A Tale of Friction Student Notes

Transcription

1 Nae: Date: Cla:.0 Bac Concept. Rotatonal Moeent Kneatc Anular Velocty Denton A Tale o Frcton Student Note t Aerae anular elocty: Intantaneou anular elocty: anle : radan t d Tanental Velocty T t Aerae tanental elocty: T t r Intantaneou tanental elocty: : lenth o arc d T Becaue, r = crcle radu, = arc lenth, r,then the anular and tanental r elocte are related a: T d r or T r A Tale o Frcton Leon Student Note

2 Nae: Date: Cla: d dt I anular acceleraton dened a:, and tanental acceleraton a: at, then d and a T d r The anular and tanental acceleraton are then related a: a T. Rotatonal Knetc Enery and Moent o Inerta o a Rd Body For a nle partcle, the knetc enery or lnear oeent dened a: K = ½. Slarly, or a nle partcle, the knetc enery or rotatonal oeent en by the orula K = ½ I, where I = r known a oent o nerta o the partcle, and r the dtance ro the pont o rotaton. For a yte o partcle, t oent o nerta the u o the nddual oent I = r. Th denton can be etended to copute the oent o nerta or a contnuou rd body: r I r d For the pecal cae analyzed n th project, the oent o nerta or a old hooeneou phere : r I, r = radu o the phere.3 Anular Moentu and Torque o a Rd Body Fro the law o leer, the torque dened a F d, where F the perpendcular orce appled on the leer, at a dtance d ro the ulcru. Etendn th denton, takn the orce F a a ector F non- perpendcular to the ector r that e the poton o the pont where the orce appled wth repect to the rotaton pont, the torque dened a the ector product: r F Where the torque repreented a a ector perpendcular to the plane dened by ector r and F. The antude o th ector en by the product o the antude o the ector r and F, r and F te the ne F r o the anle between the: r F n When the anle 90 o, r and F are perpendcular, the ornal law o leer obtaned. A Tale o Frcton Leon Student Note

3 Nae: Date: Cla: For a nle partcle, the lnear oentu dened a the ector: P, and by denton, the orce or Newton econd law or a nle partcle when t a chane wth the te en by: d F When the a o the partcle contant: d F a Slar to the lnear oentu, the anular oentu o a partcle dened a: L r p r and lar to the orce denton or lnear oeent n ter o the lnear oentu, the orce producn a rotatonal oeent, or torque, dened a: dl For a partcle o contant a n a crcular oeent r contant, and the tanental elocty, the antude o the torque en by: dt r r at r a orula lar to Newton econd law, or a rotatn rd body wth contant a. or I. Force o Frcton or a Sphercal Rd Body Rolln wthout Slppn on an Inclne Surace A phercal rd body o a rolln down an nclne urace. The orce producn th oeent are the weht o the body and the orce o rcton. Becaue the body rolln ntead ldn, the rcton to conder n th proble tatc rcton. So, the coecent to ue n the rcton orce calculaton the tatc rcton coecent μ I. The body roll becaue a torque produced by the rcton orce and the coponent o the body weht parallel to the nclned urace. A Tale o Frcton Leon Student Note 3

4 Nae: Date: Cla: All orce on the rolln body can be analyzed by un a ree-body dara: The orce that ake th phercal old o radu r and a roll down the nclne plane are thoe alon the -a. Un Newton econd law to decrbe the old lnear oeent: The orce that ake the old rotate the torque produced by the rcton orce: where I = phere oentu o nerta, α = phere anular acceleraton, r = phere radu. Subttutn 3 n : F a n r I For a hooenou phercal object: I a r r r The orce o rcton can be epreed a: 3 a r. Subttutn th alue, and α = a/r n : a n a a n a a a n a n a n A Tale o Frcton Leon Student Note 4

5 Nae: Date: Cla: Subttutn the obtaned acceleraton alue n 3, the orce o rcton can be epreed n ter o the phere weht and the anle o the nclne: n 4 n By denton, the tatc rcton orce proportonal to the noral orce, or orce the urace apple on the phere to balance the coponent o the phere weht perpendcular to the urace. F where μ the tatc rcton coecent and F n the antude o the noral orce. For th proble, the alue o the noral orce can be ound n the ree-body dara. F n n co Subttutn th alue n 6 co Cobnn equaton 4 and 6, an epreon or the coecent o tatc rcton can be ound: co n co n n co tan Equaton tate that or a phercal body rolln on an nclne, the coecent o tatc rcton a uncton o the nclned urace anle, peccally, o the tanent o th anle or lope o the nclned urace. 3. Force o Frcton or a Sphercal Rd Body Rolln wthout Slppn on a Varable Slope Path A phere rolln on a path wth arable lope can be ualzed a rolln on a equence o nclned tanent traht urace. I the path en a a uncton y =, derentable at eery pont, the lope o any thee nclne, dened a = tan, can be ound un the derate : dy d A Tale o Frcton Leon Student Note

6 Nae: Date: Cla: Then: tan Un the aboe epreon n, the coecent o tatc rcton or a phere rolln on a arable lope path en by: 8 Epreon 8 tate that the tatc rcton coecent are ro pont to pont on the path. Un equaton 8 n 6, the tatc rcton orce or a rolln phere o a : Un tronoetry, t poble to epre co θ alo n ter o. Becaue tan θ =, then: Then: 9 Soln wth rht tranle: co θ = arctan coarctan tan opp adj co adj hyp opp n hyp Un th epreon or co θ n 9: 0 Equaton 8 and 0 ee to be adequate to odel the proble o a phere rolln on a arable lope path wth rcton, but oethn need to be ed. Both equaton e neate alue or oe ecton o the path, and by denton, the tatc rcton coecent alway pote. The proble coe ro the n o the derate and the poblty o neate lope. For the nclned plane, the eleaton anle alway pote. Becaue n th proble we approached the path wth arable lope a a equence o nclne, the lope en by ut alway be pote. Wth th retrcton, equaton 8 and 0 need to be en n ter o the abolute alue o : A Tale o Frcton Leon Student Note 6

7 Nae: Date: Cla: 4. Work-Enery Theore or a Sphere Rolln on a Varable Slope Path wth Frcton The work-enery theore tate that the echancal enery knetc enery + potental enery o an olated yte under only conerate orce rean contant. Where E the echancal enery o the yte, K the knetc enery ½, U the potental enery h, and the ndee and ndcate the enere at the end and bennn o the proce, repectely. Enery cannot be created nor detroyed n an olated yte, but t can be nternally conerted to any other or o enery. When a non-conerate orce uch a rcton condered n a yte, the work-enery theore tate that the work done by the non-conerate orce, ΔW, equalent to the chane n echancal enery: 3 E K E K U 0 W U or K E K U Under non-conerate orce, ΔE no loner zero, but another key derence et. Meanwhle or conerate yte, the work done by conerate orce depend only on the ntal and nal poton. For non-conerate yte, the work done by non-conerate orce, lke rcton, depend on the path or trajectory or on the te thee orce aect the yte. In a yte under conerate orce, the work on a cloed loop zero, whle n a yte under nonconerate orce t not. E By denton, echancal work the product o the dplaceent te the orce coponent alon the dplaceent: U F W F F co A Tale o Frcton Leon Student Note

8 Nae: Date: Cla: A Tale o Frcton Leon Student Note 8 For the arable lope path y =, the work done by the orce o rcton : Takn derental dplaceent alon the path: 4 I the derental arc d dened a: the arc d can be et a a uncton o : Subttutn n n 4: Becaue d > 0, d d, then: Becaue the rcton orce alway act aant the oeent, the work done by rcton alway neate. Then: dy d d d dw W d d d dy dy d d d d dw d d d dw d dw

9 Nae: Date: Cla: A Tale o Frcton Leon Student Note 9 Aan takn all dplaceent alon the path: 6 For the entre path, the total work the u o the work done alon the lttle dplaceent the path had been dded nto. Subttutn equaton 6 n 3: Then: becaue the heht o the object on the path en by the alue o uncton that, h =. Un the aboe equaton, t poble to nd the elocty o the rolln body at the end o eery lttle dplaceent, known the alue o t elocty at the bennn o the dplaceent, and o the correpondn potental enery chane: 8 h = h = W h h U U K K U K W h h 4

10 Nae: Date: Cla: h 0 = 0 0 h = h = h 3 = 3 3 y = 0 3 A arable lope path can be approated by a equence o all nclne, and on each one, the echancal-enery theore can be appled, and the work done by the rcton orce can be ealy etated. A Tale o Frcton Leon Student Note 0