Investigation of Atwood s machines as Series and Parallel networks
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1 Ivestiatio of Atwood s achies as Series ad Parallel etworks Jafari Matehkolaee, Mehdi; Bavad, Air Ahad Islaic Azad uiversity of Shahrood, Shahid Beheshti hih school i Sari, Mazadara, Ira ehdisaraviaria@yahoo.co ABSTRACT This paper shows that Atwood s achies are echaical systes which ca be cosidered i cobiatio of series ad parallel. By the cocept of equivalet ass for series ad parallel, it will be easy to aalyze ay probles i relatio with etworks of Atwood s achies. We defied specific relatio separately for series ad parallel etwork. These relatios are ot like relatios of resistors ad capacitors ad spris etworks.. Itroductio The cocepts of series ad parallel are very failiar, because we ecouter the i various parts of physics. For exaple resistors ad capacitors etworks i electricity ad spris etworks i echaics. I priary echaics we soeties cosider the coectio of ass ad spri as series or parallel. I spri etworks our criterio for distiuishi betwee series ad parallel were displaceet of spris fro orii. Ideed cosideri equivalet spri costats reduces difficulty i ay probles. We tried to defie the ew coceptio of series ad parallel i Atwood's achies. Our criterio for series ad parallel cobiatios of Atwood s achies are tesio of stris of the syste. I this paper we assue that ass of all pulleys ad stris ca be iored. Also the frictio betwee of pulleys ad stris are iorable. At the first step with a exaple we try to show that aalysis of Atwood's achies depeds o the aer of coectio of asses to stris ad pulleys. I the fiure () with chae of the asses (, ), whether icrease or decrease, the tesio of stri will be cosistetly equal to zero ad acceleratio
2 of ad will be equal to (~9/8 /s ). Fiure(): The syste Atwood's achies that tesio of stri is cosistetly equal to zero. This exaple shows that, arraeet of stris, asses ad pulleys perfors iportat role i deteriatio of acceleratio of syste. Therefore we ca itroduce series ad parallel etworks with several exaples.. Series Network I the fiure () a pulley ad two asses (, ) are hu fro a stri with tesio of T. T T T T ad T are tesio of stris hu fro pulley, where T T T ad () Accordi to our priary assuptio we iore the ass of stris ad pulleys. So we itroduce the equivalet ass for the syste is t or () By equivalet ass we ea the ass of a sile body that ca be substituted for the cobiatio with o chae i the operatio of the rest of the syste. Also by cosideri equivalet ass it ca be equivalet syste for Atwood's achies that akes less uber of equatios. The equivalet ass ca be calculated i aother way. I fiure (), fiure (-b) shows the equivalet ass for syste show i fiure (-a). We suppose that the two systes have equal dowward acceleratio (a'). Fiure (): A siple Atwood's achie, that it cosisti of two suspeded asses coected by a stri that passes over a pulley. I this fiure we ca write: TT T ()
3 T T Exaple (): I the followi fiure k, k ad k, suddely we release the asses. Calculate acceleratio of? (a) (b) Fiure (): A Atwood's achie (a) ad equivalet syste for it (b). Evidetly by Newto's secod law for ad ad we have: T a' T ( ' ) a a () T ( ' ) a a I equatios show above, (a) represets acceleratio of ad. By solvi the equatios we coclude Which is siilar to relatio (). I this relatio ad equality ( ) is for while asses are equal. I the Atwood's achie i series etwork priary tesio of stri, T, ca be divided to saller values,t,t,,t (for a etwork with uber of asses ), so that T T T T (5) If we hai fro a stri of Atwood's achie, the aother Atwood's achies i successio, the we have series etwork. Fiure () : A Atwood's achie i series. At first we should calculate the equivalet ass for ad, k Now, we have a Atwood's achie with two asses which are k ad, k so acceleratio of ad, are equal to /7 ad /7 respectively. If we wated to accout acceleratio or separately, the we use with equatios that siilar to secod ad third equatios of relatio () so acceleratio of ad are equal to /7 ad -/7 respectively. aybe you feel sice this exaple is solved i Fowles [], usi a stadard Laraia approach, its ot eed to equivalet ass cocept, but ore icrease the ubers of pulleys ad asses ore icrease the uber of equatios ad will be ore difficult.
4 Exaple (): Cosider a ifiite Atwood's achie like followi fiure. Each of the asses is equal to. suddely we release the asses. Calculate acceleratio of above ass?[] x fo of (x) ( ) x For the ifiite Atwood's achie So, equivalet ass is equal to. The the acceleratio of is /. See to fiure (6). Fiure (5): A ifiite Atwood's achie. By the coceptio of the equivalet ass we solve this proble. We suppose above syste ade of uber of asses () that are cobie i series. If we cosider x as ass of body hu fro th pulley, we ca cosider f(x) as equivalet ass for asses of hai fro th ad (-)th pulley the accordi to relatio (): f ( x) x x x Evidetly the ext pulley coects to x, so: 6x f ( f ( x)) 5x.. Ad we have for asses Fiure (6): equivalet syste for fiure (5) the acceleratio of is /.. Parallel Network I this cobiatio tesio of stri is equal everywhere. This eas that relatio (5) ca be writte as T T.T For a syste ade of uber of asses,,.. that are cobie i parallel, the equivalet ass is t (6)... Ad while the asses are equal: So t by the equivalet ass i parallel etwork, will be easy accouti acceleratio of asses ad tesio of stris. Oe syste
5 which ca be cosidered for parallel etwork is T T T Fiure (7): a parallel etwork for, ad T T T. The siplest fiure for the syste show above is Fiure (9): equivalet arraeet, for fiure (8). Acceleratio of is equal to / ad acceleratio of is equal to /. Now, we cosider the ext cobiatio. Look at the fiure (0): Fiure (8): the siplest cobiatio for parallel etwork. Sice leth of stri is costat, it ca be easily cocluded that acceleratio of side asses are equal ad opposite of the acceleratio of iddle ass. If we cosider side asses as a sile ass,, equivalet arraeet for fiure (8) is Fiure (0): the parallel syste with two iddle asses. For aalyze fiure (0) we suppose that the stri is fixed ad steady. I fact, we suppose that the acceleratio of side asses is equal to zero. T/ T T/, Fiure (): a) a syste icludi the ad. b) A equivalet syste for (a). 5
6 By writi Newto's secod law for ad, tesio of stri is calculated T Therefore equivalet ass,,, is equal to Note that iserti i equatio (6) have the sae result. Now, we ca iaie equivalet arraeet for fiure (0) Fiure (): The parallel syste with three iterediate asses. Aai we suppose that the stri is fixed ad steady. Therefore a a a 0 that a, a, a are acceleratios of,, respectively. T/ T/6 T,,,, Fiure (): Two equivalet systes for fiure (0). Acceleratio of each ass ca be calculated ( ) a ( ) a a ( ) ( ) ( ) ( ) O coditio i which ad ad are equal acceleratio of side asses is equal to /5 ad acceleratio of ad is equal to /5 ad /5 respectively. Next parallel syste icludes three iterediate asses, ad ad, ad two side asses. Fiure (): a) a syste icludi,, b) a equivalet syste for (a). We assue that equivalet ass is,,. By writi Newto's secod law tesio of stri is calculated as follows T 9 So equivalet ass,,, is equal to 9 Note that iserti i equatio (6) has the sae result. 6
7 Now we ca iaie equivalet cobiatio for fiure () as followi The equivalet ass for,,, is 6,,, That by putti i equatio (6) the sae result will be obtaied. The equivalet cobiatio for fiure (6) is,, Fiure (5): Two equivalet systes for fiure (). If we assue that The acceleratio of side asses will be equal to /7 ad acceleratio of iterediate asses will be equal to -/7 ay be aother exaple akes bored you the ext exaple without further explaatio. I this cobiatio we iaie a syste with four iterediate asses ad two side asses. Fiure (6): A syste with four iterediate ad two side asses. Fiure (7): A equivalet syste for fiure (5). By icreasi the uber of iterediate pulleys acceleratio of side asses icreases slihtly ad acceleratio of iterediate asses teds to zero. If we had ifiite Atwood's achie i which all the asses are equal ad cobied i parallel for, acceleratio of side asses will be as follows a That if to /.,,, the acceleratio will ted 7
8 .... Fiure (8): A i fiite Atwood's achie. Coclusio Atwood's achie i series ad parallel cobiatio ca be cosidered as a basic field of echaics. Because cosideri this atter deepes attitude to probles related to this systes. Also the cocept of the equivalet ass ad the acceleratio of each of the asses could be derived with the ethod preseted i the paper, ad directly copared to the ore stadarad approach. Refereces. Robert Resik, David Halliday & Keeth S.Krae, Physics (volue ), Fifth Editio, Johwiley & Sos, 00.. Grat R.Fowles, Aalytical Mechaics 7 th ed, pp Mauree, David, Shpyrkv, Ole Copetitios Physics Bosto Uiversity Aerica,
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