Answers to Odd Problems in Intermediate Dynamics

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1 Answes to Odd Pobems in Intemediate Dynamics Patick Hami San Jose State Univesity San Jose, Caifonia Januay, 9 Capte cm/sec at.8 above te oizonta. (a) 44ft= sec (b) mp..5 v = k b e bt (bt + ) x = x + k b e bt t + b + t.7 t =. sec m m.5..9 (a) ( 4a 4b ) (b) x cm = y cm = 4. R cm = ( 8 R 8 R 8 R) b a b a (c) (677 8). p 6 a mg.9 (a) 7 mp (b) 4 mp.. 5 watt.. 4 m/s..5 v i =v f = 89 te steetcas wi un about pecent faste. Capte. 6 m/s.. (a) v = v +kv x = x t + k [n( + kv t)] (b) As t!, v!..5 v = vx + v y = [(g x =yg) + (yg)] = = tan (v y =v x ) = tan (y=x) km..9 (a) = p a (b) a = 98 m/s, v = 79 m/s, = 844 min.. = = (=).9 v = 4 sin ^ + (4 4 cos ) ^ a = ( cos 6)^+ sin ^ b

2 . ^( _ ) + ^( + _ + ) m/s..7 + z cos + p z.9 x = a cos!(t ) cos (t ) y = a cos!(t ) sin (t ) z = a sin!(t ).. = + z = tan z = = sin = z = cos. (a) An inwad spia. (b) p b + (akt bkt ) Capte.!A sin!t! A cos!t. 8 4 N..5 M M M M.7 (a)a = m m g m +m T = mm m +m g (b) g(m m )=(m + m I=R ). T=T = (7 6 ) = = 79 Peiods di e by about.8% coud be detetected easiy.. (a) ()Book/Eat (gavitationa) Book/Tabe (eectostatic) Tabe/Suface (eectostatic) Tabe/Eat(gavitationa) ()Rocket/Eat (gavitationa) Rocket/Pume (moecua coisions - fundamentay eectostatic), Pume/Atmospee (moecua coisions) () Donkey/Road (fiction) Donkey/Cat (mecanica). (b) Unbaanced foce on inne wa opposite to nozze. (c) Road on donkey s ooves..5 5L=6 5L=6 i.7 (a)v = F t m (t=m x = F m ) m t n( m t) (b) F m t +.9 T bot = 56 NT mid = 7 NT top = 59 N Euation T (x) = g( + 7x). 58 m/s m/s, 77 m..7 v(t) = b n e bv + A bm t.9 v(t) = p mg=b tan p gb=mt. x(t) = m b og cos p p gb=mt = m b og e gb=mt +e (a) v T = p mg=d (b) = (m=d) n v + vt =v T (c) v g = v T v = p v + v T. (a) 8 m (b) 5 s..5 (a) 5587 m/s. p (b) v = v T exp( gx=v T ) (c) 8 m/s..7 y = mg b t + m b e bt=m.9 (a) 9 m/s, (b) 47 s p m/s..4 x =8GM p gb=mt

3 .45 (a)x(t) = D ep t + k (b)v = m x.47 6X s ' 4 min. Capte 4 C D e p t 4. L = ma _ + ma _ sin mga cos 4. L = m _z kz C 4.5 Euations of motion (M + m)x + M cos M _ sin = M + Mx cos + mg sin =. Constant of te motion (M + m) _x + M _ cos g sin 4.7 s = [m=(m+m)] cos and X m = m+m 4.9 L = M + m _s + (M + 4m)gs sin g sin [m=(m+m)] cos cos 4. (a) L = (M + m) X _ + m X s cos + m _s + mgs sin ks (b) p X = (m + M) X _ + m _s cos p s = m X _ cos + m _s. (c) (m + M) X + ms cos = mx cos + ms mg sin + ks = 4. (a) L = m _x + mg( x) k(d x) (b) mx + kx = kd mg o m k x = x C (c) p k=m 4.5 a p + sin 4.7 y = c p x c + d 4.9 = y p + y 4. i 4. H = p m mg cos _p = mg sin _ = p m. Capte J. 5. kr 5.5 (a) = = (b) ds = ^d+^d da = dd = = a(sin u cos v + cos u sin v) = a(cos u sin v + sin u cos v) = ds = a(sin u cos v + cos u sin v) = du^e u +a(cos u sin v + sin u cos v) = dv^e v + dz^k. 5.9 = ^e =a cos sin + sin +^e =a sin cos + cos +^e =a sin R A

4 5. (b) f = ^ 5.5 (a) T = (î+ˆ±+ˆk) ) 46 K/m. (b) 45 K. (c) Te ate of tempeatue decease is getting age te fute we move fom te oigin. 5.7 v max = m n (Q+Q ) 4 a N kg. 5. = G Mem jej R e 5.7 (a) F = A ^{x+^ y+^kz (x +y +z ) = (b) W = A= p x + y + z A oi p b +a at y = (c) F = Q 5.9 Q Q Q 4" 4" Gm 5. m 5. (b) 44 m/s, (c) +577 and x = 77 m. p 5.5 x(t) = v m x a sin p a m t y(t) = v p m y b sin Capte 6 6. (a) 5 m/s (b) 45 m/s. 6. v = m+m p m g 6.7 (a) v = v m = m 4= (b) F = + 4kvm k(vm) K = = 6.9 cos mm m M +m W () b m t =4 ti K wee k = d and K = m= = i m 4= + 4kvm t K = 6. kmv. 6.5 v(t) = gt + mg m k n m kt kg/s, 6 kg/s. 6.9 t =66 s = m p times te eigt fom wic it was dopped. 6. m/s., km/s. Capte 7 7. (b) Tota toue is zeo so angua momentum is constant. (c) Patices ave te same inea momentum, so angua momentum is independent of te coice of oigin. 7.5 = (=)v x F e t ^{ 7.7 d = DR ad/s. 7. M(R + R) 4

5 MR 7.7 Numbe of tuns =! I p + =4amg t = + p p! I + amg 7.9 (a)w = d mg (b)! f = p! i + gd=a (c) F = mg 7. (a)! = 9v= (b)! = Capte 8 8. Enegy. 8. Yes. Capte m/s. 9. G 9.5 GM R L v L p Z + R p (Z L) + R i Gm 9.7 d 9.9 G^ 9. Outside = GM g = GM ^ Inside g = GM R ^ = GM 9. (a) = R Gad GM R s = GM GM s = p z +a a + sin i i R (b) = min. 9.7 (a) Fo spee outside and inside, g = GM g = GM (b)fo se outside and inside, g = GM R 9.9 g = G (towads pane). 5k 9. 4G Fo te second ed, te mass density is a deta function, zeo eveywee except at te oigin. Capte. n (Å) E(eV) = (GM) = T 5= V V.7 (a) = m m +m = + m m +m m m (b)l = (m + m ) R _ + m +m _ + G mm (c) _ + G mm = d dt _ = (d) = G (m+m) 5

6 (e) 4 = G(m +m ) a.9 (a) = (a=)( + cos ) (b) =8 p ma 5 =K. km.. e = (v max v min ) = (v min + v max ) Capte. b =4m = k.. k = 4 4 N/m, b = kg/s..5 t p k b 6= m m.7 x + v =. x(t) = x + A!! d + c A!! d +A!! d cos!d t + (! d b=m) sin! d t.. x(t) = Ae t cos (! t + ) + F 4a m ( e at ).5! = k m+m m m.9 Noma mode feuencies! = m+m k m m,! = Capte. Wok = I _ f = T i sin4 m.5 (a) L = m _ ma f _ sin sin ft + m A f sin ft (b) = mg( Af cos + A cos i ft) sin cos ft + g Af cos ft + g (c)! =.7 (a) 85 s, (b) m T = p m=k 4k. pg F sin max.5 48 ad/sec..7 T = E mg cos cos.9 H = Capte p m + E mg p m sin + mg cos _p = _ = A! d b=m t e c t p m sin. m+m. a = a + ( +!) v +! +! (! E ).5 a cent = a co = V R ^x.7 d = g sin cos + = 79 cm..9.5 km Sout and.5 km East of te expected taget. It is in igt. seconds onge tan expected. m k m 6

7 . 767 km/.. m..5 ad..7 (a) v^ (b) v^ (c) (d) d = m (Not).. (a)! = Capte 4 k m (b) (t) = a + cos(! o t) v g e 4. tan = (tan tan ) T = [mg= (sin + cos tan )] T = [mg= (sin + cos tan )] cos cos o Mg at a point 86 aong te od. = (a) CW otation and inea acceeation aong te x axis. (b) newtons acting at a distance y = 866 m fom x axis. 4.5 cg = (a=) + y =4 =y = m K/m. Capte Te otation as a magnitude of 57 Te axis of otation is in te euatoia pane and diected at an ange of west of Geenwic. = p 5 5= p = p 4 5 = p = p 4 5 = p A (a) A e ection toug te x-axis. (b) A otation of 8 about te z-axis. 5. = p and = = 5 = p 5 Capte 6 = p 5 = p 5 p p = 5 = 5 = p 5 = p Ma 5Ma =p = p =p = p A i 6.9 T = I _ + _ sin + I _ + _ cos L = I _ sin sin ^e + I _ sin cos ^e + I _ + _ cos ^e 6. (a) p = I cos! p = I! (cos cos ) V () = I! I + mgd cos + sin I! E = mgd cos + I! 7

8 (b) cos = 4 cos + 4 i = 6. _ = 8 ad/s. Capte 7 7. y(x t) = A sin x L cos p F= L t 7.5 y(x t) ' L 4R + P L n n R ( )n cos nx L cos! nt wee! n = nc=l i @y = (b) m + b _ m + F m =L m = (c) m = e bt= (A m sin! mt+b m cos! mt) wee! m = 7.9 (a) Q n + F n Q n = ( b) Q n = A n cos F nt+b n sin Capte 8 m L F F nt 8. Assume x and x ae measued in opposite diections fom te euiibium points. Mass matix is m and te potentia matix is k 6 8.! = k m 8 () = p 84m 6 () = p 6m 6 k 8.5 m k m k m 8.7 (a) L = m( _ + _ ) k + k k + k m k m = + k m k m = 8.9 (t) = C cos(! t + ) + C cos(! t + ) + K sin!t (t) = C cos(! t + ) C cos(! t + ) + K sin!t m + M M 8. M = M m + M k V = k! k = m k m m m+m 8. (a) L = m P N i= _ mg P N i i= i k P N i= (b) m i + mg i + k i k i+ + i = Capte 9 (c)! = g + 4 k ka m sin b 4 i i + i+ i i 9. (a) x = 75 m, t = 5 ns. (b) x = 74 m, t = 7 9 s. 9.5 (a) x = 7 8 m, t = 87 sec. 8

9 (b) x = m, t = 44 s. p c c cm. 9.5 (a) 8c (b) 55s 9.7 (b)f = 5 (c) 964 (d) Te same amount of igt is concentated into a smae cone. 9.9 v = p c = 77c T = 44m c 9. T = 5 9 J, p = 499 kg m/s. 9. (a) p f = m c (b)v f = 55c (c) cassicay, v f = 4c 9.5 g. 9.7 Factiona cange in aea = 6 9

c) (6) Assuming the tires do not skid, what coefficient of static friction between tires and pavement is needed?

c) (6) Assuming the tires do not skid, what coefficient of static friction between tires and pavement is needed? Geneal Physics I Exam 2 - Chs. 4,5,6 - Foces, Cicula Motion, Enegy Oct. 10, 2012 Name Rec. Inst. Rec. Time Fo full cedit, make you wok clea to the gade. Show fomulas used, essential steps, and esults with