~v = x. ^x + ^y + ^x + ~a = vx. v = v 0 + at. ~v P=A = ~v P=B + ~v B=A. f k = k. W tot =KE. P av =W=t. W grav = mgy 1, mgy 2 = mgh =,U grav
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1 PHYSICS 5A FALL 2001 FINAL EXAM v = x a = v x = 1 2 a2 + v 0 + x 0 v 2 = v a(x, x 0) a = v2 r ~v = x ~a = vx v = v 0 + a y z ^x + ^y + ^z ^x + vy x, x 0 = 1 2 (v 0 + v) ~v P=A = ~v P=B + ~v B=A ^y + vz ^z ~F o = m~a ~ F1!2 =, ~ F 2!1 w = mg f k = k W = F ~ ~s KE = 1 2 mv2 P = F ~ ~v F=kx KE 1 + U 1 = KE 2 + U 2 F = mv2 r f s s W o =KE P av =W= W grav = mgy 1, mgy 2 = mgh =,U grav W elas = 1 2 kx2 1, 1 2 kx2 2 =,U elas F (x) =,U=x ~p = m~v ~ J =~p = ~ F ~p o = P n i=1 m i~v i ~r cm =(m i ~r i )=(m i ) ~F ex = M~a cm! ==! = v=r =!= = ! a an = r a rad = v 2 =r =! 2 r I =m i r 2 i K = 1 2 I!2 I p = I cm + Md 2 = I ~ = ~r F ~ = rf sin rf ~L = ~r ~p ~L = I~! L ~ =~ F =(Gm 1 m 2 )=(r 2 ) PE G = Gm 1 m 2 ( 1 r 1, 1 r 2 ) G =6:7 10,11 N-m 2 /kg 2
2 PE G =,(Gm 1 m 2 )=r F =,kx! = p k=m x() =A sin(! + )! =(2)=T =2f! = p g=l Circle: A = r 2 ; C =2r Sphere: A =4r 2 ; V =4=3r 3 Cylinder: A =2r 2 +2rh; V = r 2 h ~a ~ b = a x b x + a y b y + a z b z ~a ~ b = ab cos ab j~a ~ bj = ab sin ab sin 60 o = cos 30 o = p 3=2 sin 45 o = cos 45 o =1= p 2 sin 30 o = cos 60 o =1=2 =,bp b 2,4ac 2a
3 PROBLEM 1 [25 POINTS] In wha follows, you may assume ha he acceleraion due o graviy is exacly 10 m/s 2. You may also ignore he eecs of fricion. Remember o specify wheher he work done is posiive or negaive. An objec of mass 20 kg is dropped (released a res) from a ower of heigh 125 m, a ime =0. a) A wha ime does he objec srike he ground, 125 m below? b) Wih wha speed does i srike he ground? c) Wha is he oal work done by graviy in his period of ime? Insead of being dropped from a ower of heigh 125 m, he objec is hrown upward from he ground a ime = 0. The upward velociy of he objec, immediaely afer i is hrown, is 50 m/s. The horizonal velociy, immediaely afer i is hrown, is also 50 m/s. d) A wha ime does he objec reurn o srike he ground? You may ignore he heigh of he person hrowing he objec. e) Wih wha speed does i srike he ground? f) Wha is he oal work done by graviy in his period of ime?
4 PROBLEM 2 [20 POINTS] In he following, assume ha he acceleraion due o graviy is 10 m/s 2. A 30 kg board of lengh 10m ress on wo blocks. One block is a he lef end of he board, while he oher is 4 meers from he righ end of he board (see diagram). A 50 kg person seps ono he lef end of he board. a) Wih he person sanding on he lef end of he board, wha is he magniude of he force exered on he board by he righ-hand suppor? b) If he person begins o walk slowly owards he righ, how far does he ge before he board ips?
5 PROBLEM 3 [20 POINTS] A 10 kg mass lies a res on he end of a relaxed massless spring wih spring consan ofk = 5 N/m (see diagram). A 2 kg blob of puy, moving a a speed of 3 m/s, his and sicks o he 10 kg mass, causing he spring o compress. All eecs of fricion may be ignored. a) Wha is he maximum amoun of compression (in meers) of he spring? b) How much ime elapses beween he ime of impac of he puy and he ime a which he spring is a is poin of maximum compression?
6 PROBLEM 4 [25 POINTS] Two aseroids, one of mass kg and he oher of mass kilograms, are observed o be ying apar from each oher (see diagram). The heavier aseroid, on he lef, is raveling lef a 1 m/s. The ligher aseroid is raveling righ a 2 m/s. The aseroids are separaed by 100 meers. The value of he graviaional consan isg =6:7 10,11. (a) How far apar do he aseroids ge before being pulled back owards each oher by heir muual graviaional aracion? (b) A he poin in ime a which he wo aseroids are a heir maximum separaion, a space-based lab saion is half-way beween he wo aseroids. If he spacelab has a mass of 10 5 kg, wha is he magniude and direcion of he graviaional force exered by he aseroids on he spacelab? Assume ha he lab is oo ligh o aec he answer o par a).
7 PROBLEM 5 [25 POINTS] A fricionless pulley of mass 12 kg and radius 0.5 meer has a massless sring draped over i. On one end of he sring is a 10kg mass, while on he oher end is a 6kg mass (see diagram). Assume ha he acceleraion due o graviy is 10 m/s 2. The wo masses are held a res and hen released wihou inroducing any iniial velociy. a) How fas are he masses moving a he poin a which he heavy mass has descended by exacly 2 meers? b) Through wha angle has he pulley urned 2 seconds afer he masses are released?
8 PROBLEM 6 [20 POINTS] A plywood disk of mass 7 kg and radius 2 m roaes on fricionless bearings abou a verical axis hrough is cener. On he disk is a circular railroad rack of radius 1.5 m, on which a baery-operaed rain of mass 1.2 kg runs. Iniially, boh he rain and he disk are a res. The rain hen acceleraes for some amoun of ime, unil is speed relaive o he racks (no relaive o he ground) is 0.6 m/s. Wha is angular speed does he disk develop?
9 PROBLEM 7 [25 POINTS] A small ball of mass 4 kg is suspended by wo 2m-long srings which are separaed by 2m a heir poin of aachmen (see diagram I). a) Find he ension in each sring. The (massless) pole from which he srings are suspended is now secured verically o a device which can be used o roae he pole abou is axis (see diagram II). b) Wha is he minimum amoun ofwork ha mus be done on he pole (in geing i o roae) so ha he boom sring is fully exended (i.e., is ensioned by pull of he roaing ball)?
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