SOLUTIONS TO CONCEPTS CHAPTER 12
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1 SOLUTONS TO CONCEPTS CHPTE. Given, 0c. t t 0, 5 c. T 6 sec. So, w sec T 6 t, t 0, 5 c. So, 5 0 sin (w 0 + ) 0 sin Sin / 6 [y sin wt] Equation of displaceent (0c) sin (ii) t t 4 second 8 0 sin 4 0 sin sin 0 sin - 0 sin -0 cceleation a w ( 0) c/sec. 9. Given that, at a paticula instant, X c 0.0 V /sec 0 sec We now that a T a seconds ain, aplitude is iven by v v ( ) 500 ( ) c.. 0c ecause, K.E. P.E. So (/) ( y ) (/) y y y y y 4. v a 0 c/sec () a 50 c/sec () y 0 5 c fo the ean position..
2 Chapte c sec ain, to find out the positions whee the speed is 8/sec, v ( y ) 64 5 ( 4 y ) 4 y 64 y.44 y. 44 y. c fo ean position (.0c)sin [(00s ) t + (/6)] 0. a) plitude c. 00 sec T sec 0.06 sec. We now that T T 4 4 T 0 5 dyne/c 00 N/. [because T 00 sec ] b) t t 0 c sin (/) c. fo the ean position. 6 We now that sin (t + ) v cos (t + ) 00 cos (0 + /6) sec.7/s c) a /s 6. 5 sin (0t + /) a) a. displaceent fo the ean position plitude of the paticle. t the etee position, the velocity becoes 0. 5 plitude. 5 5 sin (0t + /) sin (0t + /) sin (/) 0t + / / t /0 sec., So at /0 sec it fist coes to est. b) a [5 sin (0t + /)] Fo a 0, 5 sin (0t + /) 0 sin (0t + /) sin () 0 t / / t /0 sec. c) v cos (t +/) 0 5 cos (0t + /) when, v is aiu i.e. cos (0t + /) cos 0t / / t /0 sec. 7. a).0 cos (50t + tan 0.75).0 cos (50t ) d v 00 sin (50t ) dt sin (50t ) 0 s the paticle coes to est fo the st tie 50t t.6 0 sec..
3 Chapte dv b) cceleation a cos (50t ) dt Fo aiu acceleation cos (50t ) cos (a) (so a is a) t.6 0 sec. c) When the paticle coes to est fo second tie, 50t t.6 0 s. 8. y, y (fo the two iven position) Now, y sin t t sin t sin t t t t t 6 ain, y sin t t sin t sin t t / t t t 4 So, t t N/ t t 4 6 t T sec [Tie peiod of pendulu of a cloc sec] So, Tie peiod of siple pendulu Tie peiod of spin is T p T s [Fequency is sae] F. (ecause, estoin foce weiht F ) (poved). 0. T 0.4 sec 0.5. Total foce eeted on the bloc weiht of the bloc + spin foce. 0.5 T N/ 0.5 Foce eeted by the spin on the bloc is F N aiu foce F + weiht N.. T 4 sec. T 4 K K.
4 Chapte 4 5 N/ 4 ut, we now that F 4 50 Potential Eney (/) (/) J. 5c 0.5 E 5J f 5 So, T /5sec. Now P.E. (/) (/) 5 (/) (0.5) 5 60 N/. ain, T a) Fo the fee body diaa, + 0 () esultant foce 0.6. [ /( ) fo spin ass syste] b) - Fo to be sallest, should be a. i.e. is aiu. The paticle should be at the hih point. c) We have The tow blocs ay oscillates toethe in such a way that is eate than 0. t liitin condition, 0, X ( ) So, the aiu aplitude is 5. a) t the equilibiu condition, ( + ) sin ( )sin b) ( + ) sin (Given) ( ) a ( + ) K F when the syste is eleased, it will stat to ae SH whee a a When the blocs lose contact, P 0 So sin ( )sin P So the blocs will lose contact with each othe when the spins attain its natual lenth..4
5 c) Let the coon speed attained by both the blocs be v. / ( + ) v 0 / ( + ) ( + ) sin ( + ) [ + total copession] (/) ( + ) v [(/) (/) ( + ) sin ( + ) sin ( + ) (/) ( + ) v (/) ( + ) sin (/) ( + ) sin v ( sin. ) 6. Given, 00 N/, and F 0 N a) n the equilibiu position, copession F/ 0/ c b) The blow ipats a speed of /s to the bloc towads left. P.E. + K.E. / + / v (/) 00 (0.) + (/) J Chapte c) Tie peiod sec 00 5 d) Let the aplitude be which eans the distance between the ean position and the etee position. So, in the etee position, copession of the spin is ( + ). Since, in SH, the total eney eains constant. (/) ( + ) (/) + (/) v + F [because (/) + (/) v.5] So, 50( + 0.) c e) Potential Eney at the left etee is iven by, P.E. (/) ( +) (/) 00 (0. +0.) J f) Potential Eney at the iht etee is iven by, P.E. (/) ( +) F() [ distance between two etees] 4.5 0(0.4) 0.5J The diffeent values in (b) (e) and (f) do not violate law of consevation of eney as the wo is done by the etenal foce 0N. 7. a) Equivalent spin constant + (paallel) T b) Let us, displace the bloc towads left thouh displaceent esultant foce F F + F ( + ) ( ) cceleation (F/) Tie peiod T displaceent cceleation ( ) paallel - (a) F The equivalent spin constant + c) n seies conn equivalent spin constant be. So, + T ( ).5
6 Chapte 8. a) We have F F cceleation F Tie peiod T displaceent cceleation F / F / plitude a displaceent F/ b) The eney stoed in the spin when the bloc passes thouh the equilibiu position (/) (/) (F/) (/) (F / ) (/) (F /) c) t the ean position, P.E. is 0. K.E. is (/) (/) (F /) 9. Suppose the paticle is pushed slihtly aainst the spin C thouh displaceent. K Total esultant foce on the paticle is due to spin C and Total esultant foce + cceleation +. due to spin and. displaceent Tie peiod T cceleation [Cause:- When the body pushed aainst C the spin C, ties to pull the bloc towads XL. t that oent the spin and ties to pull the bloc with foce and espectively towads y and z espectively. So the total foce on the bloc is due to the spin foce C as well as the coponent of two spin foce and.] 0. n this case, if the paticle is pushed aainst C a by distance. Total esultant foce actin on an is iven by, F + z 90 y [ecause net foce & a F a Tie peiod T. K and K ae in seies. cos0 0 C 0 Let equivalent spin constant be K 4 K K K 4 K 4 K K KK Now K 4 and K ae in paallel. KK K K So equivalent spin constant + 4 KK K K + F T ( ).6
7 b) fequency T ( ) F F( c) plitude ).,, ae in seies, Tie peiod T Now, Foce weiht. t spin, Siilaly PE (/) Siilaly PE ( ) and and PE.7. When only is hanin, let the etension in the spin be l So T l. When a foce F is applied, let the futhe etension be T ( +l) Divin foce T T ( + l) l cceleation K T displaceent cceleation Chapte 4. Let us solve the poble by eney ethod. nitial etension of the spi in the ean position, Duin oscillation, at any position below the equilibiu position, let the velocity of be v and anula velocity of the pulley be. f is the adius of the pulley, then v. t any instant, Total Eney constant (fo SH) (/) v + (/) + (/) [( +) - ] Cosntant (/) v + (/) + (/) - Cosntant (/) v + (/) (v / ) + (/) Constant ( /) Tain deivative of both sides eith espect to t, dv dv dv v v 0 dt dt dt d d a ( and a ) dt dt a T T
8 Chapte 5. The cente of ass of the syste should not chane duin the otion. So, if the bloc on the left oves towads iht a distance, the bloc on the iht oves towads left a distance. So, total copession of the spin is. y eney ethod, () + v + v C v + C. Tain deivative of both sides with espect to t. dv d v + 0 dt dt a + 0 [because v d/dt and a dv/dt] a Tie peiod T 6. Hee we have to conside oscillation of cente of ass Divin foce F sin cceleation a F sin. Fo sall anle, sin. a L [whee and L ae constant] a, So the otion is siple Haonic Tie peiod T Displaceent cceleation L L 7. plitude 0. Total ass + 4 (when both the blocs ae ovin toethe) T sec. 5 Fequency Hz. 00N/ ain at the ean position, let bloc has velocity v. KE. (/) v (/) whee plitude 0.. (/) ( v ) (/) 00 (0.) v /sec () fte the bloc is ently placed on the, then let, + 4 bloc and the spin be one syste. Fo this ass spin syste, thee is so etenal foce. (when oscillation taes place). The oentu should be conseved. Let, 4 bloc has velocity v. nitial oentu Final oentu v 4 v v /4 /s (s v /s fo equation ()) Now the two blocs have velocity /4 /s at its ean poison. KE ass (/) v (/) 4 (/4) (/) (/4). When the blocs ae oin to the etee position, thee will be only potential eney. PE (/) (/) (/4) whee new aplitude. / c. 400 So plitude 5c. 8. When the bloc oves with velocity V and collides with the bloc, it tansfes all eney to the bloc. (ecause it is a elastic collision). The bloc will ove a distance aainst the spin, aain the bloc will etun to the oiinal point and copletes half of the oscillation..8
9 Chapte So, the tie peiod of is The bloc collides with the bloc and coes to est at that point. The bloc aain oves a futhe distance L to etun to its oiinal position. Tie taen by the bloc to ove fo N and N is L L L V V V v L L So tie peiod of the peiodic otion is V 9. Let the tie taen to tavel and C be t and t espectively 0. Fo pat, a sin 45. s sin45 Let, v velocity at v u a s v 0. sin 45 sin45 v t /s v u a 0 0. sec 0 0c C ain fo pat C, a sin 60, u, v 0 0 (.44) t 0.65sec. (.7) 0 So, tie peiod (t + t ) ( ) 0.7sec 0. Let the aplitude of oscillation of and be and espectively. a) Fo law of consevation of oentu, () [because only intenal foces ae pesent] ain, (/) 0 (/) ( + ) 0 + () [loc and ass oscillates in opposite diection. ut stetched pat] Fo equation () and () So, espectively. b) t any position, let the velocities be v and v espectively. Hee, v velocity of with espect to. y eney ethod Total Eney Constant (/) v + (/) (v v ) + (/) ( + ) Constant (i) [v v bsolute velocity of ass as seen fo the oad.] ain, fo law of consevation of oentu,.9
10 Chapte...() v (v v ) (v v ) v () Puttin the above values in equation (), we et v + v + constant v + Constant. v + constant Tain deivative of both sides, dv v d + e 0 dt dt d a + 0 [because, v ] dt a ( ) ( ) So, Tie peiod, T ( ). Let be the displaceent of the plan towads left. Now the cente of avity is also displaced thouh n displaced position +. Tain oent about G, we et (l/ ) (l/ + ) ( )(l/ + ) ()\ So, (l/ ) ( )(l/ + ) + + (+ ) v y ( ) l ( ) () ( ) Now F ( ) Siilaly F Since, F > F. F F a a Tie peiod.0
11 Chapte. T sec. T 0 0 l c ( 0). Fo the equation, sin [ sec t] sec (copain with the equation of SH) T T sec. We now that T l. Lenth of the pendulu is. 4. The pendulu of the cloc has tie peiod.04sec Now, No. o oscillation in day 400 ut, in each oscillation it is slowe by (.04.00) 0.04sec. So, in one day it is slowe by, 400 (0.04) sec 8.8 in So, the cloc uns 8.8 inutes slowe in one day. 5. Fo the pendulu, T T Given that, T sec, 9.8/s T Now, T T (9.8) 9.795/s L 5. a) T 0. 5 (0.7) n (0.7)sec, the body copletes oscillation, n second, the body will coplete f (0.7) ties 4 b) When it is taen to the oon T oscillation (0.7) whee cceleation in the oon f T 5 (0.577) ties..
12 Chapte 7. The tension in the pendulu is aiu at the ean position and iniu on the etee position. Hee (/) v 0 l( cos ) v l( cos) Now, T a + ( cos ) [ T +(v /l)] T in T in ain, T in cos. l ccodin to question, T a T in L + cos cos v 4 cos cos /4 cos (/4) 8. Given that, adius. Let N noal eaction. Divin foce F sin. cceleation a sin s, sin is vey sall, sin cceleation a N Let be the displaceent fo the ean position of the body, / a (/) (a/) (/) cos So the body aes S.H.. sin T Displaceent cceleation / 9. Let the anula velocity of the syste about the point os suspension at any tie be So, v c ( ) ain v c [whee, otational velocity of the sphee] v c () y Eney ethod, Total eney in SH is constant. So, ( )( cos) + (/) v c +(/) constant ( ) ( cos) +(/) ( ) +(/) constant w ( ) cos) + ( ) constant 5 d 7 Tain deivative, ( ) sin ) d dt 0 dt sin 0 7 ( ) ( ) sin 5 7 ( ) 5sin 5 7( ) 7( ) 5 constant 7( ) So the otion is S.H.. ain 5 T 7( ) 40. Lenth of the pendulu 40c 0.4. Let acceleation due to avity be at the depth of ( ) d (-d/) /s ( )cos.
13 Chapte Tie peiod T sec. 4. Let be the total ass of the eath. t any position, 4 4 So foce on the paticle is iven by, G G F X () So, acceleation of the ass at that position is iven by, G a a w G G P Q So, T Tie peiod of oscillation. a) Now, usin velocity displaceent equation. V ( ) [Whee, aplitude] Given when, y, v, ( ) [because ] [Now, the phase of the paticle at the point P is eate than / but less than and at Q is eate than but less than /. Let the ties taen by the paticle to each the positions P and Q be t & t espectively, then usin displaceent tie equation] y sin t We have, sin t t /4 & sin t t 5/4 So, (t t ) / t t ( / ) Tie taen by the paticle to tavel fo P to Q is t t ( / ) b) When the body is dopped fo a heiht, then applyin consevation of eney, chane in P.E. ain in K.E. G G v v Since, the velocity is sae at P, as in pat (a) the body will tae sae tie to tavel PQ. c) When the body is pojected vetically upwad fo P with a velocity, its velocity will be Zeo at the hihest point. The velocity of the body, when eaches P, aain will be v tie ( / ) to tavel PQ. sec., hence, the body will tae sae.
14 Chapte 4. 4/. 4/ a) F Gavitational foce eeted by the eath on the paticle of ass is, C / F G G G G 4 b) F y F cos F F sin G G G G F N c) F G [since Noal foce eeted by the wall N F ] G d) esultant foce Divin foce G e) cceleation ass So, a (The body aes SH) G F a w G G w T 4. Hee divin foce F ( + a 0 ) sin () F ( a cceleation a ( + 0) a0 ) sin (ecause when is sall sin /l) ( a0) a. acceleation is popotional to displaceent. So, the otion is SH. Now ( a0 ) G L (+a 0)sin a 0 T a 0 b) When the elevato is oin downwads with acceleation a 0 Divin foce F ( a 0 ) sin. ( a0 ) cceleation ( a 0 ) sin T a 0 c) When ovin with unifo velocity a 0 0. Fo, the siple pendulu, divin foce a a L a 0 a 0 (+a 0)sin T displaceent acceleation.4
15 Chapte 44. Let the elevato be ovin upwad acceleatin a 0 Hee divin foce F ( + a 0 ) sin cceleation ( + a 0 ) sin ( + a 0 ) (sin ) a 0 T a 0 Given that, T / sec, l ft and ft/sec a0 L a 0 a a + a 6 a 6 4 ft/sec 45. When the ca ovin with unifo velocity T 4 () When the ca aes acceleated otion, let the acceleation be a 0 T a 0.99 Now T T 4.99 a 0 a0 / 4 Solvin fo a 0 we can et a 0 /0 s 46. Fo the feebody diaa, l v / T v () 4 v a, whee a acceleation The tie peiod of sall accellations is iven by, T / 4 v 47. a) l c 0.0. T second v / 4 b) When the lady sets on the ey-o-ound the ea ins also epeience centepetal acceleation T v / v / a v 4 8 /s esultant cceleation a /s Tie peiod T second..5
16 48. a).. about the pt C.G. + h + H + (0.) T sec. b) oent of in isetia about C.G. + + Tie peiod c) ZZ (cone) a a n the C, l + l a l a a (l dis. between C.G. and pt. of suspension) Chapte O 0c 0c C.G l C.G l T a a a d) h /, l / Dist. etween C.G and suspension point... about, C.G. + h T 49. Let suspension of point. Cente of Gavity. l l/, h l/ oent of inetia about is C.G. + h T 4 c 4 l 8a n Let, the tie peiod T is equal to the tie peiod of siple pendulu of lenth. T. So, Lenth of the siple pendulu 50. Suppose that the point is distance fo C.G. Let ass of the disc., adius Hee l.. about C.G. + /+ ( / + ) T ().6
17 Chapte dt Fo T is iniu 0 d d d T d d So puttin the value of equation () T 5. ccodin to Eney equation, l ( cos ) + (/) const. (0.) ( cos) + (/) C. ain, / (0.) + (0.) () Whee oent of netia about the pt of suspension Fo equation Diffeentin and puttin the value of and is d dt (0.)( cos ) 0.08 d (C) dt.8c c d (0.) sin + dt 0.08 d dt 0.08 sin [because, 0/s ] So T 0.89sec. Fo siple pendulu T sec % oe t is about 0.% lae than the calculated value. 5. (Fo a copound pendulu) a) T The of the cicula wie about the point of suspension is iven by + is oent of inetia about..7
18 0.5 50c. (ns) Chapte b) (/) 0 ( cos) (/) ( cos ) / ( cos ) 0. ad/sec [puttin the values of and ] v c/sec. c) cceleation at the end position will be centipetal. a n () (0.) 00. c/s The diection of a n is towads the point of suspension. d) t the etee position the centepetal acceleation will be zeo. ut, the paticle will still have acceleation due to the SH. ecause, T sec. nula fequency T (.4) So, anula acceleation at the etee position, [ adious] So, tanential acceleation () c/s of the cente of the disc. / T T 4 K K KT [whee K Tosional constant] K T Tosional constant T 54. The. of the two ball syste (L/) L / t any position duin the oscillation, [fi-] Toque So, wo done duin the displaceent 0 to 0, W d 0 / L Fi- 0 y wo eney ethod, (/) 0 Wo done 0 / 0 0 L Now, fo the feebody diaa of the od, T ( L) () L 0 L 4 0 L.8
19 Chapte 55. The paticle is subjected to two SHs of sae tie peiod in the sae diection/ Given, c, 4c and phase diffeence. esultant aplitude a) When 0, ( b) When 60 ( c) When ( cos 4 4cos0 7 c 4 4cos c 4 4cos90 5 c 56. Thee SHs of equal aplitudes and equal tie peiods in the sae diction cobine. The vectos epesentin the thee SHs ae shown it the fiue. Usin vecto ethod, esultant aplitude Vecto su of the thee vectos + cos 60 + cso 60 + / + / So the aplitude of the esultant otion is. 57. sin 00 t w sin (0t + /) So, esultant displaceent is iven by, + [sin (00t) + sin (0t + /)] a) t t 0.05s, [sin ( ) + sin ( /)] [sin 5+ sin (/ + /)] [( 0.707) + ( 0.5)].4c. b) t t 0.05s. [sin ( ) + sin ( /)] [sin 5+ sin ( + /)] [+( )] 0.7 c. 58. The paticle is subjected to two siple haonic otions epesented by, 0 sin wt s s 0 sin wt and, anle between two otions 45 esultant otion will be iven by, ( s scos 45) 0 0 { sin wt s sin wt s sin wt(/ )} [ 0 +s 0 0 s 0 ] / sin wt esultant aplitude [ 0 +s 0 0 s 0 ] / 0 0 Y Y Y.9
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