Worked Examples. v max =?


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1 Exaple iction + Unifo Cicula Motion Cicula Hill A ca i diing oe a eicicula hill of adiu. What i the fatet the ca can die oe the top of the hill without it tie lifting off of the gound? ax? (1) Copehend the Poble If the ca die oe a cicula hill, it tael in a cicle. Appaentl, if the ca tael too fat it cannot eain on a cicula path that follow the hill. One o oe of the foce eponible fo the ca taing on the cicle ut be liited in oe wa. Let ue a fee bod diaga to ee if we can deteine what foce ae eponible fo the ca cicula otion. N Gound on Ca W Eath on Ca he net foce on the ca ut point to the cente of the cicle (becaue it in cicula otion). A the ca tael fate and fate, the hill need to poide le and le noal foce to keep the ca fo penetating it. Eentuall the ca could be going fat enough that the noal foce i zeo if it went an fate that thi citical peed, the weight won t be enough to keep the ca going in a cicle coinciding with the hill and the wheel will lift off of the gound. Since the ca tael in a cicle, we ll alot cetainl ue the elationhip between centipetal acceleation and tangential peed. () epeent the Poble in oal e (Decibe the Phic) We hae been gien o identified the following quantitie: the a of the ca the peed of the ca the adiu of the cicle the ca' following (i.e. the hill' adiu) agnitude N N he noal foce fo the gound on the ca diection up agnitude g W he weight fo the eath on the ca diection down
2 Exaple iction + Unifo Cicula Motion We alo know the following elationhip that ight be ueful: Ma (Newton' Second Law) a Ca ca net towad cicle cente (Ca' in unifo cicula otion) (3) Plan the Solution We can appl Newton Second Law to the ca at the top of the hill to elate the foce on the ca to it acceleation. We know the fo of the ca acceleation ince it in cicula otion. We can et the noal foce equal to zeo; thi hould happen when the ca i going a fat a it can and till ta on the cicle. (4) Execute the Solution We ll ue the fee bod diaga to wite Newton Second Law fo the ca at the top of the hill. We chooe downwad a poitie becaue we know that the ca i acceleating downwad (i.e. towad the cente of the cicle) Appl Newton Second Law in the diection. Cicula otion iplie a / towad the cente of the cicle. At the axiu peed, the noal foce will be zeo. he entie downwad foce towad the cente i upplied b the ca weight alone. net, N W Ca ( N ) a + Ca + g ( 0) N Gound on Ca W ax Eath on Ca + g g ax (5) Intepet and Ealuate the Solution A a quick check, let a the hill had a adiu of aound 100. he fatet peed the ca 1 k 3600 k could go would then be ( )( 100 ) (about 70 ile pe h h hou). hi ee at leat eaonable fo the peed of a ca lifting off the top of a teep hill.
3 Exaple iction + Unifo Cicula Motion Plagound Slipping Soe childen ae plaing with a lage ock on the plagound. he keep putting it on a egoound pinning at 1 eolution ee econd. he e ting to find out how fa awa fo the cente the can place the ock befoe it lide to the outide. You look up the coefficient of tatic fiction fo ock on etal and find it aound 0.3. What i the fathet ditance fo the cente of the egoound that the ock can it and till not lide? (1) Copehend the Poble Let tat with a ketch to ee what going on. he ock i itting on a egoound (a flat, pinning dik) pinning at a known ate. 1 e ec op View Side View We know that the ock i taelling in a cicle. he quetion i how lage of a adiu can the cicle hae befoe the ock ut lip? Since it taelling in a cicle at contant peed, we know that the acceleation of the ock i elated to it elocit (a /). In addition, we know how long it take the ock to ake a coplete eolution ( econd). We need to know what foce poduce the ock acceleation towad the cente of the cicle. Let figue thi out b dawing the fee bod diaga fo the ock (we ll ue the ide iew ). f N loo on ock loo on ock W Eath on ock hee ae thee foce on the ock, the noal foce fo the floo of the egoound, the weight fo the eath, and the tatic fiction foce fo the floo of the egoound. Since the tatic fiction i the onl foce in the hoizontal diection, we know it ut point towad the cente of the cicle in ode to pull the ock towad the cente (a deanded b the ock centepointing acceleation). he tatic fiction foce ha a axiu tength it can poide, gien b f tatic, ax µ N. A we oe the ock outwad, the tatic fiction foce equied to keep it going aound the cicle inceae. hee will appaentl be a axiu adiu at which the tatic fiction foce hit thi axiu alue. If the ock i placed an fathe awa, it will lip.
4 Exaple iction + Unifo Cicula Motion () epeent the Poble in oal e (Decibe the Phic) We ae gien o hae identified the following quantitie a ueful: the tie it take the egoound to ake one eolution the a of the ock agnitude N N the noal foce fo the floo on the ock diection up agnitude g W the weight foce fo the eath on the ock diection down agnitude the tatic fictional foce fo the floo on the ock diection towad cente the ditance fo the cicle cente to the ock (the adiu) We alo hae the following elationhip that ight be ipotant. Ma Newton' Second Law f net tatic, ax a c µ N tatic fiction axiu centipetal acceleation (3) Plan the Solution Since it in unifo cicula otion, we know the ock acceleation in te of it elocit and adiu. We can ue Newton Second Law to elate thi acceleation to the foce on the ock. B ineting the axiu alue fo the tatic fictional foce, we hould be able to find the laget adiu at which the ock won t lide. (4) Execute the Solution Daw the fee bod diaga fo the ock. We chooe towad the cente of the cicle a poitie, a thi i the diection the ock actuall acceleate. Appl Newton Second Law in the x diection. Since the ock i in unifo cicula otion, we know how it acceleation i elated to and. f N net, x a f + N + W a x, x x x x W x
5 Exaple iction + Unifo Cicula Motion Since we don t know the ock elocit, we need to wite it in te of the adiu and the eolution tie (a.k.a. the peiod). he ock tael one cicufeence in tie. eplace the elocit in ou Newton Second Law xequation with thi expeion fo. Now we hae the adiu in te of thing we know (peiod and a ) and the tatic fiction foce. he axiu adiu i whee the tatic fiction foce i at it axiu. We need the noal foce fo the floo on the ock. We can find thi fo appling Newton Second Law in the diection. We know the acceleation of the ock in the diection i zeo. We can inet thi noal foce alue into the expeion fo the axiu adiu to get ou anwe. (Note that the a cancel out) Now we can inet the nueical alue fo the poble tateent. ditance π tie π f ax f, ax ( µ N ) net, f + N + W ( 0), ( g ) a 0 + N + 0 ax ax µ µ g ( ) ( )( ) (. ) N g ( g ) c ax c he ock will lide if it i placed oe than 30 c (about 1 ft) fo the cente of the egoound. If we place the ock anwhee cloe to the cente than thi, it will not lide on the egoound. he ock would then jut tael aound the cente in a cicle with the et of the egoound.
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