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in.al6rrlltati*l.a*. P,T.O.
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|b) Find rhr- ifir'rrrr: nf
.ii:'r l !
,',r =1+ ., -,1i.-r 5 iJ
b-l' fi arixs-JorrJrrn urellrr:d. f 101
bt'J I
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Rzd Rr It-r3.:4 Rr- 3Rl
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R,4 Rr + R?-
&5 -*+ Rr-
I D-+o -\ -f1 o-\o
.3.\
-1I
o
ot
krt Rrx -lb'--+ fipx -tl,o
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39 oI 50
Rr+ f,r \ o oo
lqlr,
- rrlla
zlp
o-\1r"1
tl r- |
-tlt o JF*---r Fz- rK: o
o
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o \ lr a
4 :1 n1
1.{c}Thcvclocilyo[atrajnrr,hichstartsfromrcslisgivcninthefollorr.ingr;rblc.Thctimeisinminutesandvtlocityisinkrn/hour.
t n 4 tr B to t2 l4 r6 l8 20
l6 28.8 40 .r6.{ 5r.2 32.O 17.t) I a, o
gsffi atc a-PPioxirnatc lY tbe
'lsirnPson's 5 rule'
total dist:rncc run in 3O minutes by using composite
lrol
rt O {,&" +- 'o '*l*&J .,
S?,ne 4+e try)tdor'gt *t i o li in^'f,'J
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ta,'t\'
a) lr^' 9o ."f*'h4
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Sr.a'r oA) ."di".i" I) \$ 3f tl o +q t t ?.t b +€ + t-+ o+ 1- + I
lYh, 4:rrlrc
lo1o
l
1\,,lr
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)
'Yl X htin rlaa. Ix3 h*a,
krnhot,rf
? :8 EL}, \ ho.rt6ol{
!l'"l.*z*<Y
-{dl A spherc of radius a and mass M ro}ls dou-n a rough plane inclincd at an
angle a to the horizontal.Ifxthc
v'bc the distanci of thc poinr of c.,ntact of thc sphcre from a fixed point onplane, find the accclcration b1. using Hamilton's equations. tlSI
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fr stt"" '* ,e"rd ,t*llal la.r--tr AiJ'"**\te. frg:e *-
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Yjnc,h<- *0d
-'- T= J- Y,,Af + \.x tx nu*xz L5 * t*.t
Ato
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L= aO
T = _l, \^;,t'lo \= -rJ= - faol r,
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H.;i^tl!rq; ?h-;- - L=
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?7 L"rfr
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2 s )r-= )'t rc-'*=
-h*
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42 of 50
t
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8. (al The ends A and.B o[ a rod 2O
until stesd)' stati Prevails' The
and 60u re$Pcctive'l!" Find the
crn long, have thc temperature at 3O" and 80'
temPeratures of the enels are changed to 40'
tcmperaturc.di sritrurion irr thc rsd at timf, t.
2Ll18i
2 Ir D+gA
^| % J*do 2\-tvtl=
=) 4*.o *D*"
C1r* Cz-af a*o 0tt 3o
e# e= !o rta toAL
1o- 'o+ Lz*$9r'lo{f t-t-
f'-tH{l,
-
Npr Ldt U(e ' t) " X (r) 4 xu a Tt l
x LT
(04=o a X']=,o a , ,xlnll fl* n "7 ..
;{ {o) = \o 4 4o - q>': :'i',a,tu4; \n
X(rr) o 6o i 60"' roa'+ q--a 4t: I
Y(:r)- r t\ o
t'oo d f {Ll- covt }a^{* "
't"
), ^{,-e xlt, r d x(r)- h;e.{lr tr,it*
T(r)* e.'l* -* "'+ f'"t"a| T t&) r-+ao ^8 **'-fr o ,lrT",
fO )-{, xn=1x ' x(r)*c-roebtlq+c+-twr{*-
A,ti.ou-(eo,t\* 6o
rr(b, t) - \"3o
&*tr^1*i^"t- "
ut*,o)- *+7
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44 ol 50
.b( 2DS.z\
I tb) Solte tlrr: initinl \ia lrte problem
u' 2tu ul0l rth order classical Runge-
[15:intenal 10, 0 .1 l. Usc the fnul,hith h o. ? o11 tlrc
soltt t10n.1(u tta mcthod. colnpa rc with the exact
trt , - Lhf ui'') ' I Wo o'L
u+t{" r {,) zOY-t = kr)
,,a= -A'Oq
Kr.o t'tvso bt
k1- t+
uto.r) =
(
t(
Lt h- . Uot-)?
,l-t-tr) ? - Q'Ol5$ql L
=- - o' ol \3a 1*b$+ h"
I'+L, tl."lr)
k\f l-t"1;to.xcl r 3 ?2
ut") fe
tr.r-tt'.kt'
t+ ( lo'', 'lo'') a' -.o'
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!'+(.h-r*g, t''r' *+')
ot3) r \;- -o, lozrT{a -q. o33I?-Yf
1.1 ro
tf'L 1 tlo.u 4 ? -o,11$51+k ?
\r tct) qto .z) * I.lt :.tlrrt'Li) tI., l=
€
Sri*rrir = &-- 12t "lt 4 -! - bLt c-61,
c=t
rA
)c o*L ) u=1-I rtr-
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Page 27
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{5 ol 50
8. lc| Prort that liquid rnotio$ is possible rvhcn vclociry at (x, y, z| is given b1'
.- 3x" - rr 3.xr. 3xzu = =;l-,- = ?.$ = J+,
rsherc rr = .r, . r-) * z'itttcl ihc stt"o* lincs nrt
lhe lntersection of thc surfaccs, lxr + y, + zr)r = "(y'
il z'l', hi the planes
passing through Ox. Is rhis irotarionll? [r"]
\l: 3*L- trL V ? 3)t4*vJ.2*
Y{0qo\*fr'0
Y' .l"q2 r,t
t v'?-o ' a7q.O
f\A"hw; ta
eL ,2Y =b_tg
7tf,1,51"-l*
TE 7'7
Aq. =.7* Y C0p
Du,r*onil b"
,d5e, - l9't^1L
4 ---*Ys rI fiE\<dx
," J! -t tutd+
t l5'x -r9r (r.hittrl)= o [: '* i'"',--:*1' J ', |."r-*f*di' Y1 t n*\rl],t
j*n sg..a6 *, S* b0 +- V
drl+
A ?a s+-'N'Y-
9z*z-
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3r-(4.L* z"') dz
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