Chapte 6 NEWTON S nd LAW AND UNIFORM CIRCULAR MOTION 103 PHYS 1 1 L:\103 Phy LECTURES SLIDES\103Phy_Slide_T1Y3839\CH6Flah 3 4 ting Quetion: A ball attached to the end of a ting i whiled in a hoizontal plane. At the point indicated, the ting beak. Looking down on the ball fom aboe, which path doe it take? 103 PHYS D. Abdallah M. Azzee 1
Thu fa we hae applied Newton law, F = ma to linea motion. Now we ll apply it to otational motion 103 PHYS 3 Paticle moing with unifom peed in a cicula path with adiu ha an acceleation a c : a c (Deiation: ee Chapte 4.4) The acceleation point towad the cente of the cicle! Centipetal acceleation 103 PHYS 4 D. Abdallah M. Azzee
Newton law along the adial diection (along ): F ma m c The diection of the centipetal foce point towad the cente of the cicle. 103 PHYS 5 Conical Pendulum 103 PHYS 6 D. Abdallah M. Azzee 3
Example 6. Conical Pendulum A mall object of ma m i upended fom a ting of length L. The object eole with contant peed in a hoizontal cicle of adiu, a hown in the Figue. Find an expeion fo Whee i the acceleation ecto? FREE BODY DIAGRAM F x m a x m T in θ m (1 ) F y m a y 0 T coθ m g 0 T coθ m g ( ) ta n θ g g ta n θ = L inθ = L g inθ coθ T co mg a c L L T T in mg 103 PHYS 7 EXAMPLE 6.3 A ball of ma 0.5 kg i attached to the end of a ting 1.5 m long and whiled in a hoizontal plane at contant peed. If the cod can withtand a maximum tenion of 50 N, what i the maximum peed of the ball befoe the ting beak? F ho m T ma c m Tmax max 1. m/ m T What happen when inceae the length of the cod Read the et of the example 103 PHYS 8 D. Abdallah M. Azzee 4
TRY to ole 103 PHYS 9 103 PHYS 10 D. Abdallah M. Azzee 5
EXAMPLE 6.4 what i the max peed A ca take a bend on a flat, hoizontal oad. If the adiu of the bend i 35 m and the coefficient of tatic fiction between the tie and dy paement i 0.5, what maximum peed can the ca afely hae? F f ma m f μ n μ mg,max c f m,max max f,max max μ g 150m 13.1 m/ 47 km/h m 103 PHYS 11 Sample Poblem A ca of ma m = 1600 kg taeling at a contant peed = 0 m/ aound a flat, cicula tack of adiu R = 190 m. Fo what alue of between the tack and the tie of the ca will the ca be on the ege of liding off the tack? f m R Poitie diection towad the cente μ μ mg m mgr gr (0 m / ) 0.1 In eal ituation, why i a heaie ca (9.8 m / )(190 m) le lippey? 103 PHYS 1 D. Abdallah M. Azzee 6
K:\103 Phy LECTURES SLIDES\103Phy_Slide_T1Y3839\CH6Flah 103 PHYS 13 EXAMPLE 6.5 The banked exit amp A ciil enginee wihe to deign a cued exit amp fo a highway in uch a way that the ca will not hae to ely on fiction to ound the cue without kidding. Suppoe the deignated peed fo the amp i to be 48 km/h (13.4 m/) and the adiu of the cue i 50 m. At what angle hould the cue be banked? 103 PHYS 14 D. Abdallah M. Azzee 7
103 PHYS 15 m Fx max m n inθ= (1) F ma 0 y n coθ-mg=0 n coθ mg () y x n in a n y n co (1) /() tanθ= g mg θ 1 tan 0.1 g 103 PHYS 16 D. Abdallah M. Azzee 8
103 PHYS 17 EXAMPLE 6.6 Let Go Loop-the-Loop A pilot of ma m in a jet aicaft execute a loop-the-loop, a hown in the Fig. In thi maneue, the aicaft moe in a etical cicle of adiu.70 km at a contant peed of 5 m/. Detemine the foce exeted by the eat on the pilot (A) at the bottom of the loop and (B) at the top of the loop. Expe you anwe in tem of the weight of the pilot mg. 103 PHYS 18 D. Abdallah M. Azzee 9
At Bottom: F ma m c a c n B m mg m nb mg mg 1 g =.91 m g At Top: n T mg m m nt mg mg 1 g =0.913 mg a c 103 PHYS 19 Q: what i n at point A At A-point: n A m n A 103 PHYS 0 D. Abdallah M. Azzee 10
K:\103 Phy LECTURES SLIDES\103Phy_Slide_T1Y3839\CH6Flah 103 PHYS 1 Poblem: A ball (m = 0.5 kg) on the end of a =1 m long ting wing in a etical plane. At the top of it wing it peed i = 4.5 m/, and the tenion T in the ting i cloet to 1) Zeo ) 5 N 3) 10 N 4) 15 N 5) 0 N F y T m T mg ma mg mg T m Could be zeo! m 103 PHYS D. Abdallah M. Azzee 11
Poblem: A ball (m = 0.5 kg) on the end of a =1 m long ting wing in a etical plane. At the bottom of it wing it peed i = 4.5 m/, and the tenion T in the ting i cloet to 1) Zeo ) 5 N 3) 10 N 4) 15 N 5) 0 N F y T mg ma m T m mg T mg 4.9N 10.1N 15.0 N mg 103 PHYS 3 Sample Poblem In a 1901 cicu pefomance, Allo Dae Deil Diaolo intoduced the tunt of iding a bicycle in a loop-the-loop (Fig. 6-10a). Auming that the loop i a cicle with adiu R =.7 m, what i the leat peed Diaolo could hae at the top of the loop to emain in contact with it thee? 103 PHYS 4 D. Abdallah M. Azzee 1
SOLUTION: N mg m( ) R Poitie diection towad cente If N=0, then g R (9.8 m / )(.7 m) 5.1 m / Theefoe, he mut maintain at leat 5.1 m/ at the top of the loop. Othewie, he ll fall off the tack. 103 PHYS 5 103 PHYS 6 D. Abdallah M. Azzee 13
Sample Poblem Een ome eaoned olle-coate ide blanch at the thought of iding the Roto, which i eentially a lage, hollow cylinde that i otated apidly aound it cental axi. Befoe the ide begin, a ide ente the cylinde though a doo on the ide and tand on a floo, up againt a cana-coeed wall. The doo i cloed, and a the cylinde begin to tun, the ide, wall, and floo moe in union. When the ide' peed eache ome pedetemined alue, the floo abuptly and alamingly fall away. 103 PHYS 7 The ide doe not fall with it but intead i pinned to the wall while the cylinde otate, a if an uneen (and omewhat unfiendly) agent i peing the body to the wall. Late, the floo i eaed back to the ide' feet, the cylinde low, and the ide ink a few centimete to egain footing on the floo. (Some ide conide all thi to be fun.) Suppoe that the coefficient of tatic fiction between the ide' clothing and the cana i 0.40 and that the cylinde' adiu R i.1 m. (a) What minimum peed mut the cylinde and ide hae if the ide i not to fall when the floo dop? 103 PHYS 8 D. Abdallah M. Azzee 14
SOLUTION: f mg 0 μ N mg 0 N m R mg N μ Poitie diection towad the cente gr (9.8 m/ ) (.1 m) 7.17 m / 7. m / μ 0.40 (b) If the ide' ma i 49 kg, what i the magnitude of the centipetal foce on he? (7.17 m / ) N m (49 kg) R.1 m 100 N 103 PHYS 9 D. Abdallah M. Azzee 15