Chapte 6 NEWTON S nd LAW AND UNIFORM CIRCULAR MOTION Phyic 1 1 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? Phyic 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 Phyic 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 Phyic 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. Phyic 5 Phyic 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 Q = L inθ = L g inθ coθ T coθ Phyic 7 mg a c θ L L T θ T inθ mg θ 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 Phyic 8 D. Abdallah M. Azzee 4
TRY to ole Phyic 9 Phyic 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 =,max µ mg c f = m,max max f,max max = = µ g = 150m = 13.1 m/ 47 km/h m Phyic 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 µ 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? µ Phyic 1 D. Abdallah M. Azzee 6
Phyic 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? Phyic 14 D. Abdallah M. Azzee 7
Phyic 15 m Fx = max = m n inθ= (1) F = ma =0 y n coθ-mg=0 n coθ = mg () y x n inθ a n θ y n coθ (1) /() tanθ= g mg θ 1 = tan = 0.1 g o Phyic 16 D. Abdallah M. Azzee 8
Phyic 17 EXAMPLE 6.6 Let Go Loop-the 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. Phyic 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 Phyic 19 Q: what i n at point A At A-point: n A = m n A Phyic 0 D. Abdallah M. Azzee 10
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 mg mg T = T + mg = ma = m Could be zeo! m Phyic 1 Poblem: A pendulum coniting of a m = 0.5 kg ball on the end of a =1 m long ting i eleaed fom hoizontal. At the bottom of it wing it peed = 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 Phyic D. Abdallah M. Azzee 11
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? Phyic 3 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. Phyic 4 D. Abdallah M. Azzee 1
Phyic 5 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. Phyic 6 D. Abdallah M. Azzee 13
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 µ 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? Phyic 7 SOLUTION: f mg = 0 µn 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 Phyic 8 D. Abdallah M. Azzee 14