Solution Mechanics Mj

8/11/2019 Solution Mechanics Mj

1/18

1 . Range = r = (u^2 * sin2)/g = 20 2 . Now, as the system is not underany horionta! "or#e , dis$!a#ement o" %& o" $!an' and a!! = 0 m*r m*d = 0 ( d= dis$!a#ement o" !o#') d = r (dis$!a#ed a#'ward), there"ore,min. +ength o" $!an'= r r ( sothat a!! wi!! "a!!

on edge ) = 0 2 -nswer (1)

grade 2

2 . #onsider the "d,

as the rod is mass!ess,hen#e torue a!an#e aout a gies mg*!#os = N* ! sin ,hen#e N = mg,(moment o" nertia = 0) net horionta! "or#e on rod =0,hen#e "or#e on u$$er !o#' y rod = mg(right) as rod is mass!ess,net erti#a! "or#e =0, #onsider "d o" u$$er !o#' , R = 2mg , " =R and ma = N " or ma=mg2mg a= g2g

,#onsidering a=0(i.e whenthe !o#'s ust egin to s!ide,their a##e!eration wi!! e same) , = 3 -nswer = (2) 4rade 5

6 . +et the angu!ar e!o#ity aout %& = w(#!o#'wise) then u = u/ w!/2 ( measuring distan#e u$owards ) u/2 = u/ w!/2 w= 6u/2!,!et the $oint e at distan#e 7,then u/ w7 = 0,hen#e 7= +/8 -nswer 5 2 4rade 5 1

. t is oious that the !o#' has ero "ri#tion at = 0 ( horionta! and) at = 90( erti#a!).:etween these the gra$h "o!!ows "ig 2-nswer 5 (2)4rade1;.


2/18

!et uni"orm a##e!eration "or !o#' i= ai(i=1,2),thena1=0.1g and a2= (0.10.0;) g =0.0;g then na! e!o#tity o" oth aresame,!et this ha$$en a"ter time t then 10.1g t = 0.0;gt,hen#e 1= 0.1;gt=1.;t.>en#e t= 2/6s -nswer 5 2 4rade 6

? @ime o" "!ight = 2u*sin / g...= 2 s.t is oious "rom the time intera!

etween shots and time o" Aight that the #orre#t g is 1 -nswer 5 1 4rade ( 1)

B.%onsider "d o" !o#'s mg@N1 = 0C N1= mg@ DN1 = ma ...and "or 6m !o#', EDN1 = 6ma =FE = DN1 = D(mg@)(dout) G(

9 . %onsider the diagram as shown then #onsidering e!o#ities a!ong the!ine oining the two odies,


3/18

H17 = Ho/ 2 * (11) = 0 and H2 7 = H17 = Ho/ 2( :y using #oeI#ient o" restitution = 1) :y !aw o" #onseration a!ong J dir. Ho/ 2 = H1 y H2 y +aw o" #onseration o" energy giesHo^2 = 0 H1y^2 Ho^2/2 H2y^2 Now,this gies either H1y = 0 or H2y = 0,ut "rom the $ro!em,H2y =0,hen#e H1y = H0/ 2 >en#e the e!o#ities are "or ueen =Ho/ 2 i

"or styri'er = Ho/ 2

shi"ting to origina! system ,we get the reuired #roos $rodu#t-nswer 5 2

4rade ;

10 %onsider the "d

then #onsidering the 2 !o#'s together as a system,we hae 2mgsin= (D1 D2) mg#os( "or stati# eui,a!an#ing "or#es a!ong thein#!ine,net a##e!eration eing 0hen#e tan =(D1 D2) /2-nswer 5 6


4/18

4rade 1

11 t is #!ear that the e!o#ities wi!! "a!! !inera!y "or u$$er !o#' and wi!!in#rease !inear!y at the same rate,as the assuming uni"orma##e!eration,...and na!!y a #ommon e!o#ity is rea#hed-nswer 2

4rade 2

12 @he on!y "or#e whi#h $roides #entri$eta! a##e!eration has to e thetension in the rod,#!ear!y @ = mw^2 (!/2)-nswer 5 none o" these,wrong ans is gien in oo' @ #annot e ero4rade 5 116 K!ongated !ength o" s$ring = 6r.e!ongation = ( 6 1) r.

Now on !eaing,the e!o#ity o" $oint the $arti#!e = 0 hen#e net #entri$eta! "or#e = 0 = '( 6 1 ) r #os 60 N mg #os 60 ,onsustituting a!ues we get the ans-nswer

grade 2

1 :y wor' Knergy th. %hange in L.K. = wor' done = area under the gra$h =0.; *20 * ; = ;0M

hen#e na! L.K. = ;0 0.; * 2;* = ;0 ;0 = 100 M

-nswer grade 1

1; reuired re!atie e!o#ity = Hr 5 Hm = 10i10'10 -nswer 1 grade 018 as the re!atie motion etween the !o#'s is 0,hen#e the s$rings wi!! eunstret#hed .

-nswer 5 grade 1

1? !et the e!o#ity at : e ,then ^2 = 2gr#o80


5/18

now,net #entri$eta! "or#e = N mg#os = m^2/r,this gies N on $utting the a!ue .answer 6

grade 2

1B $eed de#!ines "rom ma7imum to minimum and then again in#reaseswhi!e the ody returns ,!inear!y with time. >ad it eing e!o#ity,it wo$u!dhae turned negatie, as the ody returs( ie g$a$h wou!d hae een in y"or 1 ha!" and y in the other ha!h. -##e!eration is #onstant (=g) hen#e astraight !ine $ara!!e! to time a7is wou!d hae een otained.Oistan#e #oered is oious!y 0-nswer 1grade 1

19 "rom the "d

Bg @ = Ba2 @ ;g(1/ 2) = ; a1

#onstraint euation is a1( 1/ 2) a2 = 0 onso!ing,we #an get a1 and a2 -nswer 14rade 6


6/18

20 +et the time ta'en y : e t then t = 2h/g and distan#e #oeredhorionta!!y = ; 2h/g= range o" $arti#!e - = 100^2 sin 120/g ( using range= u^2 sin 2 /g) so!e "or hanswer 2

grade 1

21 !et the e!o#ity o" man a!ong 7 and y dir. :e H7 and Hy res$e#tie!y then !et the reuired time e t,then in time t... 2 = Hyt and distan#e moed y tru#' = H7t = Bt Hy= 2/t and H7 = B/t so net s$eed = H = H7^2 Hy^2) "or minimum s$eed .so!e dH/dt =0

-nswer 1 4rade 6

22 the $arti#!e trrown u$wars wi!! !oose Lineti# energy and gain $otentia!

energy,reerse ha$$ens "$r other $arti!#e.hen#e e!o#ities #annot e same.%onsider a "rame o" re"eren#e $ara!!e! and $er$ to in#!ine ,then,dis$!a#ement a!ong the in#!ine "or !ower $arti#!e = o ,a##e!eration a!ong thein#!ine= gsin we get 0 = u#ostgsint^2/2 hen#e t =2u#os/gsin.imi!ar ananysis ho!ds "or u$$er $arti#!e and we get samea!ue o" time-nswer 24rade

26.in#e net dis$!a#ement = area under the gra$h = 0 hen#e wor' done =0 -nswer 6

grade 0

2..H7 = d7/dt = 2t.>en#e integra! o" this wi!! tt^2,whi#h is an inerted$arao!a

2;

using s = ut 3 at^2,"or oth $arti#!es ,the dis$!a#ement o" oth the$arti#!es shou!d e the same a"ter time t = 2s . Pn sustituting,and#o!!e#ting #oeI#ients o" i and se$arate!y,and so!ing,we get a=B-nswer 4rade 128 -s the initia! e!o#ities a!ong 7 dite#tion is same "or oth the $arti#!es,and there is no "or#e to #ause a##e!eration , hen#e the e!o#ities a!ong 7dire#tion wi!! remain un#hanged ,"urther,the distan#e #oered a!ong 7 dir. =2 * radius ,hen#e time ta'en wi!! e the same

-nswer 5 2$ossi!y,answer gien is wrong 4rade 5 1

2? . 7= my^2( distan#e "rom 7 a7is = y #oordinate),so dis$!a#ement a!ong ydir. = (6 2 (1)) / 2 = 6 = y,now sustitute in aoe e.-nswer


7/18

4rade 2

2B t = (2h/g) ,as the $arti#!es do not hae an initia! e!o#ity a!ong y dir.( using h = 1/2gt^2) Now sustitute in the aoe e7$ression and na!!y #a!#u!ate t2mtm.29 @he #onstraint euation "or motion o" !o#'s : and - is 7a7a 777 =0( as the !ength o" tota! !ength o" the string#onne#ting - and : is #onstant) their e!o#ities are a!oso re!ated thus.>en#e Ha= 6/2 H = 0.9m/s u$wards-nswer 5 14rade

60 %onsider the diagram,

!et H$= $ersonQs e!o#ity

a,,#= He!o#ity o" wind wrt $erson in the three #ases res$e#tie!y Hw/g= e!o#ity o" wind w.r.t ground ,whi#h wi!! e the same in a!! #ase , asthe ground is stationary medium now Hw/g = H$i a = (2H$/ 2) i / 2 = (6H$ # #os )i # sin,where determines the reuired dire#tion ,now magnitude o" Hw/g a!soremains same in a!! #ases,hen#e, we #an nd a re!ation etween a, and #,now so!e "or

-nswer 64rade

61 euation o" motion is y = usin t 1/2gt^2 7= (u#os ( or ))t,where is used when $!at"orm moes "ordward,sin#e here,we need to 'now thehorionta! #om$onent o" e!o#ity wrt ground


8/18

hen#e range = (u#os ( or ) )* (2usin )/g,time o" "!ight wi!! e thesame "or oth #ases e!iminate u "rom R1/R2= (u#os )/(u#os ),and sustitute in any o" theaoe e,to get -nswer

grade 6

62 :eta=: and -!$ha = - then we must hae tan : = dy/d7= (dy/dt)/(d7/dt)= Hy/H7= (usin-gt)/u#os- ,where u = initia! e!o#ity,so!e "or u-nswer 6 grade 666 @he string a"ter "o!!oing a #ir#u!ar $ath o" radius R wi!! reo!e arountthe !ower $oint in a radius o" R6R/?= R/?

"urther the tota! energy wi!! remain same hen#e mgR= energy at theu$$ermost $oint= eergy at the $oint where the string ma'es an ang!e with the horionta! .now,net #entri$eta! "or#e= m^2/(R/?) = @ mgsinand mgR= 1/2m^2 mg(R/?)(1 sin ) so!e "or the #ase when the strings!a'ens i.e @= 0.-nswer 2grade

6 dout

6; !et a!$ha= -,+et angu!ar e!o#ity = w ,and #m= u then = u wr and"or ro!!ing without s!i$$ing, =wR and He!o#ity o" $oint o" #onta#t o" oinwith the $!an' $er$ to the $!an' = *u sin - = wRsin- = H/(Rr) R sin-=7(say) ,hen#e re angu!ar e!o#ity = 7/(R#ot(-/2)) ( R#ot (-/2)= distan#e o"7ed $oint "rom the $oint o" #onta#t o" oin with the $!an')-nswer 2!ee! ;

68 dout

6?#onsider the $roe#ti!e motion y = 7tan 3 g (g7/use#)^2 then at y=asin80,this has two so!utions 7=(Ra)/2 and (Ra)/2 ( ie $oints $oints K and: res$e#tie!y. imi!ar!y "or $oints % and O we otain simi!ar euation. Nowe!iminate etween the e . sing the re!ation "or sum and $rodu#t o" roots


9/18

-nswwer 1grade ;

6B we hae angu!ar a##e!eration = dw/dt= wdw/d ( where = ang!e o"rotation) = :sin ,so!e this di"". e.-nswer 6grade 1

69 +et -: ma'e an ang!e with the horionta!,then tan = ( h/7)

and H-= wr (no s!i$$age etween the string and $u!!eys),a!ong S "urther the #onstraint is that the e!o#ity o" the $oint o" #onta#t o" -:with the $ointed end o" the wa!!($oint T) in a dir. Uer$ to rod . = 0 and -T =#!et the re angu!ar e!o#ity aout - = thenHT $er$. = H- #os # = 0 ,"ro these e,we otain the desired ans

-nswer 2grade ;

0we hae @ &g = &a,a= net a##e!eration then h= 1/2at^2 hen#e ,time wi!!e minimum when a##e!e. s ma7imum hen#eanswer 1grade 1

1 #onsider the diagram


10/18

then y de" o" e e = (H2sin()H1 sin )/(u1#os() u2 #os ) = ( H2sin H1 sin ) /( u1#os u2 s#os) y symmetry , u1= u2 and H1 = H2 ( :y symmetry) hen#ee= tan

ans 2

grade 2

2 time=t = usin/g and dis$!a#ement= d=u^2sin2/g i (usin)^2/2g ,age!o#ity = d/t,modu!us o" this is the answer

answer6

grade 1

6 the dis$!a#ement o" $arti#!e 1 wrt 2 = (u#ost (u#os t d) ) i (1/2gt^2 5 (1/2gt^2d))

hen#e net distan#e^2= (modu!us o" dis$!a#ement)^2OC now use the#ondition dO/dt=0

to get the re timeanswer 6

grade

!et the uni"orm a##e! . = a


11/18

then #onsider the e o" motion is @ (m(+!)g)/+ &(g)= (m(+!)/+ &) aso!e"or a

answer 1

grade 2

; sing energy #onseration 1/2m^2= 3 () (w^2) where = moment o"

nertia! aout - = (m!^2)/6, where != 2a.hen#e #a!#u!ate wanswer

garde 2

8"or #om$!eting the #ir#!e,the string shou!d not s!a'en hen#e @ mg#os =m^2/R

!aw o" #onseration o" energy gies 1/2mu^2 mgR(11/ 2) = mgR( 1 R#os ) 3 m^2

use @= 0 as the re !imiting #ondition

-nswer 2

grade 6

? #onsider "d


12/18

Eor #onstant s$eed, E#os uN =0

N= mg E sin

so!e "or E and use wor' done = E#os * 8 ( dis$!a#ement)

answer 2

grade 2

B %onsidering u$$er Auid, 1= 2g(h/2)

now as the the u$$er Auid #an e re$!a#ed y a Auid o" density 2$ andHo!ume !!ing ha!" the height = (h/2) so that the tota! $ressure at the !ower$oint remains the same.n the new #ase, de$th o" se#ond ho!e e!ow watersur"a#e= h/2 h/2= h hen#e H2= 2gh,hen#e the ratio is 1/ 2

answer

grade 6

9 we hae 1/2mHe^2 4&m/r = 0( r = radius u" earth)

and mHo^2/(R) = 4&m/ ( R)

!et the e!o#ity o" stri'ing= H then

1/2mH^2 4&m/(Rr) = 1/2mHo^2 4mm/(R)

these e gie the a!ue o" H

answer

grade 6

;0 .#onsider "or#e = (d/d7) = (e7$ (7^2))*(27)C "or sma!! 7 the e7$onentia!term= 0

hen#e "or#e is $ro$otiona! to 7 ( with $ro$otiona!ity #onstant= 2) =F >&

hen#e #hoi#e

grade 1


13/18

;1 .

;2 sin#e the dis$!a#ement o" the $arti#!e is same and "or#e is #onseratie (inde$endent o" $ath) hen#e wor' done = 6*a *a= ?*a = same "or oththe $aths answer (1) rating 1

;6


14/18

"or Ho -nswer 2 Rating 280 !et the e7erted y the s#ooter on the man in etri#a! dir = ;00 in horionta! dir. @he "or#e aries,the time intera!s t1 and t2,the wor'"or#e wi!! e o" some magnitude (= or res$e#tie!y ) " hen#e net "or#ewi!! e

;00 "or t1 ;00 "or t2 ;00 "or t6

#!ear!y magnitude o" "or#e to time intera! t2 is !ess than the other #ases answer rating 6

81%onside the "or#e e,!et the heaier ody e o" mass & then

&g@ = &aC @ mg = ma Chen#e a = (&m)g/ ( &m ) and @ = &g 5 & ( &m ) g/( & m ) = 2&mg/( &m ) as & is ery !arge hen#e @ V 2&m/(&) g = 2mg Now on $u!!ey = 2@ = mg answer 6rating 2

82 !et initia! e!o#ity o" a!! 1 e u1, y de"n o" e H1 = (u1 ( 0eu1)) /2= (1e)u1/2 H2 = (u1 eu1) / 2= (1e)u1/2 now t = WR/1

!et the reuired time e t,then re!atie e!o#ity = H2 H1 = eu1 and re!atie distan#e = 2WR ( R = radius o" #ir#!e) then @ = 2WR/eu1 = (2/e)*WR/u1 = 2t/eanswer 1

rating 86 -s the system is in e initia!!y ,hen#e #omined mass o" man and !adder =& m then mass o" !adder = &m now !et the dis$!a#ement o" !adder e 7 ( wrt ground)dis$!a#ement o" man wrt ground = !7 dis$!a#ement o" !o#' wrt ground = 7dis$!a#ement o" %& = ( (&m )7 m (!7) &(7))/ (&m m &) = m!/2&

8 dout

8; m$u!se = #hange in momentum ,!et the mass o" ea#h ody = 1 unit ( asthey are identi#a!) then initia! e!o#ity o" - =U = - na! e!o#ity o" - = U 5 M = H- then initia! e!o#ity o" : = 0= : na! e!o#ity o" : = H = H:then H (UM ) = U 0 hen#e H = M

now e = ( H: 5 H-) /( - :) = ( M 5 U M )/( U0) = 2M/U 1 answer 2 rating 2


15/18

88 :y !aw o" #onseration o" momentum a!ong horionta! dir.,here !et a!$ha= -and H = e!o#ity o" motion o" gun&H = m(#os- H) (&m) 0 ( -s the s$eed o" she!! wrt earth = #os-H) hen#e so!e "oe H answer 1rating 28? +et mass o" !o#' = & then as a$$!ying !aw o" #onseration o" -ngu!ar momentum( M ) ust e"oreand a"ter the #o!!ision ,we getnitia! M = &(a/2)

Eina! M = w ( = moment o" inertia o" the !o#' aout an a7is $assingthrough U through one side =2/6*ma^2 and w = re angu!ar e!o#ity )so!e "or wanswer 1rating 6

8B #onsider the "o!!owing e mg@ = ma ( "or !o#') @ =ma( "or rod)

dout G(

89 the ma7imom s$eed o" the inse#t = w- ,where w =(L/&) .-s the mirror isin#!ined at an ang!e 80 , hen#e re ma7imum re!atie s$eed in the mirror =tan 80 * w- answer 6

rating 2

?0 assuming uni"orm e!o#ity gradient throughout the de$th o" the riered, we hae d(H)/d7 = (20)/1 =2 and "or#e = '-* (dH/d7) where ' = #oeI#ient o" is#osity, - = area o" #rossse#tion

answer 5 2rating 1

?1 #onsider the "rame o" re"eren#e o" the esse!,!et o!ume o" the !o#' = Hdenstity o" !o#' = d1, o" water = d2 then net downward "or#e = Hd1g Hd2g Hd1a ( where the 1stterm is dueto graity, se#ond due to "or#e o" ouyan#y and third is the net $seuo "or#e )

"rom this #a!#u!ate net a##e!e.(=7) o!e "or time using h = 1/27t^2

?2 !et the area o" #ross se#tion o" the tues e - then #onsider the horionta! se#tion u" the tue , mass o" this se#tion =$-+ !et the diX o" height e h ( right #o!umn !e"t #o!umn ) ,then the "or#e dueto this $ressure diX . = $-+a i.e $g(h2h1)- = $gh- =$-+a , hen#e h = +a/g

answer 2rating 2


16/18

?6 "or#e = (d/d7) = aosina7 V a^2o 7 whi#h is $ro$otiona! todis$!a#ement hen#e w = ao ,#!ear!y the 1ststatement is in#orre#t as the time o"os#i!!ation = 2W/w answer 2 rating 2

? "or 7F 0., "or#e is gien as (d/d7) = 2L7 ,dire#ted towarrds the origin hen#e w=2L time $eriod = 2W/w ,whi#h is inde$endent o" am$!itude

.?Bso!utionG "or#e o" ouyan#y and is#osity a#ts on a ody "a!!!ing "ree!yhtrough air

there"ore "or#es a#ting on that dro$ are 1."or#e o" graity 2."or#e o" ouyan#y

6."or#e o" is#osity

at steady dro$ has e!o#ity at whi#h 876.17r77Y=/676.17r^67g(density o" honeydensity o"air)

hen#e ansG(1)?9so!utionG

.B2so!utionGwor' done on ody =#hange in energy

we ha,

/676.17R^6=2?7/676.17r^6 =Fr=R/6Cenergy o" 2? u!es=2?7776.17r^2energy o" 1stu!e=776.17R^2

wor' done=776.17R^2(61)hen#e, ansG(1)

B so!utionGto ha node at 7=0C y=0C

y the su$er$osition $rin#i$!e new wae has euation a#os('7t) any o" the gien euation

and adding euation 6. we get eation 2asin'7sinwt whi#h gies y=0 at7=0C

hen#e ansG(6)

.B;so!utionGdo$$!er eXe#t

o Ge!o#ity o" oserer sG e!o#ity o" sour#e


17/18

Ho a!ong 7a7is=Ho/; Hs a!ong ya7is=6/;Hogt

EQ=E7(2 o /2s)sin#e oserer is moing towards the dete#tor with Ho/; and they ha sameerti#a! e!o#itythere"ore EQ wi!! a!ways e the same E7((2/;)/2)=?E/;hen#e ansG().B8so!utionGits a "ree end there"ore the disturan#e wi!! moe u$wardhen#e ansG().B?so!utionG"reuen#y =n/2+srt(@/Z)

here @=&gGtension in the stringnow , there is a #hange in tension !ets su$$ose mQ

then new "reuen#y=2n/2+srt(mQg/ Z)they are eua! there"ore mQ=&/hen#e ansG&/

.BB

so!utionG

in its 6rdharmoni# !enth o" string +=6!emda/2C

"rom the euation = 2 #m sin[(0.8 #m1)7\#os[;0076.1 s1)t\

'=276.1/!emda=0.8

=F!enmda=276.1/0.8=10.8?

+=6710.8?/2=1;.?#m

hen#e ansC()

B9

so!utionGa##. @o do$!er eXe#t EQ=Eo7(2/(2))

EQ=2E/6

a$$arent "reuen#y heard "rom the sour#e=2E/6

a##. @o do$!er eXe#t1EQ=Eo7(2/2))

1EQ=2E "reuen#y heard a"ter the reAe#tion "romwa!!

:eat "reuen#y re#orded 2E2E/6=E/6

hen#e ansG(6)

90


18/18

1.so!utionG eta=107!og(/.)

" eta1 and eta2 are the sound !ee!s at intensities 1 ]2

eta1eta2=107(!og(1/0)!og(2/0))

=107(!og(1/2)

=107.;

=2

hen#e ansG(2)

.91so!utionG!east #ount=main s#a!e diision/no. P" ernier s#a!e diisions=0.0;

=F0.0;=main s#a!e diision/200 =F@he distan#e etween #onse#utiethreads on the

s$herometer s#rew is=0.0;7200=1mmhen#e ansG(2)

.92so!utionGerror in radius o" s$here=2

(de!r/r)=2/100C

o!ume o" s$here=/6 76.17r^6

@hen $er#entage re!atie error in the

measurement o" o!ume= 67 (de!r/r)7100C =67(2)=8

Solution Mechanics Mj

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