plce of ind F A C U L Y O F E D U C A I O N Deprtent of Curriculu nd Pedoy Physics Dynics: Atwood Mchine Science nd Mthetics Eduction Reserch Group Supported by UBC echin nd Lernin Enhnceent Fund 0-04
he Atwood Mchine
he Atwood Mchine he Atwood Mchine is pulley syste consistin of two weihts connected by strin. We will ssue no friction nd tht both the strin nd pulley re ssless. If the sses of the two weihts re different, the weihts will ccelerte uniforly by. Our xis is defined such tht positive indictes tht ccelertes downwrds, while ccelertes upwrds.
he Atwood Mchine I Let W be the weiht of, nd W be the weiht of. Let the tension of the strin be. Assue tht >. Which of the followin is the correct free body dir (force dir) of? A. B. W C. W W D. W E. W W
Solution Answer: A Justifiction: here re only two forces ctin on, the force of tension due to the strin pullin it up nd the force of its own weiht pullin it down. A. W C nd D re incorrect becuse itself is not force. B nd E re incorrect becuse W is not force ctin directly on. Any force due to weiht ust point in the direction of, which is downwrds.
he Atwood Mchine II Accordin to Newton s second lw, F which of the followin expressions is true? A. B. C. D. E. W
Solution Answer: B Justifiction: Fro the force dir nd tkin downwrds s positive ccelertion for, W herefore, by Newton s second lw, F F (Add forces in the se direction s, subtrct forces in the opposite direction of )
he Atwood Mchine III Which of the followin is the correct expression obtined by pplyin Newton s second lw for? Press for hint A. B. C. D. E. W
Solution Answer: A Justifiction: his question is solved in the se wy we s the previous question, for. It is ood ide to strt by drwin the free body dir for. W F (Add forces in the se direction s, subtrct forces in the opposite direction of ) Notice tht the nitude of is the se for both nd. However, in the cse of, it is ctin in the se direction s the ccelertion rther thn opposin it.
he Atwood Mchine IV Wht is the ccelertion of the two weihts? E. D. C. B. ) ( A.
Answer: E Justifiction: We hve lredy found two expressions for the ccelertion, lthouh they contin the unknown quntity. Since is equl lon ll points on the strin, it is the se in both equtions. Addin the two equtions will eliinte : Solution ) ( ) (
he Atwood Mchine V Now tht we hve found n eqution for the ccelertion, we cn nlyze soe how the Atwood Mchine behves. Under wht conditions will be netive? A. B. C. D. E.
Solution Answer: E Justifiction: Recll tht positive iplies tht ccelertes downwrds, while ccelertes upwrds. If is netive, this ens tht will ccelertes downwrds. We know intuitively tht will ccelerte downwrds if >. his is lso reflected in the forul we derived. 0 since 0
he Atwood Mchine VI How should the sses of the weihts be chosen such tht the neither weiht ccelertes? A. B. C. D. E.
Solution Answer: C Justifiction: If neither weiht ccelertes, then = 0. his occurs when =. No tter where the two weihts re positioned, the weihts will not ove. 0 0 since
he Atwood Mchine VII Suppose tht >> (one of the weihts is uch hevier thn the other). How cn the ccelertion be pproxited in this cse? A. B. C. ( ) D. E. 0
Solution Answer: D Justifiction: If >>, then we cn ke the followin pproxitions: Copred to, we cn pproxite tht = 0. We cn tret s if it were in freefll.
he Atwood Mchine VIII Dependin on the vlues chosen for the ss for nd, we cn chieve vriety of vlues for. Wht re ll the possible vlues of tht cn be obtined by vryin the sses? A. B. 0 C. D. E. 0
Solution Answer: D Justifiction: he lrest possible vlue for is obtined when = 0. 0 0 he sllest vlue of is obtined when = 0. 0 0 No tter how lre we choose, the weihts cnnot ccelerte upwrds or downwrds fster thn. Wht would hppen if this were not true?
he Atwood Mchine IX Wht is the tension on the strin? Press for hint E. D. C. B. A.
Answer: B Justifiction: We cn substitute the solution for into either one of the force equtions: For exple, usin the top eqution ives: Solution ) ( ) (
Solution Continued Answer: B Justifiction: Mny nswers cn be ruled out by doin few quick clcultions. Answer A is incorrect becuse it does not hve units of force. Answer C nd D re incorrect becuse it is possible to obtin netive vlues for tension. Suppose we let = 0. We should expect our forul to return = 0, since is in freefll. his is not true for nswer E.
he Atwood Mchine X Suppose is fixed, lthouh we re free to choose ny vlue for. How should be chosen (in ters of ) such tht =? A. B. C. D. E. 0
Solution Answer: C Justifiction: If the tension of the strin is equl to the weiht of, then is not ccelertin. Recll tht there is no ccelertion when =. Fro the forul:
he Atwood Mchine XI Wht is the force on the pulley required to hold it up lon the ceilin? (Recll tht we re ssuin ssless pulley nd strin) A. Fpulley 0.5 B. C. D. E. F F F F pulley pulley pulley pulley ( 4 )
Solution Answer: C Justifiction: he pulley ust be held up by twice the tension on the strin. Fpulley 4 Notice tht in the forul bove, F = 0 if either or is zero. Since the sses will be in freefll, no force is required to hold the ssless pulley up. Also notice tht F pulley = +, the su of the weiht of the two sses, only when the weihts re not ccelertin ( = ).