OY d Q log OX. OY d P log The two foces F, F t O ccel ech othe. Now, the emiig foces e P d Q log the sme lie bt opposite diectios. Hece the esltt is P ~ Q (diffeece) log XO. Fid the positio of the esltt Now, AB d OX meet t C. Tigles OCA d EGA e simil. OC CA OC CA, ie) EG GA P F F. OC P. AC () Tigles OCB d BLM e simil. OC CB OC CB, ie) BL LM Q F F. OC QCB. () () d () P.AC Q.CB ie) CA Q CB P ie) C divides AB extelly. Note : The effect of two eql d like pllel foces c ot be eplced by sigle foce.

24 The coditio of eqilibim of thee copl pllel foces P P+Q Q A C B R Let P, Q, R be the thee copl pllel foces i eqilibim. Dw lie to meet the foces P, Q, R t the poits A, B, C espectively. Eqilibim is ot possible if ll the thee foces e i the sme diectio. Let P + Q be the esltt of P d Q pllel to P. Hece R mst be eql d opposite to P + Q. R P + Q (i mgitde, opposite i diectio) P. AC QCB. P CB Q AC P Q CB AC R AB Hece, P CB Q AC R AB ie) If thee pllel foces e i eqilibim the ech foce is popotiol to the distce betwee the othe two. Note: The cete of two pllel foces is fixed poit thogh which thei esltt lwys psses. Poblem Two me, oe stoge th the othe, hve to emove block of stoe weighig 00 kgs. with light pole whose legth is 6 mete. The weke m cot cy moe th 00 kgs. Whee the stoe be fsteed to the pole, so s jst to llow him his fll she of weight?

25 Soltio: x 6 x A C B Let A be the weke m beig 00 kgs., B the stoge m beig 00 kgs. Let C be the poit o AB whee the stoe is fsteed to the pole, sch tht AC x. The the weight of the stoe ctig t C is the esltt of the pllel foces 00 d 00 t A d B espectively. 00.AC 00.BC m. i.e. 00x 00 (6-x) 00 00x 00x 00 o x Hece the stoe mst be fsteed to the pole t the poit distt metes fom the weke Poblem Two like pllel foces P d Q ct o igid body t A d B espectively. P ) If Q be chged to, show tht the lie of ctio of the esltt is the sme s it wold Q be if the foces wee simply itechged. b) If P d Q be itechged i positio, show tht the poit of pplictio of the esltt will P Q be displyed log AB thogh distce d, whee d. AB. P Q Soltio: P Q A C D B

26 5 Let C be the cete of the two foces. The P. AC Q.CB. () () If Q is chged to foces. P, (P emiig the sme), let D be the ew cete of pllel Q P The P.AD DB.... () Q Q.AD P.DB. () Reltio () shows tht D is the cete of two like pllel foces, with Q t A d P t B. (b) Whe the foces P d Q e itechged i positio, D is the ew cete of pllel foces. Let CD d Fom (), Q. (AC+CD) P. (CB CD) i.e. Q.AC + Q.d P.CB P.d (Q + P).d P.CB Q.AC P (AB AC) Q (AB CB) (P Q).AB[P.AC Q.CB fom ()] d P P Q. AB Q Poblem The positio of the esltt of two like pllel foces P d Q is lteed, whe the positio of P d Q e itechged. Show tht P d Q e of eql mgitde. Soltio: P Q Q P A C B A C B

27 6 Let C be the cete of two like pllel foces P t A d Q t B. P.AC Q.CB () Whe P d Q e itechged, the cete C is ot lteed (give) Q.AC P.CB. () () P Q P Q Q P P Q Poblem P d Q e like pllel foces. If Q is moved pllel to itself thogh distce x, pove tht Qx the esltt of P d Q moves thogh distce. P Q Soltio: P Q Q x A C D B B Let C be the cete of P d Q t A d B. P. AC Q. CB. () Let D be the ew cete of P t A d Q t B sch tht P. AD Q. DB () ie) PAC CD QDB BB QCB CD x BB x

28 7 P QCD Q. x sig () CD Qx P Q Poblem 5 Two like pllel foces P d Q (P>Q) ctig o igid body t A d B espectively be itechged i positio, show tht the poit pplictio of the esltt i AB P Q will be displyed log AB thogh distce AB. P Q Soltio: P C D A B Q Let C be the cete of two like pllel foces P t A d Q t B. P. AC Q. CB () Let D be the ew cete whe P d Q e itechged i positio. Q. AD P. DB.. () i.e.) QAC CD P. DA AB i.e.) QCB AB CD P. AC CD AB Q. CB Q. AB QCD. P. AC PCD. P. AB P Q. CD P Q. AB sig () P Q CD. AB P Q

29 8 Poblem 6 A light od is cted o by thee pllel foces P, Q, d R, ctig t thee poits distt, 8 d 6 ft. espectively fom oe ed. If the od is i eqilibim, show tht P: Q: R ::. Soltio P Q A B D C R P, Q, R e pllel foces ctig o the od AD t B, D, C espectively. Give, AB ft, AD 8ft, AC 6ft. BC ft, CD ft, BD 6ft. Fo eqilibim of the od, ech foce shold be popotiol to the distce betwee the othe two. P Q R P : Q : R 6 : : 6 P : Q : R : :.5 Momet of foce (o) Tig effect of foce Defiitio: The momet of foce bot poit is defied s the podct of the foce d the pepedicl distce of the poit fom the lie of ctio of the foce. p O A F N B

30 9 Momet of F bot O F x ON F x p. Note: Momet of F bot O is zeo if eithe F O (o) ON O. i.e.) F 0 (o) AB psses thogh O. Hece, momet of foce bot y poit is zeo if eithe the foce itself is zeo (o) the foce psses thogh tht poit. Physicl sigificce of the momet of foce It meses the tedecy to otte the body bot the fixed poit. Geometicl Repesettio of momet O O A F B N A F N B Let AB epeset the foce F both i mgitde d diectio d O be y give poit. the momet of the foce F bot O F x ON AB x ON. AOB Twice the e of the tigle AOB Sig of the momet If the foce teds to t the body i the ticlockwise diectio, momet is positive. If the foce teds to t the body i the clockwise diectio, momet is egtive. Vigo s Theoem of Momets The lgebic sm of the momets of two foces bot y poit i thei ple is eql to the momet of thei esltt bot tht poit.

31 0 Poof: Cse Let the foces be pllel d O lies i) Otside AB P+Q R P Q O A C B Let P d Q be the two pllel foces ctig t A d B. P + Q be thei esltt R ctig t C. sch tht P.AC Q.CB.. () Algebic sm of the momets of P d Q bot O P.OA + Q.OB P x (OC AC) + Q x (OC + CB) (P +Q).OC P.AC +Q.CB (P+Q).OC sig () R.OC momet of R bot O. ii) P d Q e pllel d O lies withi AB A C O B P RP+Q Q Algebic sm of the momets of P d Q bot O P.OA Q.OB P. (OC+CA) Q. (CB CO) (P+Q).OC + P.CA Q.CB by () R.OC momet of R bot O.

32 Cse II iii) P d Q meet t poit d O y poit i thei ple. O lies otside the gle BAD O D C Q R A P B Thogh O, dw lie pllel to the diectio of P, to meet the lie of ctio of Q t D. Complete the pllelogm ABCD sch tht AB, AD epeset the mgitde of P d Q d the digol AC epesets the esltt R of P d Q. Algebic sm of the momets of P d Q bot O. AOB +. AOD ACB +. AOD [ AOB ACB] ADC + AOD ( ADC + AOD). AOC Momet of R bot O. iv) O lies iside the gle BAD Algebic sm of the momets of P d Q bot O: AOB AOD ACB AOD ADC AOD ( ADC AOD). AOC momet of R bot O. D O C Q R A P B

33 Poblem 7 Two me cy lod of kg. wt, which hgs fom light pole of legth 8 m. ech ed of which ests o sholde of oe of the me. The poit fom which the lod is hg is m. ee to oe m th the othe. Wht is the pesse o ech sholde? Soltio R R x C A B AB is the light pole of legth 8m. C is the poit fom which the lod of kgs. is hg. Let AC x. The BC 8 x. give ( 8 x) x i.e) 8 x 0 x 6. x. i.e. AC d BC 5. Let the pesses t A d B be R d R kg. wt. espectively. Sice the pole is i eqilibim, the lgebic sm of the momets of the thee foces R, R d kg. wt. bot y poit mst be eql to zeo. Tkig momets bot B, CB R.AB 0 i.e. 5 R R 0. 8 Tkig momets bot A, R.AB.AC 0. i.e. 8R 0. R 8 8

34 Poblem 8 A ifom plk of legth d weight W is sppoted hoizotlly o two veticl pops t distce b pt. The getest weight tht c be plced t the two eds i sccessio withot psettig the plk e W d W espectively. Show tht W W W W W W b. Soltio Let AB be the plk plced po two veticl pops t C d D. CD b. The weight W of the plk cts t G, the midpoit of AB, AG GB Whe the weight W is plced t A, the cotct with D is jst boke d the pwd ectio t D is zeo. R R A C G D B W W W Thee is pwd ectio R t C. Tke momets bot C, we hve W. AC W.CG i.e. W (AG CG) W.CG W.AG (W +W ).CG i.e. W. (W+W ) CG

35 CG W W W. () Whe the weight W is ttched t B, thee is loose cotct t C. The ectio t C becomes zeo. Thee is pwd ectio R bot D. Tke momets bot D, we get W.GD W (GB GD) GD (W+W ) W.GB W. GD W W W () W W W W W W CG + GD CD b W W W W W W b b Poblem 9 The esltt of thee foces P, Q, R, ctig log the sides BC, CA, AB of tigle ABC psses thogh the othocete. Show tht the tigle mst be obtse gled. If A 0, d B C, show tht Q+R P. Soltio: A R F O E 90-C B P D C Q

36 5 Let AD, BE d CF be the ltitdes of the tigle itesectig t O, the othocete. As the esltt psses thogh O, momet of the esltt bot O O. Sm of the momets of P, Q, R bot O O P.OD+Q.OE+R.OF 0.. () I t. dbod, OBD EBC 90 C. t( 90 C) OD i.e) cot C BD OD BD OD BD cot C. () Fom t. dabd, BD cos B AB Fom( ), OD ccos B. cot C Similly OE c.cos Bcos C si C cos C ccos B. si C c Rcos Bcos C( R si C, R is the cicmdis of the ) R cosc cos A d OF R cos AcosB Hece () becomes P. Rcos Bcos C Q.Rcos Ccos A R.Rcos Acos B 0 Dividig by R cos Acos Bcos C, P A cos Q B cos R C cos 0 () Now, P, Q, R beig mgitdes of the foces, e ll positive. () my hold good, if t lest oe of the tems mst be egtive. Hece oe of the cosies mst be egtive. i.e) the tigle mst be obtse gled. If A 0 d the othe gles eql, the B C 0 Hece () becomes

37 6 P Q R 0 cos0 cos0 cos0 P Q R i.e. 0 i.e. P Q R.6 Coples: Defiitio Two eql d like pllel foces ot ctig t the sme poit e sid to costitte cople. Exmples of cople e the foces sed i widig clock o tig tp. Sch foces ctig po igid body c hve oly otto effect o the body d they c ot podce motio of tsltio. The momet of cople is the podct of eithe of the two foces of the cople d the pepedicl distce betwee them, The pepedicl distce (p) betwee the two eql foces P of cople is clled the m of the cople. A cople ech of whose foces is P d whose m is p is slly deoted by (P, p). A cople is positive whe its momet is positive i.e., if the foces of the cople ted to podce ottio i the ti-clockwise diectio d cople is egtive whe the foces ted to podce ottio i the clockwise diectio..7 Eqilibim of thee foces ctig o Rigid Body. I the pevios sectios we hve stdied theoems d poblems ivolvig pllel foces d foces ctig t poit. Hee we stdy thee impott theoems d solved poblems o foces ctig o igid body d thei coditios of eqilibim. Theoem Poof: If thee foces ctig o igid body e i eqilibim, they mst be copl. P R A B C D E Q

38 7 Let the thee foces be P,Q,R Give : They e ctig o igid body d i eqilibim. Tke A o the foce P, d B o the foce Q sch tht AB is ot pllel to R. Sm of the momets of P, Q, R bot AB 0 [ P,Q, R e i eqilibim] Now, momet of P d Q bot AB 0 [ P d Q itesect AB]. Momet of R bot AB 0, Hece R mst itesect AB t poit C Similly if D is othe poit o Q sch tht AD is ot pllel to R, we pove, R mst itesect AD t poit E. Sice BC d DE itesect t A, BD, CE, A lie o the sme ple. i.e) A lies o the ple fomed by Q d R. Sice A is bity poit o the foce P, evey poit o the foce P lie o the sme ple. ie) P, Q, R lie o the sme ple. Thee Copl Foces theoem If thee copl foces ctig o igid body keep it i eqilibim, they mst be eithe cocet o ll pllel. Poof: Let P, Q, R be the thee foces ctig o igid body keep it i eqilibim. Oe foce mst be eql d opposite to the esltt of the othe two. they mst be pllel o itesect. Cse : If P d Q e pllel (like o like) The the esltt of P d Q is lso pllel. Hece R mst be pllel to P d Q. Cse : If P d Q e ot pllel: (itesect) They meet t O. Theefoe, by pllelogm lw, the thid foce R mst pss thogh O. i.e) the thee foces e cocet. Note: A cople d sigle foce c ot be i eqilibim Coditios of eqilibim. If thee foces ctig t poit e i eqilibim, the ech foce is popotiol to the sie of the gle betwee the othe two.. If thee foces i eqilibim e pllel, the ech foce is popotiol to the distce betwee the othe two

39 8 Two Tigoometicl theoems If D is y poit o BC of tigle ABC sch tht BD m d ADC, DC BAD, DAC the ) m cot m.cot. cot ) m cot.cot B m.cot C. Poof: A. Give, m ) B m D C BD DC BD. DA DA DC Usig, sie foml i ABD, ADC, m si si BAD ABD si si m si si si si Divide by si si m cot cot cot cot m ACD DAC si.cos cos.si si cos cos.si si.si. si cot cot cot cot m cot m.cot. cot

40 9. m BD DA. DA DC si BAD si ACD si ABD si DAC si B.si C si B.si 80 C si C.si B si B.si C si C si.cos B cos si B si B si C cos cos C si Divide by si B si C si m cot B cot cot cot C m cot cot C cot B cot ( m )cot cot B mcotc Poblem 0 A ifom od, of legth, hgs gist smooth veticl wll beig sppoted by mes of stig, of legth l, tied to oe ed of the od, the othe ed of the stig beig ttched to poit i the wll: show tht the od c est iclied to the wll t gle give by cos l. Wht e the limits of the tio of : l i ode tht eqilibim my be possible? Soltio: C T A L 90 0 G R w l D B

41 0 AB is the od of legth, with G its cete of gvity d BC is the stig of legth l. The foces ctig o the od e: (i). Its weight W ctig veticlly dowwds thogh G. (ii). The ectio R t A which is oml to the wll d theefoe hoizotl. iii) The tesio T of the stig log BC. These thee foces i eqilibim ot beig ll pllel, mst meet i poit L. Let the stig mke gle with the veticl. ACB GLB. LGB 80 dalg 90, AG:GB :, Usig the tigoometicl theoem i ALB 80.cot90. cot ( )cot i.e) cot cot cot cot () Dw BD to CA. Fom t. dcdb, BD BC.si l. si t. dabd, BD ABsi si l si si () Elimite betwee () d (). We kow tht cos ec cot () si l () si cosec () l si Sbstitte () d () i () i.e. l si l cot si l cos cos l cos

42 l cos (5) Eqilibim positio is possible, if l 0 i.e. l l Also i.e. l i.e. l l cos positive d less th o l.. (6) o l (7) l [ By (6) & (7) ]. l l Poblem A bem of weight W higed t oe ed is sppoted t the othe ed by stig so tht the bem d the stig e i veticl ple d mke the sme gle with the hoizo. Show tht the ectio t the hige is W 8 cos ec Soltio: C L Let AB be the bem of weight W d G its cete of gvity. R G α θ T B BC is the stig The foce ctig o the bem e: i) Its wt. W ctig veticlly dow wds t G ii) the tesio T log BC iii) the ectio R t the hige A. A 90 0 W

43 Fo eqilibim (i), (ii) d (iii) mst meet t L. BC d AB mke the sme gle with the hoizo. They mke i.e. Let 90 with the veticl LG, BLG 90 LGB ALG Usig tigoometicl theoem i ALB, AG:GB : cot90.cot.cot90 i.e. t cot t t cot. () Applyig Lmi s theoem t L, si i.e. R W 90 si 90 R cos si W cos R cos W W 90 cos W cos cos cos si si W cos si cos cot si W cos si cos.t si [By ()] W cos cos ec W cot W.cos ec cot cot si si W.cot 9t W W cot 9 cot 8 W cos ec 8

44 Wll Poblem A solid coe of height h d semi-veticl gle is plced with its bse fltly gist smooth veticl wll d is sppoted by stig ttched to its vetex d to poit i the wll. 6 Show tht the getest possible legth of the stig is h t. 9 (The cete of gvity of solid coe lies o its xis d divides it i the tio : fom the vetex.) Soltio: O R O T C A G D W B Let A be the vetex, & height AD h. Semi-veticl gle D AC. G divides AD i the tio : Legth AO is getest, whe the coe is jst i the poit of tig bot C. At tht time, oml ectio R mst be pepedicl to the wll. Sice, the coe is i eqilibim, the thee foces T, W, R mst be cocet t O. AOG & AO D e simil. AO AO AD AG Now, OG CD. Fom h h AO AO CD CD ACD, t CD ht AD h OG h.t ()

45 Fom AOG, AO AG GO AO 9 h t 6 AO h 9. 6 h h.t 9h h.t t 6 9 A O h 6 6 A O h. 9 9h 6h 6 t t t Poblem A hevy ifom od of legth lies ove smooth peg with oe ed estig o smooth veticl wll. If c is the distce of the peg fom the wll d the iclitio of the od to the wll, show tht c si Soltio: O R A R 90 D c P G W B

46 5 Foces ctig o the od AB e i) Weight W t G ii) iii) Rectio R t A ( to the wll) Rectio R t the peg P ( to the od) Fo eqilibim, W, R,R mst be cocet t O. Fom ightgled tigle ADP (DP c) Fom Fom c si. () AP AP AOP, si.. () AO OA OGA, si.. () AG c AP OA si AP AO AG t c AG c Poblem A hevy ifom sphee ests tochig two smooth iclied ples oe of which is iclied 60 to the hoizotl. If the pesse o this ple is oe-hlf of the weight of the sphee, pove tht the iclitio of the othe ple to the hoizotl is Soltio: c si 0 M N A 60 o R B C 60 o L R A B

47 6 Let the sphee cete C est o the iclied ples AM d BN. MA mkes hoizotl d let NB mke gle with the hoizo. The foces ctig e 60 with the i) Rectio R A t A pepedicl to the iclied ple AM d to the sphee d hece pssig thogh C. ii) Rectio R B t B which is oml to the iclied ple BN d to the sphee d iii) hece pssig thogh C. W, the weight of the sphee ctig veticlly dowwds t C log CL. Clely the bove thee foces meet t C. Also ACL 60 d BCL Applyig Lmi s theoem, R A W si si 60 W si R A. () si 60 Bt W R A () Fom () d (), we hve W si W si 60 i.e. si si 60 si 60cos cos60si i.e. si cos si o si cos si i.e. si si cos o cos i.e. t o 0

48 7 Poblem 5 A ifom solid hemisphee of weight W ests with its cved sfce o smooth hoizotl ple. A weight w is sspeded fom poit o the im of the hemisphee. If the ple 8w bse of the im is iclied to the hoizotl t gle, pove tht t W Soltio: R C A L O D G B W C w Dw GL pepedicl to OC d BD pepedicl to OC. Bse AB is iclied t gle θ with the hoizotl BD. Foces ctig e i) Rectio R c ii) Weight W t G iii) Weight w t B. Sice these thee foces e pllel, d i eqilibim ech foce is popotiol to the distce betwee the othe two. W w () BD GL Now, OBD BD OB cos cos Hee, OG, 8 GL OG. si 8 dis si W w () cos si 8 8w t W

49 8 UNIT II. Fictio I the pevios sectios we hve stdied poblems o eqilibim of smooth bodies. Pcticlly o bodies e pefectly smooth. All bodies e ogh to ceti extet. Fictio is the foce tht opposes the motio of object. Oly becse of this fictio we e ble to tvel log the od by wlkig o by vehicles. So fictio helps motio. It is tgetil foce ctig t the poit o cotct of two bodies. To stop movig object foce mst ct i the opposite diectio to the diectio of motio. Sch foce is clled fictiol foce. Fo exmple if yo psh yo book coss yo desk, the book will move. The foce of the psh moves the book. As the books slides coss the desk, it slows dow d stops movig. Whe yo ide bicycle the cotct betwee the wheel d the od is exmple of dymic fictio. Defiitio If two bodies e i cotct with oe othe, the popety of the two bodies, by mes of which foce is exeted betwee them t thei poit of cotct to pevet oe body fom slidig o the othe, is clled fictio; the foce exeted is clled the foce of fictio. Types of Fictio Thee e thee types of fictio ) Stticl Fictio ) Limitig Fictio ) Dymicl fictio.. Whe oe body i cotct with othe is i eqilibim, the fictio exeted is jst sfficiet to miti eqilibim is clled stticl fictio.. Whe oe body is jst o the poit of slidig o othe, the fictio exeted ttis its mximm vle d is clled limitig fictio; the eqilibim is sid to be limitig eqilibim.. Whe motio eses by oe body slidig ove othe, the fictio exeted is clled dymicl fictio.. Lws of Fictio Fictio is ot mthemticl cocept; it is physicl elity. Lw Whe two bodies e i cotct, the diectio of fictio o oe of them t the poit of cotct is opposite to the diectio i which the poit of cotct wold commece to move. Lw Whe thee is eqilibim, the mgitde of fictio is jst sfficiet to pevet the body fom movig.

50 9 Lw The mgitde of the limitig fictio lwys bes costt tio to the oml ectio d this tio depeds oly o the sbstces of which the bodies e composed. Lw The limitig fictio is idepedet of the extet d shpe of the sfces i cotct, so log s the oml ectio is lteed. Lw 5 (Lw of dymicl Fictio) Whe motio eses by oe body slidig ove the othe the diectio of fictio is opposite to tht of motio; the mgitde of the fictio is idepedet of the velocity of the poit of cotct bt the tio of the fictio to the oml ectio is slightly less whe the body moves, th whe it is i limitig eqilibim. Fictio is pssive foce: Expli ) Fictio is oly esistig foce. ) It ppes oly whe ecessy to pevet o oppose the motio of the poit of cotct. ) It c ot podce motio of body by itself, bt mitis eltive eqilibim. ) It is self-djstig foce. 5) It ssmes mgitde d diectio to blce othe foces ctig o the body. Hece, fictio is pely pssive foce. Co-efficiet of fictio The tio of the limitig fictio to the oml ectio is clled the co-efficiet of fictio. It is deoted by i.e.) F R Note: ) depeds o the te of the mteils i cotct. ) Fictio is mximm whe it is limitig. R is the mximm vle of fictio. F ) Whe eqilibim is o-limitig, F R i.e.) R ) Fictio F tkes y vle fom zeo pto R. Agle of Fictio F R B C B C R R O F A O R A

51 50 Let OA F(Fictio), If OB R (Noml ectio) &OC be the esltt of F d R. BC OA F BOC, t.. () OB OB R As F iceses, - iceses til F eches its mximm vle R. I this cse, eqilibim is limitig. Defiitio Whe oe body is i limitig eqilibim ove othe, the gle which the esltt ectio mkes with the oml t the poit of cotct is clled the gle of fictio d is deoted by t I the limitig eqilibim, B OC gle of fictio. BC OB OA OB t R R i.e.) The co-efficiet of fictio is eql to the tget of the gle of fictio. Coe of Fictio R R O R

52 5 We kow, the getest gle mde by the esltt ectio with the oml is (gle of fictio) whee t. Coside the motio of body t O (its poit of cotct) with othe. Whe two bodies e i cotct, coside coe dw with O s vetex, commo oml s the xis of the coe, - be the semi-veticl gle of the coe. Now, the esltt ectio of R d R will hve diectio which lies withi the sfce o o the sfce of the coe. It c ot fll otside the coe. This coe geeted by the esltt ectio is clled the coe of fictio.. Eqilibim of pticle o ogh iclied ple. R F A W Let - be the iclitio of the ogh iclied ple, o which pticle of weight W, is plced t A. Foces ctig o the pticle e, ) Weight W veticlly dowwds ) Noml ectio R, to the ple. ) Fictiol foce F, log the ple pwds (Sice the body ties to slip dow). Resolvig the foces log d pepedicl to the ple, R F F W si, R W cos t

53 5 Bt i.e) F R t t F Whe, R t t Hece, it is cle tht whe body is plced o ogh iclied ple d is o the poit of slidig dow the ple, the gle of iclitio of the ple is eql to the gle of fictio. Now is clled s the gle of epose. Ths the gle of epose of ogh iclied ple is eql to the gle fictio whe thee is o extel foce ct o the body.. Eqilibim of body o ogh iclied ple de foce pllel to the ple. A body is t est o ogh ple iclied to the hoizo t gle gete th the gle of fictio d is cted o by foce pllel to the ple. Fid the limits betwee which the foce mst lie. Poof: ectio. Let be the iclitio of the ple, W be the weight of the body& R be the oml Cse : Let the body be o the poit of slippig dow. Theefoe ple. P R cts pwds log the R R W si W cos W

54 5 Let P be the foce pplied to keep the body t est. Resolvig the foces log d pepedicl to the ple, P R W si.. () Let R W.cos () P W. si. W cos W si t. cos W cos si.cos cos si W. si cos.si P W cos Cse ii Let the body be o the poit of movig p. Theefoe limitig fictiol foce R cts dowwd log the ple. R P W si W cos R W Let P be the extel foce pplied to keep the body t est. Resolvig the foce, R W cos ; P R W si P. W cos W si W cos si cos cos.si

55 5 W cos.si W cos Let P. si If P P, body will move dow the ple. If P P, body will move p the ple. Fo eqilibim P mst lie betwee P d P. i.e.) P P P.5 Eqilibim of body o ogh iclied ple de y foce. Theoem: A body is t est o ogh iclied ple of iclitio to the hoizo, beig cted o by foce mkig gle with the ple; to fid the limits betwee which the foce mst lie d lso to fid the mgitde d diectio of the lest foce eqied to dg the body p the iclied ple. P P R µr R W si α A A W si R W cos W Let α be the iclitio of the ple, W be the weight of the body, P be the foce ctig t gle with the iclied ple d R be the oml ectio. Cse i: The body is jst o the poit of slippig dow. Theefoe the limitig fictio pwds. Resolvig the foces log d to the iclied ple, Pcos R W si.. () R cts

56 55 Psi R W cos.. () R W cos Psi Pcos W cos Psi W si P cos si Wsi cos We hve W si cos P cos si t si t.cos P W cos t.si W si cos cos.si cos.cos si.si si W cos Let si P W. cos Cse ii: The body is jst o the poit of movig p the ple. Theefoe Resolvig the foces log d to the ple. Pcos R W. si. () Psi R W. cos. () R W cos Psi Pcos W cos Psi W. si P cos si Wsi cos si t.cos P W cos t.si si.cos si.cos W cos cos si.si W.si cos R cts dowwds.

57 56 Let P W.si cos To keep the body i eqilibim, P d P e the limitig vles of P. Fid the lest foce eqied to dg the body p the iclied ple We hve, P W. si cos P is lest whe cos is getest. i.e.) Whe cos i.e.) Whe 0 i.e.) Whe Lest vle of P W. si Hece the foce eqied to move the body p the ple will be lest whe it is pplied i diectio mkig with the iclied ple gle eql to the gle of fictio. i.e.) The best gle of tctio p ogh iclied ple is the gle of fictio Poblem A pticle of weight 0 kgs. estig o ogh hoizotl ple is jst o the poit motio whe cted o by hoizotl foces of 6kg wt. d 8kg. wt. t ight gles to ech othe. Fid the coefficiet of fictio betwee the pticle d the ple d the diectio i which the fictio cts. Soltio: C D 6 0 F A 8 B Let AB (8) d AC (6) epeset the

58 57 Let AB 8 d AC 6 epeset the diectios of the foces, A beig the pticle. The esltt foce 8 6 0kg. wt. d this cts log AD, mkig gle cos with the 8kg foce. 5 Let F be the fictiol foce. As motio jst begis, mgitde of F is eql to tht of the esltt foce. F 0 () If R is the oml ectio o the pticle, R 0.. () If is the coefficiet of fictio s the eqilibim is limitig, F R Poblem A body of weight kgs. ests i limitig eqilibim o iclied ple whose iclitio is 0. Fid the coefficiet of fictio d the oml ectio. Soltio: R R W si 0 W cos 0 0 W kg Sice the body is i limitig eqilibim o the iclied ple, it ties to move i the dowwd diectio log the iclied ple.

59 58 log d Fictiol foce R cts i the pwd diectio log the iclied ple. Resolvig to the ple, R W si 0 (). R W.cos0. () t, 0 Poblem A ifom ldde is i eqilibim with oe ed estig o the god d the othe gist veticl wll; if the god d wll be both ogh, the coefficiets of fictio beig d espectively, d if the ldde be o the poit of slippig t both eds, show tht, the iclitio of the ldde to the hoizo is give by ectios t the wll d god. Soltio: S t. Fid lso the B S G R θ C R E A W

60 59 AB is the ifom ldde, whose weight W is ctig t G sch tht AG GB. Foces ctig e,. Weight W. Noml ectio R t A. Noml ectio S t B. R 5. S Whe the ldde is o the poit of slippig t both eds, fictiol foces S, R ct log CB, AC espectively. Sice the ldde is i eqilibim esltt is zeo. Resolvig hoizotlly d veticlly, S R.. () R S W. () R R R W W W R By Vigo s theoem o momets, tkig momets bot A S. BC S. AC W. AE S. ABsi S. ABcos W. AG. cos S. si S.cos W.. cos AB AG W S W S.si S. cos t W S W W t I the pevios poblem, whe of fictio. Poblem show tht 90, whee is the gle

61 60 Soltio: I the pevios poblem, we hve poved t Pt t, we get t t ; t t t t 90 cot t90 i.e.) 90 Poblem 5 A ifom ldde ests i limitig eqilibim with its lowe ed o ogh hoizotl ple d its ppe ed gist eqlly ogh veticl wll. If be the iclitio of the ldde to the veticl, pove tht t whee is the coefficiet of fictio. Soltio: S S L B S R G R C R A W Whe the ldde AB is i limitig eqilibim, five foces e ctig s mked i the fige.

62 6 Let ) Weight of the ldde W ) Noml ectio R t A ) Noml ectio S t B ) Fictiol foce R 5) fictiol foce S R, S be the esltt ectios of R, R d S, S espectively. We hve foces R, S, W. Fo eqilibim, they mst be cocet t L. I LAB, LG A 80 ; ALG B LG 90, AG : GB : By tigoometicl theoem i LBA, (+) cot80.cot90. cot t.cot t cot t t cot t i.e.) t t Poblem 6 A ifom ldde ests with its lowe ed o ogh hoizotl god its ppe ed gist ogh veticl wll, the god d the wll beig eqlly ogh d the gle of fictio beig. Show tht the getest iclitio of the ldde to the veticl is. Soltio I the pevios poblem, we hve poved, t t t t t Bt t

63 6 Poblem 7 A ldde which stds o hoizotl god, leig gist veticl wll, is so loded tht its C. G. is t distce d b fom its lowe d ppe eds espectively. Show tht if the ldde is i limitig eqilibim, its iclitio to the hoizotl is give by b t b whee, e the coefficiets of fictio betwee the ldde d the god d the wll espectively. Soltio: As i poblem 5, five foces e ctig o the ldde Hee, AG : GB : b By Tigoometicl theoem i LBA, b. cot90 b.cot 90. cot i.e.) b t b.t. cot b. b. t b b Poblem 8 A ldde AB ests with A o ogh hoizotl god d B gist eqlly ogh veticl wll. The cete of gvity of the ldde divides AB i the tio : b. If the ldde is o the poit of slippig, show tht the iclitio of the ldde to the god is give by b t whee is the coefficiet of fictio. ( b) Soltio: Pt I the pevios poblem, i b t b b t b

64 6 Poblem 9 A ldde AB ests with A estig o the god d B gist veticl wll, the coefficiets of fictio of the god d the wll beig d espectively. The cete of gvity G of the ldde divides AB i the tio :. If the ldde is o the poit of slippig t both eds, show tht its iclitio to the god is give by Soltio: Pt : b : i poblem7. t Poblem 0 t. A ldde of legth l is i cotct with veticl wll d hoizotl floo, the gle of fictio beig t ech cotct. If the weight of the ldde cts t poit distt kl below the middle poit, pove tht its limitig iclitio to the veticl is give by cot cot k cosec. Soltio: B S S S L C kl G R R R A W Foces e ctig s mked i the fige. Fo eqilibim, the thee foces mst be cocet t L, whee W be the weight of the ldde. I LAB, BC CA l; CG kl. BG BC CG l kl ( k) l R, S, W

65 6 B LG 90, LG A 80 ALG ; GA CA CG l kl l k: k BG : GA k. By Tigoometicl theoem i LBA, k k].cot80 k.cot 90 k.cot. [ cot k.t. cot k kcot t cot k k.cot k cot cot k cot cot cot cot k.cosec.cot t cot k cot.t.cot ie) t k t.t. cot t k. t si cot cot k.cosec Poblem A ifom ldde ests i limitig eqilibim with its lowe ed o ogh hoizotl ple d with the ppe ed gist smooth veticl wll. If be the iclitio of the ldde to the veticl, pove tht, t, whee is the coefficiet of fictio.

66 65 Soltio: B L 90 S G R R C R A W Sice the wll is smooth, thee is o fictiol foce. Foces ctig o the ldde e i) its weight W, ii) Fictiol foce R iii) R t A iv) S t B. Fo eqilibim, the thee foces W, R, S mst be cocet t L. whee R is the esltt of R d L G A 80, ALG, B LG 90; BG : GA :. ABC By Tigoometicl theoem i LAB, cot80.cot90. cot.cot 0 cot t t i.e) t t t R. I tigle LAB, Poblem A pticle is plced o the otside of ogh sphee whose coefficiet of fictio is. Show tht it will be o the poit of motio whe the dis fom it to the cete mkes gle t with the veticl.

67 66 Soltio: A R R B O W Let O be the cete, A the highest poit of the sphee d B the positio of the pticle which is jst o the poit of motio. Let The foces ctig t B e: ) the oml ectio R ) limitig fictio R AOB ) Its weight W, Sice the pticle t B is i limitig eqilibim, Resolvig log the oml OB, R W cos. () Resolvig log the tget t B, R W si.. () t.6 Eqilibim of Stigs Whe ifom stig o chi hgs feely betwee two poits ot i the sme veticl lie, the cve i which it hgs de the ctio of gvity is clled ctey. If the weight pe it legth of the chi o stig is costt, the ctey is clled the ifom o commo ctey. t.7 Eqtio of the commo ctey: A ifom hevy iextesible stig hgs feely de the ctio of gvity; to fid the eqtio of the cve which it foms.

68 67 Let ACB be ifom hevy flexible cod ttched to two poits A d B t the sme level, C beig the lowest, of the cod. Dw CO veticl, OX hoizotl d tke OX s X xis d OC s Y xis. Let P be y poit of the stig so tht the legth of the e CP s Let ω be the weight pe it legth of the chi. Coside the eqilibim of the potio CP of the chi. The foces ctig o it e: (i) Tesio T 0 ctig log the tget t C d which is theefoe hoizotl. (ii) Tesio T ctig t P log the tget t P mkig gle Ψ with OX. (iii) Its weight ws ctig veticlly dowwds thogh the C.G. of the c CP. Fo eqilibim, these thee foces mst be cocet. Hece the lie of ctio of the weight ws mst pss thogh the poit of the itesectio of T d T o. Resolvig hoizotlly d veticlly, we hve Tcos Ψ T o () d Tsi Ψ ws () Dividig () by (), t Ψ ws T 0 Now it will be coveiet to wite the vle of T o the tesio t the lowest poit, s T o wc () whee c is costt. This mes tht we ssme T o, to be eql to the weight of kow legth c of the cble.

69 68 The t Ψ ws wc s c S ctψ () Eqtio () is clled the itisic eqtio of the ctey. It gives the eltio betwee the legth of the e of the cve fom the lowest poit to y othe poit o the cve d the iclitio of the tget t the ltte poit. To obti the cetesi eqtio of the ctey, We se the eqtio () d the eltios dy ds dy si Ψ d t Ψ which e te fo y cve. dx Now dy dψ dy ds. ds dψ si Ψ d dψ c t Ψ si csec Ψ csec Ψ t Ψ y ʃ csec Ψ t Ψ dψ + A csec Ψ + S If y c whe Ψ 0, the c csec0 + A A 0 Hece y csec Ψ (5) y c sec Ψ c ( + t Ψ) c + s (6) dy t Ψ s y c dx c c dy y c dx c Itegtig, cos h - Whe x 0, y c y c x c + B i.e. cos h B o B 0 cos h - y c i.e. y ccos h x c x c (7) (7) is the Ctesi eqtio to the ctey. We c lso fid the eltio coectig s d x.

70 69 Diffeetitig (7). dy csih x. dx c c sih x c Fom (), s ct Ψ c. dy dx csih x c (8) Defiitios: The Ctesi eqtio to the ctey is y ccosh x c. cosh x c is eve fctio of x. Hece the cve is symmeticl with espect to the y-xis i.e. to the veticl thogh the lowest poit. This lie of symmety is clled the xis of the ctey. Sice c is the oly costt, i the eqtio, it is clled the pmete of the ctey d it detemies the size of the cve. The lowest poit C is clled the vetex of the ctey. The hoizotl lie t the depth c below the vetex (which is tke by s the x xis) is clled the diectix of the ctey. If the two poits A d B fom whee the stig is sspeded e i hoizotl lie, the the distce AB is clled the sp d the distce CD (i.e. the depth of the lowest poit C below AB) is clled the sg..8 Tesio t y poit: We hve deived the eqtios T cos Ψ T 0 () Ad T si Ψ ws () We hve lso pt T 0 wc () Eqtio () shows tht the tesio t the lowest poit is costt d is eql to the weight of potio of the stig whose legth is eql to the pmete of the ctey. Fom the eqtio (), we fid tht the hoizotl compoet of the tesio t y poit o the cve is eql to the tesio t the lowest poit d hece is costt. Fom eqtio (), we dedce tht the veticl compoet of the tesio t y poit is eql to ws i.e. eql to the weight of the potio of the stig lyig betwee the vetex d the poit. ( s e CP)

71 70 Sqig () d () d the ddig, T T 0 + w s w c +w w (c +s ) w y sig eqtio (6) of pge 77 T wy () Ths the tesio t y poit is popotiol to the height of the poit bove the oigi. It is eql to the weight of potio of the stig whose legth is eql to the height of the poit bove the diectix. Impott Coolly: Sppose log chis is thow ove two smooth pegs A d B d is i eqilibim with the potios AN d BN hgig veticlly. The potio BCA of the chi will fom ctey. The tesio of the chi is lteed by pssig ovet the smooth peg A. The tesio t A c be clclted by two methods. O oe side (i.e. fom the ctey potio), Tesio t A w.y whee y is the height of A bove the diectix. O the othe side, tesio t A weight of the fee pt AN hgig dow w. AN yan I othe wods, N is o the diectix of the ctey. Similly N is o the diectix. Hece if log chi is thow ove two smooth pegs d is i eqilibim, the fee eds mst ech the diectix of the ctey fomed by it.

72 7 Impott Fomle: The Ctesi coodites of poit P o the ctey e (x, y) d its itisic coodites e (s, Ψ). Hece thee e fo vible qtities we c hve eltio coectig y two of them. Thee will be C 6 sch eltios, most of them hvig bee ledy deived. We shll deive the emiig. It is wothwhile to collect these eslts fo edy efeece. (i) The eltio coectig x d y is y ccosh x () c (ii) (iii) (iv) d this is the Ctesi eqtio to the ctey. The eltio coectig s d Ψ is s ct Ψ () The eltio coectig y d Ψ is ycsecψ () The eltio coectig y d s is y c +s. () (v) The eltio coectig s d x is s csih x c (vi) We hve y ccosh x c d y csec Ψ, sec Ψ cosh x c x c cosh -(secψ) log(secψ + sec Ψ log(secψ + t Ψ) x clog (secψ + tψ) (6) This eltio c lso be obtied ths: dx dx. ds dψ ds dψ cos Ψ. d dx (ct Ψ ) sice cos Ψ fo y cve dψ ds cos Ψ. CsecΨ csecψ

73 7 Itegtig, x ʃ csec Ψ dψ + D clog (secψ + Ψ) + D At the lowest poit, Ψ 0 d x 0 0 clog (sec0+t0 + D i.e. 0 D x clog (secψ + t Ψ) (vii) The tesio t y poit wy (7), whee y is the distce of the poit fom the diectix. (viii) The tesio t the lowest poit wc (8) sih - x log(x+ x + ) cosh - x log(x+ x ).9 Geometicl Popeties of the Commo ctey: Let P be y poit o the ctey y ccosh x. c PT is the tget meetig the diectix (i.e. the x xis) t T. gle PTX Ψ PM (y) is the odite of P d PG is the oml t P. Dw MN to PT. Fom ΔPMN. MN PMcosΨ ycosψ

74 7 csecψ cos Ψ ccostt i.e. The legth of the pepedicl fom the foot of the odite o the tget t y poit of the ctey is costt. Agi t Ψ PN MN PN C PN C t Ψ S c CP PM NM + PN y c +s, eltio ledy obtied. If is the dis of cvte of the ctey t P, P ds dψ d dψ (ct Ψ) csec Ψ Let the oml t P ct the x xis t G. The PG. cos Ψ PM y PG ρ PG y cosψ csecψ. secψ csec Ψ Hece the dis of cvte t y poit o the ctey is meiclly eql to the legth of the oml itecepted betwee the cve d the diectix, bt they e dw i opposite diectios. Poblem A ifom chi of legth l is to be sspeded fom two poits i the sme hoizotl lie so tht eithe temil tesio is times tht t the lowest poit. Show tht the sp mst be l log(+ Soltio: Tesio t A wy A Ad tesio t C w.y C sice T wy t y poit Now w.y A.w.y C y A y C c Bt y A ccosh x A c cosh x A c c

75 7 o x A c cosh - log (+ ) x A clog (+ ) () We hve to fid c. y A c +s A, s A deotig the legth of CA. c + l (s totl legth l) i.e. c c + l o c l ( ) c l () Sbstittig () i (), x A sp AB x A l log (+ ) l ) log (+ ) Poblem A box kite is flyig t height h with legth l of wie pid ot, d with the vetex of the ctey o the god. Show tht t the kite, the iclitio of the wie to the god is t - h d tht its tesios thee d t the god e w(l +h ) d w(l h ) whee w is the l h h weight of the wie pe it of legth. Soltio: Y A h C l L c O M X