Uniform Circular Motion

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1 Unfom Ccul Moton Unfom ccul Moton An object mong t constnt sped n ccle The ntude of the eloct emns constnt The decton of the eloct chnges contnuousl!!!! Snce cceleton s te of chnge of eloct:!! Δ Δt The eloct chnges f ethe: ts ntude chnges ts decton chnges Lets t to detemne the ntude nd decton of the cceleton, n tems of the constnt ntude of the eloct, nd the dus of the ccle. ntl! Acceleton n Unfom Ccul Moton fnl! Let s detemne the ege cceleton s the object (the geen bll) moes fom poston mked ntl to tht mked fnl. These two postons wee chosen so tht the lne fom the cente of the ccle to those postons mke n ngle wth the s. The othe two ngles mked come fom the fct tht the eloct ectos n unfom ccul moton e tngent to the ccle t ll tmes.

2 Acceleton n Unfom Ccul Moton ntl! fnl! Dstnce the bll tels fom ntl to fnl tme c length of ccle d ( ) Snce the bll tels t constnt speed, we cn get the tme fom ntl to fnl: d t The components of the ntl eloct: cos sn The components of the fnl eloct: f cos f sn 3 Acceleton n Unfom Ccul Moton Fom the boe nfomton, we cn get the ege cceleton s the object moes fom the ntl to the fnl poston: Δ Δt g, 0 g, Δ Δt sn sn sn Ths s just the ege cceleton. To get the nstntneous cceleton, we need to let the ngle shnk to zeo. sn Usng the fct tht: lm 0 0 We fnd tht t the top of the ccle, the nstntneous cceleton ponts downwd - towd the cente of the ccle. In fct, f we pck n othe pont ound the ccle, we wll fnd tht the cceleton n unfom ccul moton hs the ntude: nd ponts towd the cente of the ccle. Ths mens tht the cceleton ecto s NOT constnt! 4

3 Acceleton n Unfom Ccul Moton We fnd tht t the top of the ccle, the nstntneous cceleton ponts downwd - towd the cente of the ccle. In fct, f we pck n othe pont ound the ccle, we wll fnd tht the cceleton n unfom ccul moton hs the ntude:!! " " "! " nd ponts towd the cente of the ccle. Ths mens tht the cceleton ecto s NOT constnt!! " 5 Knemtcs of Unfom Ccul Moton Fo n object mong t constnt speed n ccle of dus : Acceleton: Snce ths cceleton ponts to the cente of the ccle, t s clled centpetl (fo cente-seekng ) cceleton. Dstnce teled n eoluton: D π Tme (o peod) fo eoluton: T π 6 3

4 Emple: Centpetl Acceleton: A c clms tht ts ltel cceleton s 8.9m/s (ths s 0.9g s). Ths epesents the mmum cceleton tht c cn undego sdews to ts moton wthout skddng. If the c s telng t constnt speed of 00 m/h (~45m/s), wht s the mnmum dus of cue t cn negotte wthout skddng?? 45m/s ( 45) m Note: If the speed nceses, the dus necess nceses b the sque If the cue s bnked, we wll fnd the the mnmum dus necess s smlle Wht s podng the cceleton? 7 Dnmcs of Unfom Ccul Moton Newton s st Lw: A bod moes n constnt speed n stght lne unless net foce cts on t to chnge ts moton If n object s mong n ccle, thee must be foce ctng on t.!! Newton s nd Lw: F m We usull thnk of ths s A net foce on n object of mss m cuses n cceleton. We cn tun ths thnkng ound. In unfom ccul moton, we detemned tht thee s n cceleton. If thee s n cceleton, thee must be foce cusng ths cceleton. If n object s undegong unfom ccul moton, t s beng cted on b net centpetl ( cente-seekng ) foce: F m m 8 4

5 Emple: Bll swngng n etcl ccle A bll s beng swung n etcl ccle. The mss of the bll s 50g, nd the dus of the ccle s.m. A) Wht s the mnmum speed the bll must he t the top of the ccle so tht t contnues to moe n ccle? At the top of the ccle, thee e two foces ctng on the bll: T Note tht we e defnng downwd to be poste - ell defnng nwd (towd the ccle cente) s poste. 9 Emple: Bll swngng n etcl ccle T m F m m + T + Ths lst equton ss tht s the speed of the bll nceses, T (the tenson n the stng lso must ncese (snce nd e constnt). Ths cetnl confoms wth ou epeence. But lso note tht s gets smlle, T must decese. Eentull, f gets smll enough, T becomes zeo, nd the stng wll no longe be unde tenson. The mnmum speed occus when T0: mn mn m mn g 3.8m/s 0 5

6 Emple: Swngng bll contnued B) If the speed s ctull twce the mnmum detemned n pt A, wht s the tenson n the stng t the bottom of the ccle? T m F m m + T m T + (0.5)(6.56) T (0.5)(9.80) +. T 7.34N Note tht we e now defnng upwd to be poste - ell defnng nwd (towd the ccle cente) s poste. Emple:C gong oe hump n the od Wht s the mnmum speed such tht the c wll lee the od t the top of the hump? m m F m m FN + As nceses, F N must decese. F N Mnmum occus when F N becomes zeo. m mn mn g 6

7 Emple: Puck nd weght m 0.4kg A puck s sldng on fctonless tble. It s connected to stng tht goes though hole n the cente of the tble. The othe end of the stng s ttched to weght. m.0kg Wht s the speed of the puck necess so tht the puck moes n ccle wth dus of 0.8m (nd such tht the weght m emns stton)? 3 Emple: Puck nd weght SIDE VIEW: Fee bod dgm fo puck F N T T m g m g Fee bod dgm fo weght 4 7

8 Emple: Puck nd weght (cont) Appl Newton s nd Lw to the puck (nwd s poste): F m m + T T m m Wht s T? Appl Newton s nd Lw to the weght: F m + T 0 T m g 0 Now we cn sole fo the speed: T m (0.8m)(9.80m/s )(.0kg) 0.4kg 4.4 m/s gm m 5 Emple: C on Cue: Flt Rod A c ppoches cue wth speed of 4m/s (bout 50km/h). The dus of cutue of the od s 50m. Wll the c the c skd? Assume tht fo d od the coeffcent of sttc fcton s 0.6, whle fo wet od t s 0.5. Top ew: R50m 4m/s m000kg 6 8

9 Emple: C on Cue: Flt Rod Wht s the centpetl foce tht keeps the c on the od nd mong n ccle? It s the foce of fcton pushng the tes fom the sde. It s sttc fcton - s long s the tes do not skd. Vew fom boe the c: Vew fom behnd the c: F fs F fs F N 7 Emple (cont): C on Cue: Flt Rod Detemne the mmum llowed sttc fcton fo the two od condtons: A) D od m Ffs μsfn (0.6)(9800) 5900N C wll st on the od. Note tht the fcton foce s 3900N, nd tht t wll ncese f R s smlle o s bgge. m B) Wet od F μ F (0.5)(9800) 500N fs s N C wll skd off of the od. 8 9

10 Emple (cont): C on Cue: Flt Rod Appl Newton s nd Lw to the c: m F m m + Ffs (000kg)(4m/s) Ffs 50m F 3900N fs + F F F N N F N m 0 0 (000kg)(9.80m/s ) 9800N Wht ths mens s tht 3900N s needed to keep the c tunng nd on the od. Is ths bgge thn the mmum llowed sttc fcton? 9 Emple: C on Cue: Bnked Rod Vew fom boe the c: R50m 4m/s m000kg Cues lke ths e usull bnked. Wh? 0 0

11 Emple: C on Cue: Bnked Rod Vew fom boe the c: Vew fom behnd the c: F fs F N s the ngle tht the od s bnked t. Cn ths be chosen so tht we don t need to el on fcton t ll? Yes! In the fgue boe, the -component of the noml foce cn pode the foce pushng the c towds the cente of the ccle.

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