1121 T Question 1

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1 1121 T Question 1 ( aks) You ae cycling, on a long staight path, at a constant speed of 6.0.s 1. Anothe cyclist passes you, tavelling on the sae path in the sae diection as you, at a constant speed of v b = 9.0.s 1. At the instant when she passes you, you ealise that she is a fiend of yous and you acceleate to catch up to he. You acceleate, stating fo t = 0, the tie when she passes you, with constant acceleation a = 1.5.s 2. a) i) On a displaceent-tie gaph, sketch you position, x a (t), and the position of you fiend, x b (t) as functions of tie, fo tie t < 0 (i.e. while she is still behind you). Label these sections of the gaphs x a and x b. ii) Also sketch displaceent-tie gaphs fo you you fiend fo tie t > 0. Label these sections of the gaphs x a and x b as well. iii) Showing you woking, deive both algebaic expessions and quantitative values fo the tie and distance it takes you to catch up with you fiend. b) In pat (a), you acceleated with constant acceleation and ovetook he. In this pat, you acceleate with fowads acceleation a = 1.5.s 2 fo a tie T 1, then deceleate with fowads acceleation a = 1.5.s 2 fo a tie T 2. You judge T 1 and T 2 so that when you stop deceleating, you ae tavelling alongside he at the sae speed. Daw a second displaceent-tie gaph to show this situation. Clealy ak x a, x b and the tie intevals T 1 and T 2.

2 Question 1 iii) x b = v b t x x a x a = v 0 t at2 catch up when x a = x b v b t = v 0 t at2 x b v b v 0 = 1 2 at t = 2(v b v 0 ) a x a b) x b t = 2(9.0.s s 1 ) 1.5.s 2 = 4.0 s. Distance tavelled in that tie: x b = v b t = 2v b(v b v 0 ) a = 36. x x b x a T 2 x a T 1 t x b

3 Question 2 ( aks) i) Wite Newton's second law in a fo that applies to a finite object that is not necessaily igid, but that has constant ass. If you stateent is an equation, state caefully the eaning of each te. (Fo exaple, do not let the ake wonde "what foce?" o "what acceleation?".) The sketches show successive states of a an juping vetically in the ai. He begins (sketch A) fo a stationay position with his legs bent. He then staightens his legs and ankles apidly: sketch B shows the oent at which his feet leave the gound. The lines aked "CoM" show the height of his cente of ass. Between the fist two sketches, his cente of ass ises a distance L. Sketch C shows hi at the point whee his cente of ass has its axiu height, which is a vetical distance h above its height at the point of take-off. You ay neglect ai esistance. A B CoM h CoM L CoM ii) Showing you woking, and stating any assuptions you ake, deteine the speed of the an's cente of ass at the oent (B) when his feet leave the gound. iii) Assue that the vetical acceleation a c of his cente of ass is constant between A and B. Deive an expession fo a c. iv) Using you answe to pat (i), and thinking caefully, deive an expession fo the vetical foce N exeted by the gound on his feet duing the phase A to B. v) If the an's ass is 70 kg, if L is 0.4 and h is 0.6, what is the downwads foce (assued constant) exeted by his feet duing the phase A to B? State any physical law o pinciple you use in obtaining you answe.

4 Question 2 i) Σ F ext =.a c whee Σ F ext is the total extenal foce, is the ass of the object and a c is the acceleation of the cente of ass of the object OR F ext =.a c whee F ext is the total extenal foce, is the ass of the object and a c is the acceleation of the cente of ass of the object ii) OR Σ F ext = d dt (.v c) OR Σ F ext = d dt p c etc duing the jup phase (B C), no extenal nonconsevative foces act, so the echanical enegy associated with the cente of ass is conseved U i + K i = U f + K f o U + K = 0 iii) iv) At C, K = 0, so 1 2 v c 2 = U gav = gh so v c = 2gh so v c 2 = 2gh otion in one diension with constant acceleation: 2a y L = v yf 2 v yi 2. othe notations acceptable 2a c L = v c 2 a c = v c 2 2L = gh L Σ F ext =.a c applied in the vetical diection gives N g =.a c N = (g + a c ) = g 1 + h L (diection is upwads) OR N = g 1 + h L up. v) Fo Newton's thid law, this foce F has agnitude N but is downwads. F = g 1 + h L (down). F = 1.7 kn.

5 Question 3 a) ω A sall, flat agnet, of ass, is positioned at a distance fo the cente of a steel disc that otates with angula velocity ω about a hoizontal axis, as shown. The agnitude of the agnetic foce between the agnet and the disc is F and it is in the noal diection only. The coefficients of static and kinetic fiction between the disc and the agnet ae espectively µ s and µ k espectively, and µ s > µ k. The agnet does not slide when the disc is stationay. What is the axiu value of ω at which the agnet will not slide on the disk? (Hint: at which point is it ost likely to begin to slide?) b) With espect to a vey lage sepaation, the potential enegy of a pai of asses M and sepaated by is U = GM/. The agnitude of the gavitational foce between the is F = GM/ 2, whee is the distance between thei centes and whee G is the univesal constant of gavitation. Deteine the total echanical enegy E of a sall satellite (ass ) in a cicula obit of adius aound a planet of ass M in tes of G, M, and. (Hint: what is the centipetal foce?)

6 Question 3 While it is not sliding, the agnet undegoes unifo cicula otion, so the total foce on it is a c = ω 2 towads the cente. The total foce is F f + g. The geatest fictional foce is equied at the botto of the cicle, whee, taking the upwads diection as positive: F f g = ΣF = a c = ω 2 so F f = g + ω 2 (At the top of the cicle, F f g = ω 2 so F f = ω 2 g and at inteediate angles it has inteediate values.) Fo the definition of static fiction, the axiu value of F fax = µ s N, whee N is the noal foce, so F fax = µ s N = g + ω ax 2 Hee, the only foce in the noal diection is F, so F = N, µ s F = g + ω ax 2 ω ax 2 = µ s F g ω ax = µ s F g b) E = U + K K = 1 2 v2 centipital foce = v2 so v 2 = GM so E = U + K = GM GM = F g = = GM 2 GM 2

7 Question 4 L θ o L Two light, inextensible stings, each of length L ae hung fo the sae fixed point, as shown. On one, a lup of plasticine (a soft ateial) of ass M is attached. Initially, it hangs vetically. On the othe sting, a ball of ass is attached. Initially, it is stationay, but displaced so that its sting akes and angle θ o with the vetical, as shown. The diensions of the ball and plasticene ae uch salle than L. The ball is then eleased. When its sting is vetical, it stikes the plasticine and the two eain stuck togethe. Showing all woking and stating any assuptions you ake, deive an expession fo the speed of the cobined object iediately afte the collision, in tes of the paaetes given and g, the gavitational acceleation. Ai esistance is negligible. M Question 4 a b c d θ o L L α Befoe the collision (a to b), no nonconsevative foces act, so echanical enegy is conseved. Taking the botto of the path as the zeo fo U: U a + K a = U b + K b gh + 0 = v2. v 2 = 2gh = 2gL(1 cos θ) v = 2gL(1 cos θ). h M v M M+ V H Duing the collision (b to c), extenal foces in the hoizontal diection ae negligible, so oentu is conseved in the hoizontal diection. So v + 0 = (M+)V V = M+ v = M+ 2gL(1 cos θ). M+

8 Question 5 Two cylindical jas each have adius. The thickness of the walls is negligible copaed with. When epty, the ass of each ja is and thei adius of gyation k = (to an appoxiation sufficient fo this poble). One ja is full of wate with ass M. The othe is full of honey with ass M. They ae both placed, stationay, on an inclined plane aking an angle θ with the hoizontal. Thei oientation on the plane allows the to oll to the botto along the shotest path on the plane. Fiction between jas and plane is always sufficient to ensue olling. i) The viscosity of the wate is sufficiently low that the wate does not otate. Deteine the speed of the ja of wate afte it has olled a distance s down the plane. ii) The viscosity of the honey is sufficiently high that the honey otates as a solid object, at the sae ate as the ja. Deteine the speed of the ja of honey afte it has olled a distance s down the plane. The oent of inetia of a hoop is 2. That of a disc is 1 2 2, whee tes have thei usual eaning. Hint: what is the elative velocity at the point of contact duing olling? Question 5 v = 0, ω = 0 h s θ ω θ v Because the elative velocity at the point of contact duing olling is zeo, nonconsevative foces do no wok, theefoe echanical enegy is conseved. U + K = 0 K = U 1 2 (+M)v Iω2 = (M+)gh Rolling ω = v/ so (+M)v 2 + Iv 2 / 2 = 2(M+)gh i) Fo wate, only the ja otates, so I = k 2 = 2, so (+M)v v 2 / 2 = 2(M+)gh (2+M)v 2 = 2(M+)gh v = gh 2(M+) 2+M = gh 1+/M 1/2+/M ii) Fo honey, both ja and contents otate, so I = M2 so (+M)v 2 + ( M)2 v 2 / 2 = 2(M+)gh (2+3M/2)v 2 = 2(M+)gh v = gh 2(M+) 2+3M/2 = gh 1+/M 3/4+/M

1131 T Question 1

1131 T Question 1 1131 T1 2008 Question 1 ( aks) You ae cycling, on a long staight path, at a constant speed of 6.0.s 1. Anothe cyclist passes you, taelling on the sae path in the sae diection as you, at a constant speed

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