1. xvkv †evw©-2015 · 01/03/2017 · 1. xvkv †evw ©-2015 1 bs cÖ‡kœi dËi ... l cv‡Îi...
TRANSCRIPT
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
1. XvKv †evW©-2015
1 bs cÖ‡kœi DËi
K †h mg Í msN‡l©i †ÿ‡Î MwZ kw³ msiwÿZ _v‡K Zv‡K
w¯’wZ¯’vcK msNl© e‡j|
L fi n‡”Q e ‘i RoZvi cwigvc| e¯‘ †h a‡g©i Kvi‡Y †Kv‡bv
wbw`©ó A‡ÿi mv‡c‡ÿ Zvi †KŠwYK MwZi cwieZ©‡b evav †`q Zv‡K
Zvi N~Y©b RoZv ev RoZvi åvgK e‡j| A_©vr ˆiwLK MwZi †ÿ‡Î
fi †h f‚wgKv cvjb K‡i †KŠwYK MwZi †ÿ‡Î N~Y©b RoZv ev
RoZvi åvgK †m f‚wgKv cvjb K‡i| †Kv‡bv e ‘i fi mKj †ÿ‡Î
aªæe Aci c‡ÿ wbw`©ó A‡ÿi mv‡c‡ÿ †Kv‡bv e ‘i N~Y©b RoZv wbw`©ó
wKš‘ wfbœ wfbœ A‡ÿi mv‡c‡ÿ wfbœ wfbœ|
M
gvwS AB eivei †bŠKv Pvjbv K‡i AC eivei Icv‡i †cuŠQvj|
†mÖv‡Zi †eM, u = 4kmh–1
†bŠKvi †eM, v = 3kmh–1
Ges †mÖvZ I †bŠKvi †e‡Mi ga¨eZx© †KvY, = 90
DÏxc‡Ki wP‡Î,
ABC G, ACB = = 30
sin = ABAC
ev, AC = AB
sin =
1.5km
sin =
1.5km12
= 3km
AC eivei †bŠKvi AwZµvšÍ `~iZ¡ = 3km (Ans.)
N g‡b Kwi, †mÖvZ I evZv‡mi mw¤§wjZ †eM w.
tan30 = vsin90
w + vcos 90
ev, 1
3 =
vw
ev, w = v 3
w = 3 3 km/h
†bŠKvwU‡K AD eivei Pvjbv Ki‡j; g‡b Kwi, jwä †eM R Ges Zv
cv‡oi mv‡_ †KvY Drcbœ K‡i,
R cos = w + vcos 150
ev, R cos = 3 3 + 3 – 3
2
ev, Rcos = 3 3 – 3 3
2
Rcos = 3 3
2 .......... (i)
Rsin = wsin0 + vsin 150
ev, Rsin = 0 + 3 12 = 1.5..........(ii)
(i) I (ii) bs n‡Z,
tan = 1.5
3 3
2
= 1
3
= 30
AD eivei Pvjbv Ki‡j †bŠKvwU B we›`y‡Z †cuŠQ‡e bv|
2 bs cÖ‡kœi DËi
K ej cÖ‡qv‡M †Kv‡bv e ‘i ˆ`N©¨, AvKvi ev AvqZ‡bi cwieZ©b
NUv‡bv n‡j ej AcmviY Kiv gvÎB e¯‘wU c~e©ve ’vq wd‡i Avmvi
ag©‡K w¯’wZ¯’vcKZv e‡j|
L Lvov Dc‡i wbwÿß e ‘i †ÿ‡Î Abyf‚wgK w`‡K wb‡ÿcY †e‡Mi
Dcvsk k~b¨| ZvB wbwÿß e ‘i Abyf‚wgK ~iZ¡I k~b¨ nq|
M †`Iqv Av‡Q,
wb‡ÿcY †eM, v0 = 20 ms1
wb‡ÿcY †KvY, 0 = 45
AwfKl©R Z¡iY, g = 9.8 ms2
†ei Ki‡Z n‡e, me©vwaK D”PZv, H = ?
Avgiv Rvwb, H = v0
2sin20
2g = (20 ms1)2 (sin 45)2
2 9.8 ms2 = 10.2m (Ans.)
N g‡b Kwi,
ejwU 35m Abyf‚wgK ~iZ¡ AwZµg Kivi gyn~‡Z© f‚wg n‡Z
h D”PZvq _vK‡e| G ~iZ¡ AwZµ‡g t mgq jvM‡j,
x = v0cos0t t = x
v0cso0 =
35m
20 ms1 cos45 = 2.475 sec
h = v0sin0t 12
gt2 = 20 sin45 2.475 12
9.8 (2.475)2
= 4.986 m > 3m
30
d=1.5km
A
3km jwä †eM = R
B C 30
†mÖv‡Zi †eM, u= 4kmh–1
‡bŠK
vi †
eM,
v =
3km
h–
1
= 290
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
myZivs H wdìvi K¨vP wb‡Z mg_© n‡Zb bv|
3 bs cÖ‡kœi DËi
K †Kv‡bv K…wÎg DcMÖ‡ni AveZ©bKvj wbR A‡ÿi Pviw`‡K
N~Y©vqgvb c„w_exi AveZ©b Kv‡ji mgvb Ges AveZ©‡bi w`K c„w_exi
AveZ©‡bi w`‡K n‡j, c„w_exi mv‡c‡ÿ GwU w¯’i _vK‡e| G ai‡bi
DcMÖn‡K f‚-w ’i DcMÖn e‡j|
L GKRb †`Šowe` †`Š‡oi ïiæ‡Z mvg‡bi w`‡K Szu‡K _v‡Kb|
G‡Z gvwU‡Z cv w`‡q wZwb Zxh©Kfv‡e ej cÖ‡qvM K‡ib| d‡j
cÖwZwµqv e‡ji w`KI nq Zxh©K A_©vr mvg‡bi w`‡K| cÖwZwµqv
e‡ji GKwU e„nr Dcvsk mvg‡bi w`‡K KvR K‡i| d‡j ªæZMwZ
AR©b Ki‡Z †`Šowe‡`i †ek myweav nq|
M †`Iqv Av‡Q,
mij †`vj‡Ki †`vjbKvj, T = 2 sec
†ei Ki‡Z n‡e, †KŠwYK K¤úv¼, = ?
Avgiv Rvwb, = 2T =
2 3.14162 = 3.1416 rad.s1 (Ans.)
N cÖv_wgK Ae ’vq †`vjKwUi †`vjbKvj, T1 = 2 s
cwieZ©xZ †`vjbKvj n‡e, T2 = 2 s + 2 s 50% = 3 s
cÖv_wgK Kvh©Ki ˆ`N©¨, L1 = 100 cm
cwieZ©xZ Kvh©Ki = L2
mij †`vj‡Ki wØZxq m~Î †_‡K cvB,
T1
T2 =
L1
L2
ev, L1
L2 =
T12
T22 =
22
32 = 49
ev, L2 = 94 L1 =
94 100 cm = 225 cm
myZivs DÏxc‡Ki kZ©vbymv‡i †`vj‡Ki Kvh©Ki ˆ`N©¨ n‡Z n‡e 225 cm, wKš‘ Kvh©Ki `N©¨ Kiv n‡q‡Q 150 cm| myZivs 150 cm Kvh©Ki
ˆ`‡N©¨i †`vjKwU DÏxc‡Ki kZ© c~iY K‡iwb|
4bs cÖ‡kœi DËi
K jf¨ Kvh©Ki kw³ Ges †gvU cÖ`Ë kw³i AbycvZ‡K h‡š¿i
Kg©`ÿZv e‡j|
L ywU e¯‘i w ’wZ¯’vcK msN‡l©i †ÿ‡Î msN‡l©i ci cÖ_g e¯‘i
†eM,
iif vmm
mv
mm
mmv 2
21
21
21
211
2
†`qv‡ji mv‡_ e‡ji msN‡l©i †ÿ‡Î, v2i = 0 Ges m2 » m1| myZivs
v1f = – v1i Ges v2f = 0
A_©vr †`qvj w ’i _vK‡e Ges ejwU GKB †eM wecixZ w`‡K wd‡i
Avm‡e|
M †`qv Av‡Q, cÖ_g Zv‡ii-
ˆ`N©¨ weK…wZ, LL = 10 % = 0.1
e¨vm, D = 2 mm
e¨vm n«vm D n‡j cqm‡bi AbycvZ,
= DD /
LL
ev, DD =
LL = 0.5 0.1 = 0.05
D = 0.05 2 mm = 0.1 mm
myZivs e¨vmva© n«vm, r = 0.1 mm
2 = 0.05 mm (Ans.)
N aiv hvK, Dfq Zv‡ii Avw` ˆ`N©¨ = L
cÖ_g Zv‡ii e¨vmva©, r1 = 1 mm = 10 – 3 m
wØZxq Zv‡ii e¨vmva©, r2 = 2.5 mm = 2.5 10 – 3 m
cÖ_g Zv‡ii ˆ`N©¨ e„w× = l1
wØZxq Zv‡ii ˆ`N© e„w× = l2
kZ©vbymv‡i, l1 = 3 l2
cÖ_g Zv‡ii Bqs‡qi ¸Yv¼, Y1 = F
r12 /
l1L
wØZxq Zv‡ii Bqs‡qi ¸Yv¼, Y2 = F
r22 /
l2L
Y1
Y2 =
r22
r12
l2l1
=
2
3
3
m 10
m 105.2
l2
3l2 =
6.253 = 2.083
Y1 = 2.083 Y2
myZivs cÖ_g Zv‡ii w ’wZ¯’vcKZv †ewk|
5 bs cÖ‡kœi DËi
K e„Ëc‡_ N~Y©‡bi mgq †Kv‡bv e ‘i Ici e„‡Ëi †K‡›`Öi w`‡K †h
ej wµqv K‡i Zv‡K †K›`ÖgyLx ej e‡j|
L evZv‡mi cÖev‡ni w`‡K †`Šov‡j e¨w³i mv‡c‡ÿ evZv‡mi
Av‡cwÿK †eM, evZv‡mi cÖK…Z †eM A‡cÿv Kg nq| ZvB ZLb
evZv‡mi †eM K‡g †M‡Q e‡j g‡b nq|
M †`Iqv Av‡Q, K…wÎg DcMÖ‡ni MwZkw³, EK = 3.6 109J
K…wÎg DcMÖ‡ni fi, m = 120 kg
K…wÎg DcMÖ‡ni †eM v n‡j, EK = 12 mv2
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
v = 2EK
m = 2 3.6 109J
120 kg = 7746 ms1
f‚c„ô n‡Z K…wÎg DcMÖ‡ni D”PZv h n‡j, v = GM
R + h
ev, R + h = GMv2 h =
GMv2 R
= 6.6 1011 Nm2kg2 6 1024 kg
(7746 ms1)2 6.4 106m
2 105m = 200 km (Ans.)
N K…wÎg DcMÖnwU‡K ewntwe‡k¦ Z_v gnvk~‡b¨ †cÖi‡Yi Rb¨
cÖ‡qvRbxq me©wb¤œ kw³,
hRhRhR
drrGMmdrr
GMmFdrW 2
2
hRGMm
rGMm
hR
111=
GMm
R + h
= 6.6 10-11 6 1024 120
6.4 106 + 2 105
= 7.2 109 J wKš‘ K…wÎg DcMÖ‡ni MwZkw³ 3.6 109 J, hv cÖ‡qvRbxq me©wb¤œ
kw³ A‡cÿv Kg| myZivs ejv hvq, mÂvwiZ MwZkw³ K…wÎg
DcMÖnwU‡K ewntwe‡k¦ cvVv‡bvi Rb¨ ch©vß bq|
6 bs cÖ‡kœi DËi
K mgy`ª c„‡ô 45 Aÿvs‡k 0C ZvcgvÎvq Djø¤^fv‡e Aew ’Z 760 mm D”PZv wewkó ﮋ I weï× cvi` ¯ ͇¤¢i Pvc‡K cÖgvY Pvc ejv
nq|
L cv‡Îi †`qvj Zvc mycwievnx Ges M¨v‡mi ms‡KvPb ev cÖmviY
ax‡i ax‡i msMwVZ n‡j M¨vm cwi‡ek‡K Zvc †`q A_ev cwi‡ek
n‡Z Zvc MÖnY K‡i| d‡j ZvcgvÎv AcwiewZ©Z _v‡K| G‡Z M¨v‡mi
Pvc I AvqZb cwiewZ©Z nq| ZvcMZxq G cÖwµqv‡K m‡gvò
cÖwµqv e‡j|
M g‡b Kwi, wbg¾‡bi c~‡e© †ejy‡bi AvqZb wQj = V1
n«‡`i Zj‡`‡k †ejy‡bi cÖviw¤¢K AvqZb, V2 = 1 L
n«‡`i MfxiZv, h = 40.81 m
n«‡`i c„ô‡`‡k Pvc, P1 = 105 Nm–2
n«‡`i Zj‡`‡k Pvc, P2 = P1 + hg
e‡q‡ji m~Îvbymv‡i, P1V1 = P2V2 ev, P1V1 = (P1 + hg)V2
V1 = (P1 + hg)V2
P1
= 105 Nm2 + 40.81m 103 kgm3 9.8 ms2
105 Nm2 1 L
= 5 L
AZGe wbg¾‡bi c~‡e© DÏxc‡Ki †ejy‡bi AvqZb 5 L wQj| (Ans.)
N n«‡`i Zj‡`‡k †ejybwUi bZzb AvqZb, V1 = (1L + 1L) = 2L
awi, c„ô †`‡k Avm‡j †ejy‡bi AvqZb = V2
P2V2 = P1V1
V2 = P1V1
P2 =
(P2 + hg)V1
P2
= 105 Nm2 + 40.81m 103 kgm3 9.8 ms2
105 Nm2 2L
= 10 L > 9 L
myZivs cvwbi DcwiZ‡j †ejybwU AÿZ Ae ’vq †cŠQv‡e bv| †ejybwU
†d‡U hv‡e|
2. ivRkvnx †evW©-2015
1 bs cÖ‡kœi DËi
K GKwU †f±i‡K hw` yB ev Z‡ZvwaK †f±‡i Ggbfv‡e wef³
Kiv nq, hv‡`i jwä n‡e g~j †f±i, Z‡e G wef³KiY cÖwµqv‡K
†f±‡ii we‡kølY e‡j|
L bvj †f±i n‡jv k~b¨ †f±i| Gi gvb k~b¨ e‡j Gi †Kv‡bv
mywbw`©ó w`K wbY©q Kiv m¤¢e bq| ZvB Gi w`K †h‡Kv‡bv w`‡KB
we‡ePbv Kiv †h‡Z cv‡i|
M †bŠKvi Zj I cvwbi Nl©Y ej D‡cÿYxq n‡j G‡ÿ‡Î
†bŠKvwU‡Z Z¡iY m„wó n‡e|
Avw`‡eM, u = 0 ms1
mgqKvj, t = 5 min = 300 sec
miY, s = 3.6 km = 3600 m
s = ut + 12 at2 ev, a =
2st2 [ u = 0] =
2 3600 m(300s)2 = 0.08 ms2
jwä Uvb, T = ma = 500 kg 0.08 ms2 = 40N
Zvn‡j, T = F2 + F2 + 2F.F cos60
ev, T = F 1 + 1 + 2 12 = F 3
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
F = T
3 =
40N
3 = 23.094N (Ans.)
N mgvb Uv‡b (F = 23.1N) D³ ~iZ¡ (3.6km) 5 wgwb‡Ui Kg mg‡q
†cuŠQv m¤¢e| †m‡ÿ‡Î iwk؇qi ˆ`N©¨ evov‡Z n‡e hv‡Z cÖhy³
Uvb؇qi ga¨Kvi †KvY 60 A‡cÿv Kg nq|
awi, Gevi = 55
Zvn‡j, F = 23.1N Gi Rb¨ jwä ej, R = F2 + F2 + 2F.F cos
= 2F2 (1 + cos ) = F 2 (1 + cos)
= 23.1N 2 (1 + cos 55) = 40.98N
†bŠKvq Z¡iY, a = Rm
40.98N500kg = 0.082ms–2
G‡ÿ‡Î, S = 3600m `~iZ¡ AwZµ‡g t cwigv‡Y mgq jvM‡j, S = vot
+ 12 at2 = 0.t +
12 at2 =
12 at2
ev, t2 = 2Sa
t = 2Sa =
2 3600 m0.082ms–2
= 296.3 sec = 4 min 56.3 sec < 5 min
myZivs, mgvb Uv‡b D³ ~iZ¡ 5 wgwb‡Ui Kg mg‡q †cuŠ‡Q m¤¢e|
2 bs cÖ‡kœi DËi
K N~Y©vqgvb e¯‘i N~Y©b A‡ÿi mv‡c‡ÿ N~Y©b RoZv I †KŠwYK
†e‡Mi ¸Ydj‡K H A‡ÿi mv‡c‡ÿ N~Y©vqgvb e¯‘i †KŠwYK fi‡eM
e‡j|
L Avgiv Rvwb, R e¨vmva© I M f‡ii †Kv‡bv e ‘ †_‡K r (r R)
`~i‡Z¡ †Kv‡bv we› y‡Z gnvKl© wefe,
V = – GM
r
myZivs ~iZ¡ e„w×i mv‡_ GM
r Gi gvb `~i‡Z¡i e¨ Ívbycv‡Z Kg‡Z
_vK‡e wKš‘ wefe FYvZ¥K nIqvq V Gi gvb evo‡Z _v‡K Ges
Amxg ~i‡Z¡ gnvKl© wefe k~b¨| ~i‡Z¡i mv‡_ gnvKl© wef‡ei
cwieZ©b wb‡Pi †jLwP‡Îi mvnv‡h¨ †`Lv‡bv hvq-
M †`Iqv Av‡Q,
wb‡ÿcY †eM, v0 = 25 ms1
wb‡ÿcY †KvY, 0 = 30
wKK Kivi t = 0.5 sec ci Abyf‚wgK †eM, vx = vxo = v0cos0
= 25 ms1 cos30 = 21.65 ms1
Ges Djø¤^ †eM, vy = v0sin0 gt = 25 ms1 sin30 9.8 ms2
0.5 sec = 7.6 ms1
wKK Kivi 0.5 sec c‡i e‡ji †e‡Mi gvb, v = vx2 + vy
2
= 21.652 + 7.62 ms1 = 22.94 ms1 Ans.
g‡b Kwi, e‡ji †e‡Mi w`K Abyf‚wg‡Ki mv‡_ †KvY Drcbœ K‡i|
tan = vy
vx
= tan–1 vy
vx
= tan–1
7.6
21.65
= 19.34
0.5s ci ejwUi †e‡Mi gvb 22.94ms–1 Ges GB †e‡Mi w`K n‡e
Abyf‚wg‡Ki mv‡_ 19.34 †KvY K‡i Dc‡ii w`‡K| (Ans.)
N ejwUi Abyf‚wgK cvjøv, R = v0
2 sin20
g =
(25 ms1)2 sin (2 30)
9.8 ms2
= 55.23 m
Ges wePiYKvj, T = 2v0sin0
g = 2 25 ms1 sin30
9.8 ms2 = 2.55 sec
GB mg‡q †MvjwKcvi KZ…©K AwZµvš Í ~iZ¡ = 2.55 sec 10 ms1
= 25.5m
†Mvj‡cvó †_‡K ejwUi cZb we› yi ~iZ¡ = 80 m – 55.23m
= 24.77 m
r
V
R
vy
vx
v
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
†h‡nZz †MvjwKcvi ejwU f‚wg‡Z cwZZ nIqvi Av‡MB G‡mwQj, ZvB
†MvjwKcvi ejwU ai‡Z cvi‡e|
3 bs cÖ‡kœi DËi
K hw` c„w_exi AveZ©‡bi mv‡_ wgwj‡q GKB †KŠwYK MwZ‡Z GKwU
K…wÎg DcMÖn MwZkxj nq, Z‡e Zv c„w_ex‡K 24 N›Uvq GKevi
cÖ wÿY Ki‡e wKš‘ c„w_ex †_‡K GKRb ch©‡eÿi Kv‡Q w¯’i g‡b
n‡e| Giƒc K…wÎg DcMÖn‡K f‚w¯’i DcMÖn e‡j|
L ywU mgvb f‡ii w ’wZ¯’vcK msN‡l©i †ÿ‡Î Avgiv cvB,
mu1 + mu2 = mv1 + mv2
Ges 12 mu1
2 + 12 mu2
2 = 12 mv1
2 + 12 mv2
2
ev, u1 – v1 = v2 – u2 … … … … (1)
Ges u12 – v1
2 = v22 – u2
2 … … … (2)
mgxKiY (2) †K (1) Øviv fvM K‡i
u1 + v1 = v2 + u2 … … … (3)
mgxKiY (1) I (3) †hvM K‡i
2u1 = 2v2
ev, v2 = u1
mgxKiY (1) I (3) we‡qvM K‡i
2u2 = 2v1
ev, v1 = u2
myZivs mgvb f‡ii ywU e ‘i w ’wZ¯’vcK msN‡l© e ‘Øq ci¯úi †eM
wewbgq K‡i|
M awi, f‚wg‡Z AvNv‡Zi c~e© gyn~‡Z© ejwUi †eM, v
DÏxcK †_‡K cvB-
ejwUi AwZµvšÍ `~iæZ¡, h = (20+ 3) cm
= 23cm
= 0.23m
AwfKl©R Z¡iY, g = 9.8 m/s2
m‡e©v”P D”PZvq †eM, v0 = 0ms–1
Avgiv Rvwb, v2 = v02
+ 2gh
= 02 + 2 9.8 0.23
= 0.21 ms–1
AZGe, f‚wg‡Z AvNv‡Zi c~e© gyn~‡Z© ejwUi †e‡Mi gvb 0.21 ms–1 (Ans.)
N w¯úªsej Øviv K…Z KvR k~b¨ KviY ejwU w¯úªswU‡K ¯úk© Kivi
mgq Gi †h †eM _vK‡e, w¯úªs †_‡K gy³ nIqvi mgq †m †eM cÖvß
n‡e| w¯úªs ms‡Kv‡P‡bi mgq w¯úªs ej Øviv FYvZ¥K KvR n‡e Ges
cÖmvi‡Yi mgq mg cwigvY abvZ¥K KvR n‡e d‡j †gvU K…Z KvR
k~b¨ n‡e|
w¯úÖs ej Øviv K…Z KvR = AvNvZ Kivi gyn~‡Z© ejwUi MwZkw³
= 12 mv2
= 12 0.2 kg 2401 m2s2
= 240.1J
myZivs DÏxcK †_‡K w¯úÖs ej Øviv K…ZKvR wbY©q Kiv m¤¢e|
4 bs cÖ‡kœi DËi
K KwVb I Zi‡ji ¯úk©we›`y n‡Z eµ Zij Z‡j Aw¼Z ¯úk©K
KwVb e ‘i mv‡_ Zi‡j g‡a¨ †h †KvY Drcbœ K‡i, Zv‡K D³ KwVb
I Zi‡ji ga¨Kvi ¯úk© †KvY e‡j|
L Zi‡ji mv› ÖZv Drcbœ nq AvšÍAvYweK e‡ji Kvi‡Y| wKš‘
M¨v‡mi mv›`ÖZv Drcbœ nq AYy¸‡jvi ga¨Kvi msN‡l©i Kvi‡Y|
ZvcgvÎv evo‡j Zi‡ji Avš ÍtAvYweK ej n«vm cvq Ges M¨vm
AYymg~‡ni ga¨Kvi msNl© e„w× cvq| ZvB ZvcgvÎv evov‡j M¨v‡mi
mv› ÖZv ev‡o wKš‘ Zi‡ji mv›`ÖZv K‡g|
M OQ = OAcos30 = 1 m 0.866 = 0.866 m
OP = OBcos15 = 1 m 0.966 = 0.966 m
QP = OP – OQ = 0.966 m – 0.866 m = 0.1 m
B we›`y‡Z e‡ei †eM v n‡j
v2 = 29.8 ms-2 0.1 m = 1.96 m2s-2
myZivs B we› y‡Z e‡ei MwZ kw³,
KB = 12 mv2 =
12 0.02 kg 1.96 m2s-2 = 0.0196 J
N OQ = OAcos30 = 1 m 0.866 = 0.866 m
OP = OBcos15 = 1 m 0.966 = 0.966 m
QP = OP – OQ = 0.966 m – 0.866 m = 0.1 m
A
B C
P
Q
O
vm
30
15
v
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
QC = OC – OQ = 1 m – 0.866 m = 0.134 m
PC = OC – OP = 1 m – 0.966 m = 0.034 m
myZivs C Gi mv‡c‡ÿ A we›`y‡Z e‡ei wefe kw³,
UA = mgQC = 0.02 kg 9.8 ms-2 0.134 m
= 0.02626 J
Ges MwZkw³, KA = 0 [ÔMÕ Ask †_‡K]
myZivs A we› y‡Z †gvU kw³,
EA = UA + KA = 0.02626 J + 0 = 0.02626 J
Avevi C Gi mv‡c‡ÿ B we› y‡Z e‡ei wefe kw³,
UB = mgPC = 0.02 kg 9.8 ms-2 0.034 m
= 0.00666 J
Ges MwZkw³, KB = 0.0196 J [ÔMÕ Ask n‡Z]]
myZivs B we› y‡Z †gvU kw³,
EB = UB + KB = 0.00666 J + 0.0196 J = 0.02626 J
C Gi mv‡c‡ÿ C we› y‡Z e‡ei wefe kw³, UC = 0
C we›`y‡Z e‡ei †eM vm n‡j
vm2 = 29.8 ms-2 0.134 m = 2.6264 m2s-2
myZivs C we› y‡Z e‡ei MwZ kw³,
KC = 12 mvm
2 = 12 0.02 kg 2.6264 m2s-2 = 0.02626 J
myZivs C we› y‡Z †gvU kw³,
EC = UC + KC = 0 + 0.02626 J = 0.02626 J
GLv‡b, EA = EB = EC
myZivs †`vjKwU hvwš¿K kw³i wbZ¨Zv †g‡b P‡j|
5 bs cÖ‡kœi DËi
K mgvb ev cÖvq mgvb we Ívi Ges K¤úv‡¼i mvgvb¨ cv_©K¨ wewkó
`ywU kã Zi½ GK mv‡_ GKB mij †iLvq GKB w`‡K mÂvwjZ n‡j
G‡`i DcwicvZ‡bi d‡j kã Zi‡½i ZxeªZvi †h ch©vqµwgK n«vm
e„w× N‡U Zv‡K exU ev ¯iK¤ú e‡j|
L x Gi †h gv‡bi Rb¨ cos2
x = 1 n‡e †mme we› y‡Z we¯ Ívi
m‡e©v”P 2a n‡e A_©vr †mme we› y‡Z my¯c›` we›`y cvIqv hv‡e|
myZivs my¯c›` we›`yi Rb¨,
cos2
x = 1
2
x = n GLv‡b, n = 0, 1, 2, 3, ......
ev, x = n 2 = 0,
2 , ,
32 ..........
A_©vr †h mKj we›`y‡Z x Gi gvb 2 Gi ALÛ ¸wYZK †m mKj
we›`y‡Z my¯ú›` we›`y MwVZ n‡e&
M ¯úxKvi A n‡Z †kÖvZvi ~iZ¡, r = 32 + 22 = 13
¯úxKvi A Gi Rb¨ O we› y‡Z k‡ãi ZxeªZv, I = PA =
P
4r2
= 1 103W
4 3.1416 ( 13)2m2 = 6.12 106 Wm2
N ïay ¯úxKvi A n‡Z cÖvß k‡ãi ZxeªZv †j‡fj,
B = 10 logII0
= 10 log6.12 106 Wm2
1012 Wm2 = 67.88 dB
¯úxKvi A I ¯úxKvi B Df‡qi myBP Ab Ki‡j O we›`y‡Z k‡ãi
ZxeªZv, I = 2 6.12 106 Wm2
G‡ÿ‡Î k‡ãi ZxeªZv †j‡fj, B = 10 logII0
= 10 log2 6.12 106 Wm2
1012 Wm2 = 70.88 dB
†h‡nZz 70.88 dB 2 67.88 dB A_©vr B 2B
myZivs ¯úxKvi A I ¯úxKvi B Df‡qi myBP Ab Ki‡j O we› y‡Z
k‡ãi ZxeªZv †j‡fj c~e©v‡cÿv wظY n‡e bv|
6 bs cÖ‡kœi DËi
K †h ZvcgvÎvq GKwU wbw`©ó AvqZ‡bi evqy Zvi †fZ‡ii Rjxq
ev®ú Øviv m¤ú„³ nq Zv‡K H evqyi wkwkivsK e‡j|
L GKB AvqZ‡bi ywU evqyc~Y© †ejyb‡K wfbœ ZvcgvÎvq ivL‡j V2
T2
= V1
T1 m~Îvbymv‡i †ewk ZvcgvÎvi †ejy‡bi AvqZb †ewk n‡e, KviY
Dfq‡ÿ‡Î Pvc evqygÛjxq Pv‡ci mgvb n‡e|
M †`Iqv Av‡Q.
A cv‡Îi AvqZb, V = 2 cm3 = 2 106m3
Ges Pvc, P = 3 105 Nm2
A cv‡Î M¨v‡mi MwZkw³ EA = 32 PV
= 32 3 105 Nm2 106m3
= 0.9J
x = 0
N N N N A A A A
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m
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
N cvÎ B Gi †ÿ‡Î, Pvc, P = 3.66 105 Nm2
Ges AvqZb, V = 2 106m3
cvÎ B Gi M¨v‡mi MwZkw³ = 32 PV = 1.5 3.66 105 Nm2 2
106m3
= 1.098J
GB MwZkw³ Zvciƒ‡c †`Lv w`‡e, hv ¯¯^ cv·K DËß Ki‡e|
†h‡nZz 1.098J > 0.9J
A_©vr cvÎ B Gi M¨v‡mi MwZkw³ > cvÎ A M¨v‡mi MwZkw³
myZivs cvÎ B †ewk DËß n‡e|
3. w`bvRcyi †evW©-2015
1 bs cÖ‡kœi DËi
K †Kv‡bv MwZkxj e ‘i †Kv‡bv we‡kl gyn~‡Z© ÿz`ªvwZÿz`ª mgq
e¨eav‡b †e‡Mi cwieZ©‡bi nvi‡K H we‡kl gyn~‡Z©i ZvrÿwYK Z¡iY
e‡j|
L Dc‡ii w`‡K wbwÿß e ‘i Ici wµqvkxj AwfKl© e‡ji w`K
wb‡Pi w`‡K| ZvB AwfKl©R Z¡i‡Yi w`KI Lvov wb‡Pi w`‡K| G
Z¡i‡Yi Kvi‡Y Dc‡ii w`‡K wbwÿß e ‘i MwZ‡eM n«vm cvq|
M GLv‡b, X we› yi ’vbv¼ (3, 2, 1)
Y we› yi ’vbv¼ (–2, 1, 4)
Zvn‡j,
OX = 3 i + 2 j + k Ges
OY = 2 i + j + 4k
†ei Ki‡Z n‡e, G‡`i ga¨eZx© †KvY, = ?
Avgiv Rvwb, OX .
OY = |
OX | |
OY | cos
ev, 3 (2) + 2 1 + 1 4 = 32 + 22 + 12 (2)2 + 12 + 42 cos
cos = 0
32 + 22 + 12 (2)2 + 12 + 42 = 0
ev, = cos10 = 90
AZGe, †f±i؇qi ga¨eZx© †KvY 90| (Ans.)
N OX ,
OY Z‡ji Ici j¤ GKK †f±i =
OX
OY
|OX
OY |
GLv‡b, OX
OY =
i
3
2
j21
k14
= i | |21
14 j | |3
2
14 + k
| |3
2
21
= 7 i 14 j + 7k
Avevi, |OX
OY | = 72 + (14)2 + 72 = 7 6
OX
OY
|OX
OY |
= 7 i 14 j + 7k
7 6 =
i 2 j + k
6
wKš‘ OY ,
OX Gi Z‡ji Dci j¤^ GKK †f±i =
OY
OX
|OY
OX |
my¯úóZ: OY
OX = (
OX
OY ) = 7 i + 14 j 7k
Ges |OY
OX | = |
OX
OY | = 7 6
OY
OX
|OY
OX |
= 7 i + 14 j 7k
7 6 =
i 2 j + k
6
A_©vr j¤ GKK †f±iØq gv‡b mgvb n‡jI w`‡K ci¯úi wecixZ|
OX ,
OY Gi Z‡ji Dci j¤ GKK †f±i KvMRc„ô n‡Z
j¤^fv‡e Lvov Ici w`‡K wµqv K‡i Ges OY ,
OX Gi Z‡ji
Ici j¤ GKK †f±i KvMRc„ô n‡Z j¤^fv‡e Lvov wb‡Pi w`‡K wµqv
K‡i|
2 bs cÖ‡kœi DËi
K †Kv‡bv MwZkxj e ‘i †Kv‡bv we‡kl gyn~‡Z© ÿz`ªvwZÿz`ª mgq
e¨eav‡b †e‡Mi cwieZ©‡bi nvi‡K H we‡kl gyn~‡Z©i ZvrÿwYK
Z¡iY e‡j|
L †f±‡ii gvb KL‡bvB FYvZ¥K nq bv| KviY †f±‡ii gvb
ej‡Z Avgiv cig gvb‡K eywS Avi cig gvb KL‡bvB FYvZ¥K
bq|
M †`Iqv Av‡Q, c„w_exi e¨vmva©, R = 6.4 103km = 6.4 106m
f‚c„ô n‡Z D we›`yi D”PZv, h = 200 km = 2 105m
Rvbv Av‡Q, f‚c„‡ô AwfKl©R Z¡iY, g = 9.8 ms2
†ei Ki‡Z n‡e, D Ae ’v‡bi AwfKlx©q Z¡i‡Yi gvb, g = ?
Avgiv Rvwb, g = g
R
R + h
2
= 9.8 ms2
6.4 106m
6.4 106m + 2 105m
2
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m
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
= 9.215 ms2 (Ans.)
N f‚c„‡ô, m = 10kg f‡ii e ‘i IRb, W = mg = 10 kg 9.8 ms2 = 98N
C Ae ’v‡b AwfKl©R Z¡i‡Yi gvb, g = g
1
dR
= 9.8 ms2
1
100 103m
6.4 106m
= 9.647 ms2
C Ae ’v‡b e ‘wUi IRb, W = mg = 10kg 9.647 ms2 = 96.47N
†h‡nZz 96.47N 98N
A_©vr W W
myZivs DÏxc‡Ki wP‡Î C Ae ’v‡b hw` m = 10kg f‡ii e ‘ wb‡q
hvIqv nq, Z‡e Gi Dci c„w_exi AvKl©Y e‡ji cwieZ©b (n«vm)
NU‡e|
3 bs cÖ‡kœi DËi
K †Kv‡bv e ‘i Ici ej cÖ‡qv‡Mi d‡j e‡ji wecixZ w`‡K e ‘i
miY NU‡j ev e‡ji wecixZ w`‡K mi‡Yi Dcvsk _vK‡j Zvn‡j ej
I mi‡Yi Dcvs‡ki ¸Ydj‡K FYvZ¥K KvR e‡j|
L †Kv‡bv ^‡i †hme wewfbœ myi _v‡K, Zv‡`i g‡a¨ †h my‡ii K¤úv¼
me‡P‡q Kg, Zv‡K g~j myi e‡j| Ab¨vb¨ myi, hv‡`i K¤úv¼ g~j
my‡ii K¤úv‡¼i †P‡q †ewk, Zv‡`i‡K Dcmyi ejv nq| Avevi,
Dcmyi¸‡jvi K¤úv¼ hw` g~j my‡ii K¤úv‡¼i mij ¸wYZK nq,
Zvn‡j †mB mKj Dcmyi‡K nvi‡gvwbK e‡j| G Kvi‡YB mKj
nvi‡gvwbK Dcmyi n‡jI mKj Dcmyi nvi‡gvwbK bq|
M †`Iqv Av‡Q,
X wmwjÛv‡i M¨v‡mi ZvcgvÎv, Tx = 600 K
Y wmwjÛv‡i M¨v‡mi ZvcgvÎv, Ty = 650 K
†ei Ki‡Z n‡e, G‡`i Mo eM©g~j †e‡Mi Zzjbv ev AbycvZ,
Cx2 t
Cx2 = ?
M¨vm؇qi NbZ¡ mgvb weavq Giv g~jZ GKB M¨vm A_©vr MÖvg
AvYweK fi M Gi gvb Dfq‡ÿ‡Î mgvb|
Avgiv Rvwb,
Cx2 =
3RTx
M Ges
Cy2 =
3RTy
M
Cx2
Cy2
= Tx
Ty =
600K650K = 0.9608
ev,
Cx2 = 0.9608
Cy2
Cy2 >
Cx2
AZGe, Y wmwjÛv‡ii M¨v‡mi Mo eM©g~j †e‡Mi gvb †ewk|
N `ywU cv‡Îi g‡a¨ M¨v‡mi Av`vb cÖ`vb wbf©i K‡i M¨v‡mi Pv‡ci
Dci| †h‡nZz Y cv‡Î M¨v‡mi Pvc †ewk ZvB Y cvÎ †_‡K M¨vm X cv‡Î Mgb Ki‡e hZÿY bv Dfq cv‡Îi Pvc mgvb nq| Pvc mgvb
nIqvi ci MwZZË¡ Abymv‡i Dfq cv‡Îi AYy¸‡jv BZ¯ ÍZ
wewÿßfv‡e †QvUvQzwU Ki‡Z _vK‡e d‡j Dfq cv‡Îi g‡a¨ AYyi
MgbvMgb NU‡Z _vK‡e|
4 bs cÖ‡kœi DËi
K †h ivwk Øviv Zi½ mÂviYKvix KYvi †h‡Kv‡bv gyn~‡Z©i MwZi
mg¨K Ae ’v eySvq Zv‡K `kv e‡j|
L cÖwZ †m‡K‡Û exU 6 ej‡Z eySvq, g~j kã Zi½Ø‡qi
DcwicvZ‡bi d‡j jwä Zi‡½i k‡ãi ZxeªZv cÖwZ †m‡K‡Û 6 evi
n«vm-e„w× N‡U| A_©vr cÖwZ †m‡K‡Û 6wU Zxeª kã †kvbv hvq Ges 6wU
wbtkã †kvbv hvq|
M cÖ_g Zi½wUi mgxKiY: y1 = 0.1 sin
200t
2017 x
= 0.1 sin 2017 (170t x)
G‡K Zi‡½i cÖwgZ mgxKiY, y = asin 2
(vt x) Gi mv‡_ Zzjbv
K‡i cvB,
v = 170 ms1
myZivs cÖ_g Zi½wUi Zi½‡eM, v = 170 ms1
N DÏxc‡Ki Zi½Ø‡qi DcwicvZ‡bi d‡j m„ó jwä Zi‡½i
mgxKiY, y = y1 + y2 = 0.1 sin
200t
2017 x + 0.1 sin
200t +
2017 x
= 0.1 2 sin(200t) cos
20
17 x
= 0.2cos
20
17 x sin (200t)
= A sin (200t)
0.2 cos
20
17 x = cwieZÆbkxj weÕ¦vi A
Dc‡iv³ mgxKi‡Y AMÖMvgx Zi‡½i mgxKi‡Yi b¨vq `kv †Kv‡Yi
†fZi (vt x) RvZxq †Kv‡bv ivwk AšÍfz©³ bvB ZvB GwU AMÖMvgx
Zi‡½i mgxKiY bq| GwU w¯’i Zi‡½i mgxKiY|
myZivs DÏxc‡K Zi½Ø‡qi g‡a¨ DcwicvZ‡bi d‡j w ’i Zi½ m„wó
n‡e|
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m
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
5 bs cÖ‡kœi DËi
K mgvb ev cÖvq mgvb we Ív‡ii wKš‘ K¤úv‡¼i mvgvb¨ cv_©K¨ wewkó
`ywU kã Zi½ GKB mgq GKB mij †iLvq GKB w`‡K mÂvwjZ
n‡j G‡`i DcwicvZ‡bi d‡j k‡ãi ZxeªZvi †h ch©vqµwgK n«vm
e„w× N‡U Zv‡K exU ev ¯iK¤ú e‡j|
L aªæeej I w¯úÖs ¯ú›`b Gi mv‡_ †`vjbKv‡ji m¤úK© wb¤œiƒc :
F = – kx ev ma = kx ev, mk =
xa
†`vjbKvj, T = 2miY
Z¡iY = 2
xa = 2
mk
GLv‡b Z¡iY I miY ci¯úi wecixZ w`‡K nIqvq xa ivwkwU
abvZ¥K|
M †`Iqv Av‡Q, r = 4 i 6 j + 12k
F = 2 i + 3 j 5k
†ei Ki‡Z n‡e, UK©, = ?
Avgiv Rvwb, = r
F = (4 i 6 j + 12k) (2 i + 3 j 5k)
=
i
42
j
63
k
12
5
= i
6
3
12
5 j
4
2 12
5 + k
4
2 63
= i (30 – 36) – j (–20 – 24) + k (12 + 12)
= 6 i + 44 j + 24k
U‡K©i gvb = (–6)2 + 442 + 242 = 50.48 GKK|
mvB‡Kj Pvj‡Ki U‡K©i gvb = 50.48 GKK| (Ans.)
N DE c‡_ mvB‡Kj Pvjv‡Z †M‡j mvB‡Kj Ges wb‡Ri AwfKl©‡K
AwZµg K‡i Ic‡i DV‡Z n‡e| d‡j Gmgq PvjK cÖPÛ Kó Abyfe
Ki‡eb, KviY Zv‡K weivU gv‡bi ÿgZv cÖ‡qvM Ki‡Z n‡e| EG c_
cÖvq Abyf‚wgK, Z‡e DuPz wbPz e‡j Zv h‡_ó gv‡bi Nl©Y cÖ vb Ki‡e|
d‡j G iv Ívq mvB‡Kj Pvjv‡Z †M‡j Pvj‡Ki Abyf‚wZ cy‡ivcywi
myLKi n‡e bv|
Z‡e GH c‡_ bvgvi mgq PvjK‡K c¨v‡Wj Pvc‡Z n‡e bv,
AwfK‡l©i `iæY mvB‡Kj ¯qswµqfv‡e wb‡P bvg‡Z _vK‡e| ïay
Zv‡K mvB‡K‡ji wbqš¿‡Y g‡bv‡hvM w`‡Z n‡e| Gmgq mvB‡Kj
PvjK †ek myLKi Abyf‚wZ cv‡eb|
6bs cÖ‡kœi DËi
K `ywU k‡ãi g‡a¨ hw` GKwUi ZxeªZv Ab¨wUi 100.1 ¸Y ev 1.259
nq, ZLb G‡`i ZxeªZv †j‡fj Gi cv_©K¨ nq 0.1 †ej ev 1
†Wwm‡ej|
L mij †`vjK MwZi †ÿ‡Î,
x = Asin(t + ) ev, dxdt = A cos(t + )
ev, d2xdt2 = A2sin(t + ) = 2x
d2xdt2 + 2x = 0
GwUB mij †`vjb MwZi AšÍiK mgxKiY|
M cÖ_g Zv‡ii †ÿ‡Î,
cqm‡bi AbycvZ, 1 = – DD
L
L =
0.01 mm 1m
5 mm 0.01m = – 0.2
wØZxq Zv‡ii cqm‡bi AbycvZ,
2 = DD
L
L =
0.03 mm 3m
15 mm 0.03m = – 0.2
Zvi؇qi cqm‡bi AbycvZ mgvb|
N Zvi؇qi cqm‡bi AbycvZ mgvb nIqvq Avgiv a‡i wb‡Z cvwi
Dfq Zvi GKB Dcv`v‡bi| myZivs G‡`i Amn cxob mgvb| Avgiv
Rvwb, Amn fvi = Amn cxob cÖ ’‡”Q‡`i †ÿÎdj
GLb cÖ_g I wØZxq Zv‡ii Amn fvi h_vµ‡g M1 I M2 n‡j,
M1
M2 =
d12
d22 =
2
mm 15
mm 5
=
19
M2 = 9 M1
A_©vr wØZxq Zv‡ii Amn fvi cÖ_g Zv‡ii Amn fv‡ii 9 ¸Y|
4. Kzwgjøv †evW©-2015
1 bs cÖ‡kœi DËi
K
V (x, y, z) AšÍwiKiY‡hvM¨ †f±i †ÿÎ n‡j
V †K
VGi
Kvj© e‡j|
L Avgiv Rvwb, †eM GKwU †f±i ivwk| gvb A_ev w`K A_ev
Df‡qi cwieZ©‡b †f±‡ii cwieZ©b nq| †Kv‡bv e ‘ e„ËvKvi c‡_
N~Y©b Kv‡j †e‡Mi gvb cwiewZ©Z bv n‡jI cÖwZ gyn~‡Z© w`‡Ki
cwieZ©b nq Ges †e‡Mi w`K nq †h‡Kv‡bv we› y‡Z e„ËvKvi c‡_i
¯úk©K eivei| myZivs ejv hvq, †Kv‡bv e ‘i e„ËvKvi c‡_ mg‡e‡M
Pjv m¤¢e bq|
M b`xi cÖ ’ eivei †e‡Mi Dcvsk = Vbsin37
= 10 ms1 sin37 = 6.02 ms1
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m
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
b`x cvi n‡Z mgq jvM‡e, t = d
6.02 ms1 = 36m
6.02 ms1
= 5.982 sec
b`xi cvo eivei †e‡Mi Dcvs‡ki †hvMdj = Vb cos37 + Vr
= 10 ms1 cos37 + 3 ms1 = 10.986 ms1
`~iZ¡, BC = 10.986 ms1 5.982 sec = 65.72m (Ans.)
N †bŠKvwU‡K A †_‡K mivmwi B we›`y‡Z †cuŠQv‡Z n‡j †bŠKv I
†¯ªv‡Zi †e‡Mi jwä Ges † ªv‡Zi †e‡Mi ga¨eZ©x †KvY = 90 n‡Z
n‡e| †bŠKv I † ªv‡Zi †e‡Mi ga¨eZ©x †KvY n‡j Avgiv cvB
tan90 = vbsin
vr + vbcos
ev, = vbsin
vr + vbcos
ev, vr + vbcos = 0
ev, cos = – vr
vb = –
3 ms-1
10 ms-1 = – 0.3
= 107.45
myZivs A †_‡K mivmwi B we› y‡Z †cuŠQv‡Z n‡j †bŠKvwU‡K †m&ªv‡Zi
w`‡Ki mv‡_ 107.45 †Kv‡Y Pvjbv Ki‡Z n‡e|
2 bs cÖ‡kœi DËi
K †Kv‡bv w¯úÖs‡K Gi mvg¨ve ’v n‡Z 1m cÖmvwiZ ev msKzwPZ
Ki‡Z †h cwigvY ej cÖ‡qvM Ki‡Z nq, Zv‡K w¯úÖs aªæeK e‡j|
L GKwU eo e„wói †duvUv †f‡½ A‡bK¸‡jv †QvU †duvUvq cwiYZ
Ki‡j me©‡gvU †ÿÎdj e„w× cvq| c„ôkw³i `iæY G‡ÿ‡Î A‡bK
kw³i `iKvi nq| e„nr cvwbi †duvUv n‡Z G kw³ †kvlY Kiv nq
weavq G‡ÿ‡Î ZvcgvÎvi n«vm NU‡e|
M Avgiv Rvwb,
)y2g(yvv 02y
2y
0
GLv‡b, Kvgv‡bi D”PZv, y0 = 60 m
Ges vy0 = v0 sin0 = 25sin53 = 250.7986= 19.96 ms-1
me©vwaK D”PZvq †e‡Mi Dj¤ Dcvsk k~b¨ A_©vr hLb y = ymax ZLb
vy = 0|
myZivs
0 = (19.96)2 – 29.8(ymax – 60)
ev, 19.6(ymax – 60) = 398.6
ev, (ymax – 60) = 20.34
ymax = 80.34
myZivs me©vwaK D”PZv 80.34 m (Ans.)
N aiv hvK, e› yK I Kvgv‡bi ¸wj f‚wg‡Z co‡Z h_vµ‡g t1 I t2 mgq jvM‡e|
GLv‡b,
Kvgv‡bi I e› y‡Ki Avw` D”PZv, yo = 60 m
[†h‡nZz ¸wjØq f‚wg‡Z c‡o]
y = 0
e›`y‡Ki ¸wji †ÿ‡Î,
vy0 = v0 sin0 = 25sin53 = 250.7986= 19.96 ms-1
myZivs
y = y0 + vy0 t1 – 12 g t1
2
ev, 0 = 60 + 19.96 t1 – 12 9.8 t1
2
ev, 4.9 t12 – 19.96 t1 – 60 = 0
t1 = 9.42
)60(9.4496.1996.19 2
= 9.42
1176398.496.19
=
8.9
68.9396.19
= 6.08 s or – 2.01 s
wKš‘ FYvZ¥K gvb MÖnY‡hvM¨ bq| myZivs t1 = 6.08 s
Kvgv‡bi ¸wji †ÿ‡Î,
vy0 = v0 sin0 = 25sin0 = 0
myZivs,
y = y0 + vy0 t2 – 12 g t2
2
ev, 0 = 60 – 12 9.8 t2
2
ev, 4.9 t22 = 60
ev, t22 = 12.245
t2 = 3.5 s
†h‡nZz, t1 > t2
myZivs Kvgv‡bi ¸wj Av‡M gvwU‡Z AvNvZ Ki‡e|
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m
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
3 bs cÖ‡kœi DËi
K evwn¨K e‡ji cÖfv‡e †Kvb e ‘i gvÎv ev ¯vfvweK Ae ’vi
cwieZ©b NUv‡bvi †Póv Ki‡j ev NUv‡j e ‘i Af¨š ͇i cÖwZ GKK
†ÿÎd‡j †h evav`vbKvix e‡ji D™¢e nq, Zv‡K cÖZ¨qbx ej e‡j|
L †Kvb wbw ©ó cwigvY M¨v‡mi g‡a¨ M¨vm KwYKv¸‡jvi m¤¢ve¨ †eM
wbY©q Kiv nq M¨v‡mi ZvcgvÎvi Dci wfwË K‡i| M¨v‡mi ZvcgvÎv
Gi AYy¸‡jvi MwZkw³i mv‡_ m¤úwK©Z Ges MwZkw³ AYymg~‡ni
†e‡Mi e‡M©i mv‡_ RwoZ| ZvQvov AYymg~n wewfbœ w`‡K MwZkxj
_vKvq G‡`i Mo‡eM k~b¨I n‡Z cv‡i| wKš‘ †e‡Mi eM©gvb me©`v
abvZ¥K Ges GwU mivmwi M¨v‡mi ZvcgvÎv I cv‡Îi Dci Pv‡ci
mv‡_ m¤úwK©Z| ZvB Mo †eM A‡cÿv g~j Mo eM©‡eM A‡bK †ewk
m½wZc~Y©|
M †`Iqv Av‡Q, e ‘i fi, m = 300g = 0.3 kg
f‚wg n‡Z †njv‡bv Z‡ji D”PZv h n‡j, mgh = Avw` MwZkw³ =
5.88J
h = 5.88Jmg =
5.88J
0.2 kg 9.8 ms2 = 2m
sin30 = h
AE ev, AE = h
sin30 =
2m12
= 4m (Ans.)
N M Abymv‡i h = 2 m Ges AE = 4 m| Avevi †h‡nZz AB = BC = CD = DE †m‡nZz AC = EC = 2 m, AD = 3 m Ges ED = 1 m| Avgiv
cvB,
sinA = h
AE = hC
AC = hD
AD
hC = h
AE AC = 24 2 = 1 m
hD = h
AE AD = 24 3 = 1.5 m
myZivs C I D we› yi wefe kw³ h_vµ‡g UC = (0.39.81) J = 2.94 J I UD = (0.39.81.5) J = 4.41 J|
†njv‡bv Z‡j e ‘i Z¡iY = EA eivei AwfKl©R Z¡i‡Yi Dcvsk
a = gsin30 = 9.8 0.5 = 4.9 ms-2
E †_‡K †njv‡bv Z‡j MwZkxj e¯‘i †eM C I D we› y‡Z h_vµ‡g vC I vD n‡j Avgiv cvB
vC2 = 2aEC = 24.9 ms-22 m = 18.6 m2s-2
Ges vD2
= 2aED = 24.9 ms-21 m = 9.8 m2s-2
C we›`y‡Z MwZkw³
KC = 12 mvC
2 = 12 0.3 kg 18.6 m2s-2 = 2.94 J
D we›`y‡Z MwZkw³
KD = 12 mvD
2 = 12 0.3 kg 9.8 m2s-2 = 1.47 J
C we›`y‡Z †gvU kw³
EC = UC + KC = 2.94 J + 2.94 J = 5.88 J
D we›`y‡Z †gvU kw³
ED = UD + KD = 4.41 J + 1.47 J = 5.88 J
myZivs C we› y‡Z †gvU kw³ = D we›`y‡Z †gvU kw³
myZivs †njv‡bv Zj eivei e ‘wU bvgvi mgq hvwš¿K kw³i wbZ¨Zv
m~Î †g‡b P‡j|
4 bs cÖ‡kœi DËi
K KwVb Zij ¯úk© we›`y‡Z Zij c„‡ôi ¯úk©K Zi‡ji wfZ‡i
KwV‡bi c„‡ôi mv‡_ †h †KvY Drcbœ K‡i Zv‡K ¯úk©‡KvY e‡j|
L e‡ji NvZ = F t Ges fi‡e‡Mi cwieZ©b = mv
e‡ji Nv‡Zi gvÎv = F Gi gvÎv t Gi gvÎv = MLT2 = MLT1
fi‡e‡Mi cwieZ©‡bi gvÎv = m Gi gvÎv v Gi gvÎv = MLT1
myZivs e‡ji NvZ fi‡e‡Mi cwieZ©‡bi mgvb|
M †`Iqv Av‡Q, †`vj‡bi we Ívi, A = 10 cm = 0.1m
†`vjbKvj, T = 2sec
†ei Ki‡Z n‡e, m‡e©v”P †eM, vmax = ?
Avgiv Rvwb, vmax = A = 2T A
= 2 3.1416
2 sec 0.1m = 0.31416 ms1 (Ans.)
N G‡ÿ‡Î eewU Abyf‚wgKfv‡e wbwÿß cÖv‡mi b¨vq AvPiY Ki‡e|
wb‡ÿcY †eM, v0 = 0.31416 ms1
wb‡ÿcY †KvY, 0 = 0
f‚wg n‡Z Avw` D”PZv, h = 45 cm = 0.45m
GB D”PZv †b‡g Avm‡Z t mgq jvM‡j,
h = v0sin0t + 12 gt2
ev, 0.45m = 0.31416 ms1 sin0 t + 12 9.8 ms2 t2
ev, 4.9t2 = 0.45 [GKKmg~n Dn¨ †i‡L]
t = 0.454.9 = 0.303 sec
GB mgqKv‡j AwZµvšÍ Abyf‚wgK `~iZ¡ = v0cos0 t
= 0.31416 ms1 cos0 0.303 sec
= 0.0952 m
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
= 9.52 cm
myZivs eewU mvg¨ve ’vb n‡Z 9.52 cm Abyf‚wgK ~i‡Z¡ f‚wg‡Z
cwZZ n‡e|
5 bs cÖ‡kœi DËi
K †Kvb k‡ãi ZxeªZv Ges cÖgvY ZxeªZvi Abycv‡Zi jMvwi`g‡K
H k‡ãi ZxeªZv †j‡fj e‡j| ZxeªZv †j‡fj, = log10 II0
|
L N~Y©b Kv‡j †KŠwYK fi‡e‡Mi msiÿYkxjZvi Rb¨ m~h© †_‡K
wewfbœ `~i‡Z¡ MÖ‡ni †eM wewfbœ nq| MÖ‡ni †KŠwYK fi‡eM Gi fi
†eM I m~h© †_‡K MÖ‡ni ~i‡Z¡i ¸Ydj| ZvB †KŠwYK fi‡eM
msiwÿZ ivLvi Rb¨ MÖn †_‡K m~‡h©i ~iZ¡ hZ K‡g, Gi †eM GKB
nv‡i ev‡o| GB Kvi‡YB MÖn Zvi N~Y©b Z‡j me©`v mgvb mg‡q mgvb
†ÿÎdj AwZµg K‡i|
M †`Iqv Av‡Q,
c„w_exi AveZ©bKvj, T1 = 365 day
c„w_ex I m~‡h©i ~iZ¡ R1 n‡j, cwiewZ©Z `~iZ¡, R2 = R1
2
†ei Ki‡Z n‡e, cwiewZ©Z AveZ©bKvj, T2 = ?
Avgiv Rvwb, †Kcjv‡ii 3q m~Îvbymv‡i,
T2
2
T12 =
R23
R13
T2 = T1 R2
R1
32
= 365 day
R1
2
R1
1.5
= 129.05 day (Ans.)
N Avgiv Rvwb, c„w_exi †KŠwYK †eM Ges †Kv‡bv ¯’v‡bi
Aÿvsk n‡j, c„w_exi N~Y©b‡e‡Mi `iæb AwfKl©R Z¡iY n«vm cvq
2Rcos2 cwigvY|
wbiÿ‡iLvq, = 0, 2Rcos2 =
2
T2 Rcos2
=
2 3.1416
86400 sec2 6.4 106m (cos0)2 = 0.0338
c„w_ex N~Y©biZ Ae¯’vq wbiÿ‡iLvq AwfKl©R Z¡iY, g = 9.78 ms2
c„w_exi AveZ©b eÜ n‡j wbiÿxq †iLvq Aew ’Z †Kv‡bv e ‘i
IR‡bi cwieZ©b = m 0.0338 ms2
mg 100%
= 0.0338 ms2
9.78 ms2 100%
= 0.346% (e„w× cv‡e)
6 bs cÖ‡kœi DËi
K GK †gvj Av`k© M¨v‡mi Rb¨ PV/T Abycv‡Zi gvb‡K mve©Rbxb
M¨vm aªæeK e‡j| Gi gvb 8.314 J mol–1 K–1|
L e‡ji NvZ n‡jv e‡ji gvb I wµqvKv‡ji ¸Ydj| Gi GKK
Ns ev kgms1| GwU e¯‘i fi‡e‡Mi cwieZ©‡bi mgvb| Gi gvÎv
MLT1| G¸‡jvB e‡ji Nv‡Zi ˆewkó¨|
M †`Iqv Av‡Q,
Mo eM©‡e‡Mi eM©g~‡ji Avw` gvb,
C12 = 11.2 kms1
Avw` Pvc P1 n‡j cwiewZ©Z Pvc, P2 = P1/2
†ei Ki‡Z n‡e, Mo eM©‡e‡Mi eM©g~‡ji cwiewZ©Z gvb,
C22 = ?
Avgiv Rvwb,
C2 = 3P
Av‡jvP¨ †ÿ‡Î,
C12 =
3P1
Ges
C22 =
3P2
[ NbZ¡ AcwiewZ©Z]
C22
C1
2
= 3P2
3P1 =
P2
P1 =
P1
2
P1 = 0.707
C22 = 0.707
C1
2 = 0.707 11.2 kms1 = 7.92 kms1 (Ans.)
N T1 = 27C = 300 K ZvcgvÎvq Aw·‡Rb AYyi Mo eM©‡e‡Mi
eM©g~j, C1
2 = 3RT1
M1 =
3R 300K
32 103 kg
[Aw·‡R‡bi †gvjvi AvYweK fi, M1 = 32 10–3 kg]
g‡b Kwi, T2 ZvcgvÎvq bvB‡Uªv‡Rb AYyi Mo eM©‡e‡Mi eM©g~j =
C22
C22 =
3RT2
28 103 kg
cÖkœg‡Z,
C12 =
C22 ev,
3RT
28 103 kg =
3R 300 K
32 103 kg
T2 = 300 K 2832 = 262.5 K
myZivs 262.5 K ZvcgvÎvq bvB‡Uªv‡Rb AYyi Mo eM©‡e‡Mi eM©g~j
gvb 27C ZvcgvÎvq Aw·‡Rb AYyi Mo eM©‡e‡Mi eM©g~j gv‡bi
mgvb n‡e|
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
5. PÆMÖvg †evW©-2015
1 bs cÖ‡kœi DËi
K †h cÖPÛ gv‡bi ej AwZ Aí mgq a‡i wµqv K‡i Zv‡K NvZej
e‡j|
L mylg e„ËvKvi MwZi ˆewkó¨ n‡jv :
1. G‡Z mg`ªæwZ we`¨gvb|
2. G‡Z mg‡KŠwYK †eM we`¨gvb|
3. Gi †KŠwYK Z¡iY k~b¨|
4. †K›`ªgyLx Z¡iY _v‡K |
M †`Iqv Av‡Q, iwki ˆ`N© Z_v e„ËvKvi c‡_i e¨vmva©, r = 2m
†KŠwYK †eM, = 2N
t = 2 3.1416 20 rad
60 sec = 2.0944 rads1
†Mvj‡Ki fi, m = 5 kg
†ei Ki‡Z n‡e, †K‡› Öi w`‡K Abyf‚Z ej = †K› ÖgyLx ej, Fc = ?
Avgiv Rvwb, Fc = m2r = 5kg (2.0944 rads1)2 2m
= 43.865 N (Ans.)
N N~Y©vqgvb Ae ’vq iwkwU wQ‡o †M‡j †MvjKwU Abyf‚wgK fv‡e
wbwÿß e¯‘i b¨vq AvPiY Ki‡e| Gi Dj ø¤^ †eM, vy0 = 0 Ges
Abyf‚wgK †eM vx0 = r = 2.0944 rads-1 2 m = 4.1888 ms-1
†MvjKwU f‚wg‡Z co‡Z t mgq jvM‡j,
y = y0 + vy0 t – 12 g t2
ev, 0 = 1.5 – 12 9.8 t2
ev, 4.9 t2 = 1.5
ev, t2 = 0.3061
t = 0.5533 s
G mgq AwZµvš Í Abyf‚wgK `~iZ¡,
x = vx0 t = 4.1888 ms-1 0.5533 s = 2.3177 m
myZivs cvidg©vi n‡Z `k©‡Ki ~iZ¡ 2.3177 m A‡cÿv †ewk n‡j
†MvjKwU `k©K‡K AvNvZ Ki‡e bv|
2 bs cÖ‡kœi DËi
K gnvwe‡k¦i †h‡Kv‡bv ywU KYv G‡`i ms‡hvRK mij †iLv eivei
ci¯úi‡K GKwU ej Øviv AvKl©Y K‡i, G e‡ji gvb KYv؇qi
f‡ii ¸Yd‡ji mgvbycvwZK Ges G‡`i ga¨eZ©x ~i‡Z¡i e‡M©i
e¨¯ÍvbycvwZK|
L Avgiv Rvwb, N~Y©b MwZi †ÿ‡Î,
†KŠwYK fi‡eM = RoZvi åvgK †KŠwYK †eM ev, L = I
†KŠwYK †eM GKK gv‡bi A_©vr = 1 n‡j, L = I 1 = I
myZivs GKK mg‡KŠwYK †e‡M N~Y©biZ †Kv‡bv e ‘i RoZvi åvgK
Gi †KŠwYK fi‡e‡Mi mgvb|
M AD n‡Z BC Z‡ji D”PZv h n‡j, h
AB = sin60
h = AB sin60 = 4m 3
2 = 3.464m
A n‡Z C we› y‡Z †cŠQv‡Z AwfKl© e‡ji weiæ‡× K…ZKvR,
EP = mgh = 30kg 9.8 ms2 3.464m = 1018.4J (Ans.)
N CD c‡_ †Kv‡bv Nl©Y bv _vK‡j CD Zj eivei wb‡Pi w`‡K
evjKwUi Z¡iY n‡Zv, g = g sin
n‡jv f‚wgi mv‡_ CD Z‡ji AvbwZ
sin = h
CD = 3.464m
5m = 0.6928
= sin1 (0.6928) = 43.85
g = 9.8 ms2 sin 43.85 = 6.79 ms2 < 9.8 ms2
myZivs †Kv‡bv Nl©Y bv _vK‡j CD eivei wb‡Pi w`‡K Z¡iY n‡Zv
6.79 ms2, Avi Nl©Y _vK‡j Z¡iY Av‡iv Kg n‡e|
AZGe, CD c‡_ bvgvi mgq evjKwUi Z¡iY AwfKl©R Z¡i‡Yi †P‡q
Kg n‡e|
3 bs cÖ‡kœi DËi
K †h mKj M¨vm mKj ZvcgvÎv I Pv‡c e‡qj I Pvj©‡mi m~Î †g‡b
P‡j, Zv‡`i‡K Av`k© M¨vm e‡j|
L Avgiv Rvwb, PV = nRT
P = 1 Ges n = 1 n‡j, V = RT ev, VT = R
myZivs GKK Pv‡c GK †gvj †Kv‡bv M¨v‡mi AvqZb ebvg cig
ZvcgvÎv †jLwP‡Îi Xvj Av`k© M¨vm aªæeK wb‡ ©k K‡i|
M †`Iqv Av‡Q,
XvKvq ﮋ ev‡j¦i ZvcgvÎv, 1 = 28.6C
Ges Av ª© ev‡j¦i ZvcgvÎv, 2 = 20C
evqyi ZvcgvÎvq †Møwmqv‡mi Drcv`K, G = 1.664
wkwkivsK n‡j, = 1 G(1 2)
= 28.6C 1.664 (28.6C 20C)
= 14.29C (Ans.)
GLv‡b,
y0 = 1.5 m
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
N XvKvq wkwkivs‡K (14.29C) m¤ú„³ ev®úPvc,
f = 11.99 + (13.63 11.99) 0.29
2 mm HgP
= 12.228 mmHgP
evqyi ZvcgvÎvq (28.6C) m¤ú„³ ev®ú Pvc,
F = 28.35 + (31.83 28.35) 0.6
2 mm HgP
= 29.394 mm HgP
XvKvq Av‡cwÿK Av ªZv, R = fF =
12.22829.394 100% = 41.6%
ivRkvnx‡Z wkwkivsK, = 1 G(1 2)
= 32.5C 1.625 (32.5C 22C)
= 15.437C
ivRkvnx‡Z evqyi ZvcgvÎvq (32.5C) m¤ú„³ ev®ú Pvc,
F = 35.66 + (39.90 35.66) 0.5
2 mm HgP
= 36.72 mmHgP
wkwkivs‡K (15.437C) m¤ú„³ ev®ú Pvc,
f = 11.99 + (13.63 11.99) 1.4375
2 mm HgP
= 13.169 mmHgP
ivRkvnx‡Z Av‡cwÿK Av`ª©Zv, R = f
F 100% =
13.16936.72 100%
= 35.86%
†h‡nZz 35.86% < 41.6% A_©vr ivRkvnx‡Z Av‡cwÿK Av ª©Zv Kg|
ZvB H e¨w³ ivRkvnx‡Z AwaKZi ¯^w¯ Í †eva Ki‡eb|
4 bs cÖ‡kœi DËi
K wbw`©ó K¤úv‡¼i Rb¨ †h Av`k© ZxeªZvi mv‡_ Zzjbv K‡i wewfbœ
ZxeªZvi k‡ãi k‡ãv”PZvi gvÎv wbY©q Kiv nq, Zv‡K H K¤úvs‡Ki
k‡ãi Rb¨ cÖgvY ZxeªZv ejv nq|
L †`Iqv Av‡Q, A I
B Gi ga¨eZx© †KvY, = 45
evgcÿ = A .
B = AB cos = AB cos45 =
AB
2
Ges Wvbcÿ = |A
B | = ABsin = ABsin45 =
AB
2
A .
B = |
A
B |
M †`Iqv Av‡Q, f‚c„ô n‡Z K…wÎg DcMÖ‡ni D”PZv, h = 2 106m
Rvbv Av‡Q, f‚c„‡ô AwfKl©R Z¡iY, g = 9.8 ms2
Ges c„w_exi e¨vmva©, R = 6.4 106m
K…wÎg DcMÖ‡ni Ae ’v‡b AwfKl©R Z¡iY, gh = g
R
R + h2
= 9.8 ms2
6.4 106m
6.4 106m + 2 106m
2 = 5.69 ms2 Ans.
N f‚c„ô n‡Z d = 3 106m MfxiZvq AwfKl©R Z¡iY,
gd = g
1
dR = 9.8 ms2
1
3 106m
6.4 106m = 5.206 ms2
K…wÎg DcMÖ‡ni Ae ’v‡b Ges Lwbi Af¨šÍ‡i mij †`vj‡Ki
†`vjbKvj h_vµ‡g Th Ges Td n‡j, mij †`vj‡Ki Kvh©Kix
Z¡i‡Yi m~Îvbymv‡i,
Th
Td =
gd
gh =
5.206 ms2
5.69 ms2 = 0.9565
Th < Td
myZivs Lwbi Af¨šÍ‡i †`vjKwU ax‡i Pj‡e|
5 bs cÖ‡kœi DËi
K †h MvwYwZK wP‡ýi Øviv GKwU †f±i ivwk‡K Ab¨ GKwU †¯‹jvi
ev †f±i ivwk‡Z iƒcvšÍi Kiv hvq ev †Kv‡bv cwieZ©bkxj †f±i
ivwki e¨vL¨v †`qv hvq Zv‡K †f±i Acv‡iUi e‡j|
L ej = fi Z¡iY = fi miY
mgq2
wbDUb (e‡ji GKK) = †KwR wgUvi
†m‡KÛ2 = †KwR
wgUvi/†m.2
ev, N = kgm s–2
M †`Iqv Av‡Q,
ek©vwUi wb‡ÿcY †eM, v0 = 30 ms1
ek©vwUi wb‡ÿcY †KvY, 0 = 30
AwfKl©R Z¡iY, g = 10 ms2
†ei Ki‡Z n‡e, me©vwaK D”PZv, ymax = ?
Avgiv Rvwb, ymax = v0
2sin20
2g = (30 ms1) (sin30)2
2 10 ms2 = 11.25m
(Ans.)
N ek©vi Abyf‚wgK cvjøv, R = v0
2sin20
g = (30 ms-1)2sin(230)
10 ms-2
= 77.94 m
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
Ges ek©vi DÇqb Kvj, T = 2v0sin0
g = 230 ms-1sin30
10 ms-2
= 3 s
3 s ci wkKvix †_‡K nwi‡Yi ~iZ¡,
x = x0 + vx0 t + 12 a t = 32.94 m +
12 10 ms-2 (3 s)2
= 32.94 m + 45 m = 77.94 m
myZivs ek©vwU nwiY‡K AvNvZ Ki‡e|
6 bs cÖ‡kœi DËi
K AwfK‡l©i cÖfv‡e †Kv‡bv cÖevnxi ga¨ w`‡q MwZkxj †Kv‡bv e ‘
m‡e©v”P †h †e‡M DcbxZ n‡j wbU ej k~b¨ nq Ges e ‘wU mg‡e‡M
Pj‡Z _v‡K, †m †eM‡K ejv nq cÖvwš ÍK‡eM ev AšÍ‡eM|
L wb‡¤œ ciek I Abybv` K¤úv‡¼i cv_©K¨ e¨vL¨v Kiv n‡jv-
ciek K¤úb Abybv` K¤úb
1. GKwU K¤úbiZ e ‘‡K
Ab¨ GKwU K¤úb ÿq
e¯‘i wbKU Avb‡j 2q
e¯‘wU‡Z K¤úb ïiæ nq
G‡K ciek K¤úb e‡j|
1. ciek K¤úbkxj e ‘
¯^vfvweK K¤úv‡¼i mgvb
n‡j e ‘ AwaK we¯ Ív‡i
Kw¤úZ nq| G ai‡bi
K¤úb‡K Abybv` e‡j|
2. ciek K¤ú‡b ch©ve„Ë
e‡ji ewkó¨ _vK‡Z nq|
2. wbR¯^ K¤úv¼ I ciek
m„wóKvix ch©vqe„Ë e‡ji
K¤úv‡¼i mgvb ˆewkó¨
AR©b K‡i|
M Y = 6 sin
8t
x25
= 6 sin250 (200t x)
G‡K cÖwgZ mgxKiY Y = Asin2
(vt x) Gi mv‡_ Zzjbv K‡i cvB,
Zi½‡eM, v = 200 cms1 Ges Zi½‰`N©¨, = 50 cm
K¤úv¼, f = v
=
200 cms1
50 cm = 4 Hz
N Zi‡½i we Ívi, a = 6 cm = 0.06 m
†`Iqv Av‡Q, gva¨‡gi NbZ¡, = 0.09 kgm3
Zi‡½i ZxeªZv, I = 22f2a2v
= 2 9.87 (4 Hz)2 (0.06m)2 0.09 kgm3 2 ms1
= 0.2047 Wm2
ZxeªZv †j‡fj, = 10 logII0
dB
= 10 log0.2047 Wm2
1012 Wm2 dB
= 113.1 dB < 120 dB
†h‡nZz gvbe KY© kÖæZ m‡e©v”P ZxeªZv 120 dB Gi †ewk, myZivs
DÏxc‡Ki Zi½wU kÖve¨|
6. wm‡jU †evW©-2015
1 bs cÖ‡kœi DËi
K GKwU †f±i ivwk‡K yB ev Z‡ZvwaK Dcvs‡k wef³ Kivi
cÖwµqvB n‡jv †f±i wefvRb|
L Avgiv Rvwb, †f±‡ii gvb A_ev w`K A_ev Df‡qi cwieZ©‡b
†f±i cwieZ©xZ nq| †eM n‡”Q †f±i ivwk| myZivs gvb cwieZ©b
bv n‡jI w`‡Ki cwieZ©‡b †eM cwieZ©xZ n‡e| mg ªæwZ‡Z eµc‡_
Pjvi mgq †e‡Mi gvb cwieZ©xZ bv n‡jI w`‡Ki cwieZ©b nq| Avi
†e‡Mi cwieZ©‡bi nvi‡K Z¡iY e‡j| myZivs Avgiv ej‡Z cvwi,
mij c‡_ mg ªæwZ‡Z Pjgvb †Kv‡bv e ‘i Z¡iY bv _vK‡jI eµ c‡_
mg`ªæwZ‡Z Pjgvb e ‘i Z¡iY _v‡K|
M K…wÎg DcMÖnwUi †K› ÖgyLx Z¡iY, Kÿc‡_ Gi Ici cÖhy³
AwfKl©R Z¡i‡Yi mgvb|
†`Iqv Av‡Q,
MÖ‡ni fi, M = 6 1024 kg
MÖ‡ni e¨vmva©, R = 6.4 106m
MÖ‡ni c„ô n‡Z Kÿc‡_i D”PZv, h = 700 km = 700 103m
Kÿc‡_i Ae ’v‡b AwfKl©R Z¡iY, g = GM
(R + h)2
= 6.673 1011 Nm2kg2 6 1024 kg
(6.4 106 m + 700 103 m)2
= 7.942 ms2 (Ans.)
N Kÿc‡_ cwiågYKv‡j K…wÎg DcMÖnwUi
MwZ‡eM, v = GM
R + h
= 6.673 1011 Nm2kg2 6 1024 kg
6.4 106m + 700 103 m
= 7509 ms1
K…wÎg DcMÖnwU‡K gnvk~‡b¨ wgwj‡q hvIqvi Rb¨ cÖ‡qvRbxq me©wb¤œ
kw³
hRhRhR
drrGMmdrr
GMmFdrW 2
2
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m
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
hR
GMm
hRGMm
rGMm
hR
111
= GMm
R + h
G cwigvY KvR Ki‡Z cÖ‡qvRbxq me©wb¤œ †eM ve n‡j
12 mve
2 = GMmR + h
ve = hR
GM
2= 2
hR
GM
= 1.415709 ms–1
= 8049 ms–1
myZivs ejv hvq, DcMÖnwU k~‡b¨ wgwj‡q hvIqvi m¤¢vebv †bB|
2 bs cÖ‡kœi DËi
K AwfK‡l©i cÖfv‡e k~b¨ ’v‡b f‚wgi mv‡_ Zxh©Kfv‡e Dc‡ii w`‡K
wbwÿß e¯‘‡K cÖwÿß e¯‘ ev cÖvm e‡j|
L †h †f±‡ii cv`we› y mywbw`©ó bq, Zv‡K ¯vaxb †f±i e‡j|
†h‡nZz ¯vaxb †f±‡ii cv`we› y mywbw`©ó bq, ZvB GB cv`we› y
g~jwe›`y‡Z Aew¯’Z nIqvi cÖ‡qvRb †bB| G Kvi‡YB ^vaxb †f±‡ii
cv`we›`y g~jwe›`y‡Z bq|
M †`Iqv Av‡Q,
cÖhy³ ej, F = 500N
j¤^ `~iZ¡, r = 1m
myZivs wb‡Y©q UK©, = Fr = 500 m 1m = 500 Nm (Ans.)
N g‡b Kwi, mgMÖ `‡Ði fi M Ges ˆ`N©¨
l
Zvn‡j GKK ˆ`‡N©¨i fi = Ml
Ges dx ÿz`ªvwZÿz`ª As‡ki fi = Ml dx
N~Y©b Aÿ `‡Ði cÖv‡šÍ Aew¯’Z n‡j Ges ˆ`‡N©¨i mv‡_ j¤ n‡j Gi
mv‡c‡ÿ `‡Ði RoZvi åvgK, I = lo(x2)
Ml dx
= Ml
l o
x2dx = Ml x3
3
l o
= Ml
l3
3 = Ml2
3
wKš‘ N~Y©b Aÿ `‡Ði ga¨we›`yMvgx n‡j Ges
`‡Ði ˆ`‡N©¨i mv‡_ j¤ n‡j RoZvi åvgK,
I =
l/2
l/2
x2 Ml dx
= Ml x3
3
l/2
l/2
= Ml
l3
24 + l3
24
= Ml2
12
†h‡nZz Ml2
3 > Ml2
12
A_©vr I > I
myZivs N~Y©b Aÿ ‡Ði cÖvš Íwe›`y‡Z Aew¯’Z n‡j RoZvi åvgK †ewk
n‡e|
3 bs cÖ‡kœi DËi
K †Kv‡bv hš¿ mieivnK…Z kw³i hZ Ask Kv‡R iƒcvš ÍwiZ Ki‡Z
cv‡i Zv‡K H h‡š¿i Kg©`ÿZv e‡j| A_ev †Kv‡bv hš¿ KZ…©K
K…ZKvR I mieivnK…Z kw³i AbycvZ‡K H h‡š¿i Kg©`ÿZv e‡j|
L †Kv‡bv e¯‘i Ici ej cÖ‡qv‡Mi d‡j e‡ji w`‡K mi‡Yi Dcvsk
_vK‡j e‡ji Øviv KvR ev abvZ¥K KvR nq| e‡ji Øviv Kv‡Ri
d‡j e ‘i MwZkw³ e„w× cvq| †Kv‡bv e ‘ AwfK‡l©i cÖfv‡e wb‡P
†b‡g Avmv-e‡ji Øviv Kv‡Ri D`vniY|
M h D”PZvq wefe kw³ mgh Ges MwZkw³ 0| myZivs †gvUkw³
mgh| aiv hvK, y D”PZvq wefe kw³ MwZkw³i wظY n‡e| y D”PZvq wefe kw³ mgy Ges MwZkw³ Ek n‡j kw³i wbZ¨Zvi bxwZ
†_‡K cvB,
mgy + Ek = mgh
Ek = mgh mgy
kZv©bymv‡i, 2(mgh mgy) = mgy
ev, 3mgy = 2mgh
y = 2h 3 =
2375 m3 = 250 m
myZivs f‚wg n‡Z 250 m D”PZvq e¯‘i wefe kw³ MwZ kw³i wظY
n‡e|
N †h‡nZz Kv‡mg Ges gwb‡ii fi wfbœ, ZvB Kv‡m‡gi mgvb ÿgZv
cÖ‡qvM Ki‡j gwbi GKB mg‡q KvRwU Ki‡Z cvi‡e bv| Z‡e gwbi
wKQyUv Kg ÿgZv cÖ‡qvM Ki‡j GKB mg‡q KvRwU Ki‡Z cvi‡e|
GLv‡b, e ‘mn gwb‡ii fi,m = 55 kg + 10kg = 65 kg
AwfKl©R Z¡iY, g = 9.8 ms2
AwZµvš Í D”PZv, h = 375m
mgqKvj, t = 40 min = 40 60 sec = 2400 sec
gwb‡ii CwáZ ÿgZv P n‡j,
P = mgh
t
= 65 kg 9.8 ms2 375m
2400 sec
l
dx
x
dx
x
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
= 99.53 watt
myZivs gwbi 99.53 watt ÿgZv cÖ‡qvM Ki‡j GKB mg‡q KvRwU
Ki‡Z cvi‡e|
4bs cÖ‡kœi DËi
K KwVb I Zi‡ji ¯úk©we›`y †_‡K eµ Zij Z‡j Aw¼Z ¯úk©K
KwVb c`v‡_©i mv‡_ Zi‡ji wfZi †h †KvY Drcbœ K‡i Zv‡K D³
KwVb I Zi‡ji ¯úk© †KvY e‡j|
L cvwbi AYy I KPzcvZvi AYyi ga¨Kvi AvmÄb ej A‡cÿv cvwbi
AYymg~‡ni ga¨Kvi mskw³ ej e„nËi gv‡bi| ZvB e„wói †duvUv
KPzcvZv‡K wfRvq bv| cÿvšÍ‡i cvwbi AYy I Avg cvZvi AYyi
ga¨Kvi AvmÄb ej A‡cÿv cvwbi AYymg~‡ni ga¨Kvi mskw³ ej
ÿz`ªZi gv‡bi| ZvB e„wói †duvUv AvgcvZv‡K wfRvq|
M †`Iqv Av‡Q,
B ˆKwkK b‡ji e¨vmva©, r = 0.4 mm
2 = 0.2 103m
cvwbi ¯úk©‡KvY, = 2
c„ô Uvb, T = 72 103 Nm1
AwfKl©R Z¡iY, g = 9.8 ms2
cvwbi NbZ¡, = 1000 kgm3
†ei Ki‡Z n‡e, B b‡j cvwbi D”PZv, h = ?
Avgiv Rvwb, T = rgh
2 cos
h = 2T cos
rg
= 2 72 103 Nm1 cos2
0.2 103 m 1000 kgm3 9.8 ms2
= 0.0734m
= 7.34 cm (Ans)
N ˆKwkKZvi ZË¡ n‡Z Avgiv Rvwb, Zi‡ji c„ôUvb,
T = r g h
2cos
ev, h = 2Tcos
r g
A I B b‡ji e¨vmva© h_vµ‡g rA I rB Ges cvwbi D”PZv hA I hB n‡j,
hA = 2Tcos
rA g
hB = 2Tcos
rB g
hA
hB =
rB rA
†h‡nZz rB < rA †m‡nZz hA > hB
myZivs Avgiv ej‡Z cvwi b‡ji e¨vmv‡a©i wfbœZvB b‡ji wfZi
Zi‡ji D”PvZvi wfbœZvi KviY| †h b‡ji e¨vmva© hZ Kg †m b‡j
Zi‡ji D”PZv ZZ †ewk|
5 bs cÖ‡kœi DËi
K Zi½ mÂvj‡bi Awfgy‡Li mv‡_ GKK †ÿÎd‡ji ga¨ w`‡q cÖwZ
†m‡K‡Û †h cwigvY kw³ cÖevwnZ nq Zv‡K Zi‡½i ZxeªZv e‡j|
L ˆmb¨iv eªx‡Ri Dci w`‡q gvP© K‡i †M‡j eªx‡Ri Ici cÖhy³ ej
AZ¨waK gv‡bi nq| G e‡ji K¤úv¼ eªx‡Ri ¯vfvweK K¤úv‡¼i
mgvb ev KvQvKvwQ n‡j eªxRwU‡Z Abybv` m„wó n‡e Ges GwU m‡e©v”P
we¯Ívi mnKv‡i Kw¤úZ n‡e| ZLb eªxRwU †f‡½ hvevi m¤¢vebv
_v‡K| G Kvi‡Y GK mv‡_ A‡bK¸‡jv ˆmb¨ eªx‡Ri Dci w`‡q gvP©
K‡i hvIqv mwVK bq|
M evqy gva¨‡g k‡ãi †eM, va = 352 ms1
k‡ãi K¤úv¼, f = 300 Hz
evqy‡Z Zi½‰`N©¨, a = va
f = 352 ms1
300 Hz = 1.173m
cvwb‡Z Zi½‰`N©¨ †ewk n‡e
cvwb‡Z Zi½‰`N©¨, w = a + 4.16m
= 1.173m + 4.16m
= 5.33 m
cvwb‡Z k‡ãi †eM, vw = fw
= 300 Hz 5.33m
= 1600 ms1 (Ans.)
N ÔMÕ As‡ki we‡kølY g‡Z,
cvwb‡Z k‡ãi †eM > evqy‡Z k‡ãi †eM
GLb cÖgvY Ki‡ev †h,
cvwb‡Z k‡ãi ZxeªZv > evqy‡Z k‡ãi ZxeªZv
evZv‡mi kãZi‡½i ZxeªZv, Ia = 22n2a2ava
= 2 9.87 (300 Hz)2 (0.25 102m)2 1.293 kgm3 352 ms1
= 5054 Wm2
cvwb‡Z kãZi‡½i ZxeªZv,
Iw = 22n2a2 wvw
= 2 9.87 (300 Hz)2 (0.25 102m)2 1000 kgm3 1600 ms1
= 1.7766 107 Wm2 >> 5054 Wm2 (= Ia)
MvwYwZK we‡køl‡Y †`Lv hvq,
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
Iw >> Ia
A_©vr cvwb‡Z k‡ãi ZxeªZv > evqy‡Z k‡ãi ZxeªZv
myZivs mvjv‡gi e³e¨ mwVK|
6 bs cÖ‡kœi DËi
K †h mKj M¨vm mKj ZvcgvÎv I Pv‡c e‡qj I Pvj©‡mi m~Î †g‡b
P‡j, Zv‡`i‡K Av`k© M¨vm e‡j|
L †Kv‡bv ¯’v‡b evZv‡mi Av‡cwÿK Av ª©Zv 70% ej‡Z eySvq, H
ZvcgvÎvq H ’v‡bi evZvm‡K m¤ú„³ Ki‡Z †h cwigvY Rjxq ev®ú
`iKvi Zvi kZKiv 70 fvM Rjxq ev®ú H gyn~‡Z© H ¯’v‡bi evqy‡Z
i‡q‡Q|
M †`Iqv Av‡Q,
Avw` AvqZb, V1 = 5.7 104m3
Avw` Pvc, P1 = 0.64 m HgP
Avw` ZvcgvÎv, T1 = 39C = (39 + 273)K = 312 K
P‚ovš Í Pvc, P2 = 0.76 m HgP
P‚ovš Í ZvcgvÎv, T2 = 273 K
†ei Ki‡Z n‡e, P‚ovšÍ AvqZb, V2 = ?
Avgiv Rvwb, P1V1
T1 =
P2V2
T2
V2 = P1V1 T2
P2T1
= 0.64m 5.7 104m3 273K
0.76m 312K
= 4.2 104m3 (Ans.)
N †h‡nZz 4.2 104m3 < 5.7 104m3
myZivs M¨vmwUi AvqZb n«vm †c‡q‡Q|
T cig ZvcgvÎvq n †gvj M¨v‡mi MwZkw³, E = 32 nRT
n (†gvj msL¨v) AcwiewZ©Z _vK‡j, E T
DÏxc‡Ki NUbvq, M¨v‡mi fi Z_v †gvj msL¨v (n) AcwiewZ©Z|
myZivs cig ZvcgvÎvi n«v‡m (T1 = 312K n‡Z T2 = 273K) MwZkw³I
n«vm cv‡e| A_©vr †bnv‡ji e³e¨ mwVK|
7. h‡kvi †evW©-2015
1 bs cÖ‡kœi DËi
K yB ev Z‡ZvwaK GKB RvZxq †f±i †hvM Ki‡j †h †f±i cvIqv
hvq Zv‡K †f±i¸‡jvi jwä †f±i e‡j|
L AwfKl© e‡ji †ÿ‡Î GK we›`y n‡Z Aci we› y‡Z e¯‘i Mg‡bi
d‡j K…ZKvR, c‡_i Ici wbf©i K‡i bv eis Avw` I Aš Íwe›`yi Ici
wbf©i K‡i| e ‘wU cybivq Avw` we›`y‡Z wd‡i G‡j K…Z KvR k~b¨ nq
Ges kw³i AcPq N‡U bv| ZvB AwfKl© ej AmsiÿYkxj ej bq
A_©vr msiÿYkxj ej|
M
Uªwji MwZ m„wóKvix ej = cÖhy³ e‡ji Abyf‚wgK Dcvsk|
= Fsin30 = 10 N 0.5 = 5 N (Ans.)
N
wPÎ-1 Abymv‡i hLb Uªwj‡K †Vjv n‡”Q ZLb e‡ji Dj¤ Dcvsk
Fcos30 wb‡Pi w`‡K wµqv Ki‡Q| Uªwji IRb W n‡j, wb¤œgyLx †gvU
ej = W + Fcos30
G‡Z f‚wgi cÖwZwµqv ej e„w× cvq d‡j Nl©Y ej †ewk nq KviY
Nl©Y ej Awfj¤^ cÖwZwµqvi mgvbycvwZK| Aci c‡ÿ UªwjwU‡K wPÎ-
30
Fsin30
Fcos30
Uªwji Dci cÖhy³ ej, F
30
Fsin30
Fcos30
F
wPÎ-1
30
Fsin30
Fcos30
F
wPÎ-2
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m
c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
2 Abymv‡i Uvbv n‡j e‡ji Dj¤^ Dcvsk Fcos30 Dc‡ii w`‡K wµqv
K‡i| d‡j wb¤œgyLx †gvU ej = W – Fcos30
G‡Z f‚wgi cÖwZwµqv ej n«vm cvq d‡j Nl©Y ej Kg nq| G Kvi‡Y
Uªwj †Vjvi †_‡K Uvbv mnR nq| G Kvi‡YB †`vKvb`vi mvwenv‡K
Uªwj Uvb‡Z e‡jwQj|
2 bs cÖ‡kœi DËi
K †Kv‡bv e ‘i Ici Av‡ivwcZ ch©ve„Ë ¯ú›`‡bi K¤úv¼ e ‘i
¯^vfvweK K¤úv‡¼i †P‡q wfbœZi n‡j cÖ_‡g AwbqwgZ fv‡e Kw¤úZ
nq c‡i Av‡ivwcZ K¤ú‡bi K¤úv‡¼ Kw¤úZ nq GB ai‡bi
K¤úb‡K ciek K¤úb e‡j|
L GKwU ¯‡i †h GKvwaK myi _v‡K Zvi g‡a¨ me‡P‡q wb¤œ
K¤úv‡¼i myi‡K †gŠwjK myi e‡j Ges Ab¨ myi¸‡jv‡K Dcmyi e‡j|
†h mKj Dcmy‡ii K¤úv¼ †gŠwjK my‡ii mij ¸wYZK Zv‡`i‡K
mg‡gj e‡j| myZivs mKj mg‡gjB Dcmyi wKš‘ mKj Dcmyi
mg‡gj bq|
M †`Iqv Av‡Q,
A Zv‡ii Uvb, T1 = 100N
B Zv‡ii Uvb, T2 = 200N
A Zv‡ii K¤úv¼, f1 = 50 Hz
†ei Ki‡Z n‡e, B Zv‡ii K¤úv¼, f2 = ?
Zvi¸‡jvi ˆ`N©¨ Ges fi mgvb e‡j, f T
ev, f2
f1 =
T2
T1
f2 = f1
T2
T1 = 50 Hz
200 N100 N = 70.7 Hz
N A Zv‡ii Rb¨ 1 = 12l
T1
C Zv‡ii Rb¨ 3 = 12l
T3
GLb, 3
1 =
T3
T1 ev, 3 = 1
T3
T1 = 50
250100
3 = 79 Hz
AZGe, C Zv‡ii K¤úv¼, 3 = 79 Hz
DÏxcK n‡Z, A Zv‡ii K¤úv¼, 1 = 50 Hz
A I C Zvi ywU‡K GK‡Î Kw¤úZ Ki‡j Drcbœ weU, N = 3 – 1 = 79Hz – 50 Hz = 29 Hz
cÖwZ †m‡K‡Û Drcbœ weU msL¨v Lye †ewk n‡j, k‡ãi ZxeªZvi n«vm-
e„w× G‡Zv ªæZ nq †h Zv Dcjwä Kiv hvq bv| Kv‡b GKUvbv kã
†kvbv hvq bv| cixÿv K‡i †`Lv †M‡Q †h, we‡Ui msL¨v †m‡K‡Û 15
Gi †ewk n‡j Kv‡b Zv Dcjwä Kiv m¤¢e bq|
†h‡nZz 29Hz > 15 Hz
AZGe, A I C Zvi‡K GK‡Î Kw¤úZ Ki‡j weU †kvbv hvq bv|
3 bs cÖ‡kœi DËi
K mg‡qi e¨eavb k~‡b¨i KvQvKvwQ n‡j e ‘i mi‡Yi nvi‡K Gi
ZvrÿwYK †eM e‡j|
L GKRb A¨v_‡jU js Rv¤ú †`qvi c~‡e© †ek wKQy ~i †`Šo †`b|
Gi D‡Ï‡k¨ n‡jv, MwZRoZv AR©b Kiv hvi `iæb †m Rv¤ú †`qvi
ci †ek LvwbKUv ~iZ¡ AwZµg Ki‡Z mÿg n‡eb|
M †`Iqv Av‡Q,
ey‡j‡Ui Avw`‡eM, 0 = 0 ms1
†kl †eM, v = 750 ms1
miY, s = 0.6m
†ei Ki‡Z n‡e, Mo Z¡iY, a = ?
Avgiv Rvwb, v2 = 02 + 2as
ev, 2as = v2 02
a = v2 0
2
2s = (750 ms1)2 (0 ms1)2
2 0.6m = 468750 ms2 (Ans.)
N Avgiv Rvwb, Abyf‚wgK cvj øv, g
vR 0
2
0 2sin
ey‡jUwU cÖvm we‡ePbv K‡i cvB,
Gi wb‡ÿcY †eM, v0 = 750 ms–1
GLb Abyf‚wg‡Ki mv‡_ †KvY,1 = 30 Ges 2 = 60 we‡ePbv K‡i
AwZµvš Í `~iZ¡ †ei Kwi,
1 = 30 Gi Rb¨, R1 = v0
2 sin 21
g
= (750ms–1)2 sin (2 30)
9.8ms–1
= 49708.09 m
Avevi, 2 = 60 Gi Rb¨, R2 = (750ms–1)2 sin (2 30)
9.8ms–2
= 49708.09m
GLv‡b,R1 = R2
AZGe, wfbœ wfbœ †Kv‡Yi Rb¨ GKB †e‡M wbwÿß e ‘i AwZµvšÍ
`~iæZ¡ mgvb|
6 bs cÖ‡kœi DËi
K †Kv‡bv e ‘i Ici cÖhy³ ej Øviv K…ZKvR e ‘i MwZkw³i
cwieZ©‡bi mgvb|
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
L †Kv‡bv e ‘i MwZkw³ = 12 mv2; e¯‘i fi m KL‡bvB FYvZ¥K
nq bv Ges v abvZ¥K ev FYvZ¥K hvB †nvK bv †Kb v2 me©`vB
abvZ¥K nq| ZvB †Kv‡bv e ‘i MwZkw³ FYvZ¥K n‡Z cv‡i bv|
M †`Iqv Av‡Q,
c„w_exi fi, M = 6 1024 kg
c„w_exi e¨vmva©, R = 6.4 106m
f‚c„ô n‡Z K…wÎg DcMÖ‡ni D”PZv, h = 690 km = 6.90 105m
†ei Ki‡Z n‡e, K…wÎg DcMÖnwUi ˆiwLK †eM, v = ?
Avgiv Rvwb, v = GM
R + h =
6.673 1011 Nm2kg2 6 1024 kg
6.4 106m + 6.90 105 m
= 7514.7 ms1 (Ans.)
N DÏxc‡K ewY©Z Ae¯’vq K…wÎg DcMÖnwUi ch©vqKvj T n‡j,
T = 2 (R + h)
v = 2 3.1416 (6.4 106m + 6.9 105m)
7514.7 ms1
= 5928 sec
f‚c„ô n‡Z K…wÎg DcMÖnwUi D”PZv h = 800 km = 8 105m n‡j,
Gi ch©vqKvj n‡e, T = 2(R + h)3
GM
= 2 3.1416 (6.4 106m + 8 105m)3
6.673 1011 Nm2kg2 6 1024 kg
= 6066. 56 sec
†h‡nZz 6066 sec 5928 sec
A_©vr T T
myZivs DcMÖnwU‡K c„w_ex c„ô †_‡K 800 km miv‡j †mwUi
cwiågYKv‡ji cwieZ©b NU‡e (e„w× cv‡e)|
5 bs cÖ‡kœi DËi
K ¯ú›`biZ †Kv‡bv e ‘KYvi MwZ hw` Ggb nq †h, Gi MwZc_
mij‰iwLK Ges Gi †h‡Kv‡bv gyn~‡Z©i Z¡iY, mvg¨ve ’vb n‡Z mi‡Yi
mgvbycvwZK wKš‘ wecixZgyLx nq, Z‡e H e ‘KYvi MwZ‡K mij
Qw›`Z MwZ e‡j|
L GKwU w¯úÖs Gi w¯úÖs aªæeK 2.5 N/m ej‡Z eySvq, G‡K Gi
mvg¨ve ’vb n‡Z 1m cÖmvwiZ Ki‡Z 2.5 N ej cÖ‡qvRb nq|
M †`Iqv Av‡Q,
cvnv‡oi P‚ovq cÖwZ NÈvq cÖvß Aa©‡`vjb msL¨v = 3600 30
= 3570
†h‡nZz 3570 ¸‡jv Aa©‡`vjb †`q 3600 †m‡K‡Û
2wU Aa©‡`vjb †`q 3600 2
3570 †m‡K‡Û
= 2.0168 sec
BnvB wb‡Y©q †`vjbKvj| (Ans.)
N c„w_exi e¨vmva©, R = 6400 km = 6.4 106m
f‚c„‡ô AwfKl©R Z¡iY, g = 9.8 ms2
awi, cvnv‡oi D”PZv = hm
f‚c„‡ô I cvnv‡oi P‚ovq †`vjbKvj h_vµ‡g T1 I T2 n‡j,
T1
T2 =
g2
g1 =
R2
(R + h)2 = R
R + h
ev, R + h
R = 1 + hR =
T2
T1 =
2.01682 = 1.0084
hR = 1.0084 – 1
h = 0.0084 R = 0.0084 6.4 106m = 53760 m
myZivs DÏxc‡Ki Z‡_¨i wfwˇZ cvnv‡oi D”PZv wbY©q Kiv m¤¢e|
6bs cÖ‡kœi DËi
K mgy`ªc„‡ô 45 Aÿvs‡k 273.15K ZvcgvÎvq Dj¤^fv‡e Aew ’Z
0.76m D”PZvwewkó weï× cvi` ͤ¢ †h Pvc †`q Zv n‡jv cÖgvY ev
¯^vfvweK Pvc|
L M¨vm ej‡Z Ggb c`v_© †evSvq hvi ¯vfvweK Ae ’v ev®úxq|
†hgb: nvB‡Wªv‡Rb, Aw·‡Rb| Avi ¯vfvweKfv‡e ev®ú ej‡Z
†Kv‡bv KwVb ev Zij c`v_©‡K Zvc w`‡j †h Ae¯’v cvIqv hvq Zv‡K
†evSvq|
GKB ZvcgvÎv e„wׇZ mKj M¨v‡mi cÖmviY GKB nq| ev‡®úi
†ÿ‡Î Ggb †`Lv hvq bv| †Kv‡bv M¨vmxq c`v‡_©i ZvcgvÎv Gi µvwš Í
ZvcgvÎv A‡cÿv Kg n‡j Zv‡K ev®ú e‡j| †Kv‡bv c`v_© Gi µvwš Í
ZvcgvÎv A‡cÿv AwaK ZvcgvÎvq _vK‡j Zv‡K M¨vm e‡j| mvaviY
ZvcgvÎvq M¨vm‡K Pvc cÖ‡qv‡M Zi‡j cwiYZ Kiv hvq bv, ev®ú‡K
hvq|
M †`qv Av‡Q, ﮋ ev‡j¦i ZvcgvÎv, 1 = 20C
Av`©ª ev‡j¦i ZvcgvÎv, 2 = 12.8C
20C G †MøBmvi Drcv`K, G = 1.79
wkwkiv¼, = ?
Rvbv Av‡Q,
= 1 – G(1 – 2)
= 20 – 1.79 (20 – 12.8)
= 7.112C
myZivs H w`‡bi wkwkiv¼ 7.112C| (Ans.)
N (8 – 7)C = 1C Gi Rb¨ m¤ú„³ Rjxq ev®úPv‡ci cv_©K¨
= (8.1 – 7.5) 10–3
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c`v_©weÁvb cÖ_g cÎ: m„Rbkxj As‡ki As‡Ki mgvavb
= 0.6 10–3 cvi`Pvc|
0.112C Gi Rb¨ ev®úPv‡ci e„w×
= 0.0672 10–3 cvi`Pvc
wkwkiv¼ = 7.112C [(M) DËi: †_‡K]
wkwkiv¼ 7.112C G m¤ú„³ Rjxq ev®ú Pvc,
f = (7.5 + 0.0672) 10–3
= 7.5672 10–3 Hg
evqyi ZvcgvÎv 20C G Rjxq ev®ú Pvc, F = 17.4 10–3 mHg
Avgiv Rvwb,
Av‡cwÿK Av`©ªZv, R = fF 100%
= 43.49%
Av‡cwÿK Av ©ªZv 43.49%| ZvB ejv hvq H w`b H ’v‡bi
AvenvIqv ﮋ I †iŠ‡ ªv¾¡j _vK‡e|
8. ewikvj †evW©-2015
1 bs cÖ‡kœi DËi
K hLb †Kv‡bv e ‘ GK Ae ’vb †_‡K Ab¨ †Kv‡bv Ae ’v‡b Mgb
K‡i ZLb Avw` Ae ’vb‡K cv`we›`y Ges †kl Ae ’vb‡K kxl© we› y
we‡ePbv K‡i †h †f±i cvIqv hvq Zv‡K miY †f±i e‡j|
L aiv hvK,
†bŠKvi B we›`y‡Z
¸Y †eu‡a GK e¨w³
BM eivei F e‡j
Uvb‡Q| G ej ywU
Dcvs‡k wef³
n‡e| GKwU Dcvsk
Fsin, hv †bŠKv‡K K‚‡ji w`‡K wb‡q †h‡Z _vK‡e| wKš‘ gvwS b`xi
†mªvZ‡K e¨envi K‡i ˆeVvi mvnv‡h¨ wecixZ w`‡K GKwU ej Drcbœ
K‡i hv Fsin AskwU‡K cÖkwgZ Ki‡e| Aci Dcvsk Fcos, hv
†bŠKv‡K mvg‡bi w`‡K wb‡q hv‡e|
M †`Iqv Av‡Q,
wb‡ÿcY †eM, v0 = 30 ms1
wb‡ÿcY †KvY, 0 = 30
AwfKl©R Z¡iY, g = 10 ms2
†ei Ki‡Z n‡e, me©vwaK D”PZv, ymax = ?
Avgiv Rvwb, ymax = v0
2sin2 0
2g
= (30 ms1)2 (sin30)2
2 10 ms1
= 11.25 m (Ans.)
N Abyf‚wgK cvjøv, R = v0
2 sin20
g
= (30 ms1)2 sin (2 30)
10 ms2
= 77.94m
wb‡ÿc‡Yi t mgq c‡i cÖvmwU y = 4m D”PZvq Avm‡j,
y = v0sin0t 12 gt2
ev, 4 = 30sin30t 12 10t2 [GKKmg~n Dn¨ †i‡L]
ev, 5t2 15t + 4 = 0
t = 52
4541515 2
=
10
8022515
= 10
14515=
10
04.1215 = 0.296 s or 2.704 s
e¯‘wU t1 = 0.296 s mg‡q a eivei Ges t2 = 2.704 s mg‡q b eivei
Dc‡i Ae ’vb Ki‡e| myZivs ab ~iZ¡ AwZµg Ki‡Z cÖ‡qvRbxq
mgq t = (2.704 – 0.296) s = 2.408 s
myZivs ab As‡ki ˆ`N©¨ = †e‡Mi Abyf‚wgK Dcvsk mgq
= 30cos30 2.408 = 30 0.8660254 2.408
= 62.56 m
myZivs R t ab = 77.94 t 62.56
2 bs cÖ‡kœi DËi
K †Kv‡bv e ‘i Ici cÖhy³ ej Øviv K…ZKvR e ‘i MwZ kw³i
cwieZ©‡bi mgvb|
L mij †`vj‡Ki †KŠwYK we¯ Ívi AbwaK 4 n‡j Gi MwZ mij
Qw›`Z nq| KviY mij Qw›`Z MwZi GKwU ˆewkó¨ n‡jv- GwU
mij‰iwLK MwZ| wKš‘ †KŠwYK we¯ Ívi 4 Gi †ewk n‡j mij
†`vj‡Ki MwZc_ Avi mij‰jwLK _v‡K bv| myZivs GKwU mij
†`vj‡Ki †KŠwYK we Ívi 3 n‡j Gi MwZ mij Qw›`Z n‡e|
M e¯‘wUi fi K‡g hvevi c~‡e©,
myZvi ˆ`N©¨ Z_v e„ËvKvi c‡_i e¨vmva©,
r = 90cm = 0.9m
†KŠwYK †eM, = 2N
t = 2 100
60 rad.s1 = 10.472 rad.s1
F
Fcos
Fsin
M
B
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†ei Ki‡Z n‡e, †K› ÖgyLx Z¡iY, ac = ?
Avgiv Rvwb, ac = 2r
= (10.472 rad.s1)2 0.9m
= 98.7 ms2 (Ans.)
N †L‡jvqv‡oi nvZ Øviv cÖhy³ Uvb Z_v †K› ÖgyLx ej AcwiewZ©Z
gv‡bi|
g‡b Kwi, e ‘i Avw` fi = m
Zvn‡j Avw` Ae ’vq,
†K›`ÖgyLx ej Z_v myZvi Uvb, Fc = mac = m 98.7
= 98.7m N
fi GK-Z…Zxqvsk K‡g †M‡j Aewkó fi, m = m m3 =
2m3
G‡ÿ‡Î myZvi bZzb ˆ`N©¨ r n‡j, m2r = m2r
ev, mr = mr
ev, r = mr
m =
mr2m/3 =
32 r
myZivs myZvi ˆ`‡N©¨ cwieZ©b (e„w×) = r r
r =
32 r r
r
= 12 = 50%
3 bs cÖ‡kœi DËi
K wbw`©ó ¯’v‡bi g‡a¨ mxgve× †Kvb Zi½ Ges H mxgvi g‡a¨ Gi
cÖwZdwjZ Zi‡½i DcwicvZ‡bi d‡j m„ó jwä Zi½‡K w¯’i Zi½
e‡j|
L †kÖwYK‡ÿi k‡ãi ZxeªZv 10-6 Wm-2 ej‡Z eySvq, †kÖwYK‡ÿi
†h †Kv‡bv ’v‡b k‡ãi w`‡Ki mv‡_ j¤^ GK eM©wgUvi †ÿÎd‡ji
ga¨ w`‡q cÖwZ †m‡K‡Û 10-6 J kã kw³ mÂvwjZ nq|
M †`Iqv Av‡Q,
1g myikjvKv n‡Z m„ó k‡ãi ZxeªZv, I = 107 Wm2
†ei Ki‡Z n‡e, ZxeªZv †j‡fj, = ?
Avgiv Rvwb, = 10 logII0
dB
= 10 log107 Wm2
1012Wm2 dB
= 50 dB
k‡ãi ZxeªZv = 50 dB (Ans.)
N cÖ_g myikjvKvi K¤úv¼, f1 = 450 Hz
wØZxq myikjvKvi K¤úv¼, f2 > 450 Hz
[ Gi evû ÿ‡q †M‡Q]
†h‡nZz cÖ_g myikjvKv ywU GKB mv‡_ úw›`Z Ki‡j cÖwZ †m‡K‡Û
N = 6wU exU Drcbœ nq|
myZivs wØZxq myikjvKvi K¤úv¼, f2 = f1 + N
= 450 Hz + 6 Hz
= 456 Hz
G‡ÿ‡Î Z…Zxq myikjvKvi K¤úv¼ 450 Hz n‡Z e„nËi wKš‘ 456 Hz n‡Z ÿz`ªZi|
Gi K¤úv¼ f3 n‡j, cÖ`Ë kZ©g‡Z,
Drcbœ exU msL¨v = f3 f1 = f3 f2
ev, f3 f1 = f2 f3
ev, 2f3 = f1 + f2
f3 = f1 + f2
2 = 450 Hz + 456 Hz
2
= 453 Hz
myZivs DÏxc‡Ki Z_¨mg~n n‡Z 3q myikjvKvwUi K¤úv¼ wbY©q Kiv
m¤¢e|
4bs cÖ‡kœi DËi
K †Kv‡bv Zij c„‡ôi †ÿÎdj cÖwZ eM© GKK (m2) e„w× Ki‡Z
†h cwigvY kw³i cÖ‡qvRb nq Zv‡K H Zi‡ji c„ôkw³ e‡j|
L c„w_ex‡Z w`‡bi msL¨v ej‡Z GLv‡b m~‡h©i PZzw`©‡K c„w_exi
AveZ©bKvj (T) eySv‡bv n‡q‡Q| c„w_ex I m~‡h©i ga¨eZx© Mo ~iZ¡
R n‡j, MÖn m¤úwK©Z †Kcjv‡ii 3q m~Îvbymv‡i, T2 R3
M †`Iqv Av‡Q,
A Zv‡ii Avw` ˆ`N©¨, L = 0.80m
Avw` e¨vm, D = 2 0.5 mm = 1 mm = 103m
ˆ`‡N©¨i e„w×, l = 7 mm = 7 103m
e¨v‡mi n«vm, d = 0.005 mm = 0.005 103m
†ei Ki‡Z n‡e, cqm‡bi AbycvZ, = ?
Avgiv Rvwb, = d/Dl/L
= dLDl
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= 0.005 103m 0.80m
103m 7 103m = – 0.57 (Ans)
N A Zv‡ii Dcv`v‡bi Bqs Gi ¸Yv¼,
YA = FALA
rA2lA
= 5N 0.80m
3.1416 (0.5 103m)2 7 103m
= 7.28 108Nm2
B Zv‡ii Dcv`v‡bi Bqs Gi ¸Yv¼,
YB = FBLB
rB2lB
= 6N 0.75m
3.1416 (0.6 103m)2 8 103m
= 4.97 108 Nm2
†h‡nZz 7.28 108 Nm2 > 4.97 108 Nm2
A_©vr YA > YB
A_©vr A I B ZviwUi g‡a¨ A ZviwU †ewk w¯’wZ¯’vcK|
5 bs cÖ‡kœi DËi
K GKwU ej‡K AmsiÿYkxj ejv n‡e hw` GKwU e ‘ c~Y© Pµ
m¤úbœ K‡i c~‡e©i Ae¯’v‡b wd‡i G‡j H ej Øviv K…ZKvR k~b¨ bv
nq| hw` †Kv‡bv e¯‘‡K GK we› y †_‡K Aci GK we›`y‡Z wb‡Z H
ej Øviv K…ZKvR e ‘‡K †Kvb c‡_ †bqv n‡q‡Q Zvi Dci wbf©i
K‡i Z‡e H ej GKwU AmsiÿYkxj ej|
L g‡b Kwi, ywU e ‘i fi m1 I m2 (m2 > m1) Ges MwZ‡eM v1
I v2| G‡`i fi‡eM mgvb n‡j, m1v1 = m2v2
ev, v2
v1 =
m1
m2
G‡`i MwZkw³i AbycvZ =
EK1
EK2 =
12 m1v1
2
12 m2v2
2
= m1
m2
m2
m1
2 =
m2
m1
m2 > m1
EK1 > EK2
A_©vr nvjKv e ‘i MwZkw³ †ewk|
M †`Iqv Av‡Q,
MÖnwUi fi, M = 12 1024 kg
e¨vmva©, R = 8 106m
gnvKlx©q aªæeK, G = 6.67 1011 Nm2kg2
†ei Ki‡Z n‡e, MÖ‡ni c„‡ô gyw³‡eM, ve = ?
Avgiv Rvwb, ve = 2GM
R
= 2 6.67 1011 Nm2kg2 12 1024 kg
8 106 m
= 14.146 103 ms1
= 14.146 kms1 (Ans.)
N ao = ab = bc kZ©g‡Z,
a, ob Gi ga¨we›`y
ab = bc = R2 =
8 106m2 = 4 106m
MÖ‡ni c„‡ô AwfKl©R Z¡iY g n‡j,
MÖnc„ô n‡Z d = 4 106m MfxiZvq AwfKl©R Z¡iY,
g = g
1
dR = g
1
4 106m
8 106m
= g2
Ges c„ô n‡Z h = 4 106m D”PZvq AwfKl©R Z¡iY,
g = g
R
R + h2 = g
R
R + R/22 = g
R
3R/22 =
49 g
g2 >
49 g
A_©vr g > g
myZivs a we› yi AwfKl© Z¡i‡Yi gvb, c we› yi AwfKl© Z¡iY
A‡cÿv †ewk|
6 bs cÖ‡kœi DËi
K †Kv‡bv ’v‡bi evZv‡m cÖwZ NbwgUv‡i †h cwigvY Rjxq ev®ú
Av‡Q Zv‡K H ’v‡bi cig Av ª©Zv e‡j|
L M¨v‡mi AYy¸‡jv mew`‡KB G‡jv‡g‡jvfv‡e PjvPj Ki‡Z cv‡i
Ges G‡`i g‡a¨ msNl© N‡U| M¨vm AYy¸‡jvi g‡a¨ ~iZ¡ Zi‡ji
Zzjbvq A‡bK †ewk nIqvq AvšÍtAvYweK ej †bB ej‡jB P‡j|
ZvcgvÎv e„w× †c‡j AYymg~‡ni Mo †eM e„w× cvq, d‡j msNl©I
ev‡o| msNl© evovi Kvi‡Y wewfbœ ¯ ͇ii cÖev‡n evavi cwigvY e„w×
cvq| A_©vr mv› ÖZv e„w× cvq|
M GLv‡b Mo MwZkw³ ej‡Z cÖwZwU M¨vm AYyi Mo MwZkw³
eySv‡bv n‡q‡Q|
2q cv‡Îi cÖwZ †gvj M¨v‡mi MwZkw³, E = 32 RT
= 1.5 8.314 Jmol1K1 273K
= 3403 J.mol1
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myZivs cÖwZwU M¨vm AYyi Mo MwZkw³ = E
NA
= 3403 J.mol1
6.023 1023 mol1 = 5.65 1021J (Ans.)
N cvÎ ywU n‡Z M¨vm e¨vc‡bi gva¨‡g †ei n‡e| e¨vcb nvi mgvb
n‡jB GKB mg‡q cvÎ ywU Lvwj n‡e| Avi e¨vcb nvi wbf©i K‡i
eM©g~j Mo eM©‡e‡Mi Dci|
DÏxc‡Ki Z_¨ n‡Z cvB,
1g cv‡Îi M¨v‡mi AvYweK fi, M1 = 2 g = 0.002 kg
2q cv‡Îi M¨v‡mi AvYweK fi, M2 = 32 g = 0.0032 kg
1g cv‡Îi M¨v‡mi ZvcgvÎv, T1 = 273 K
†gvjvi M¨vm aªæeK, R = 8.31 Joule mol–1 K–1
2q cv‡Îi M¨v‡mi ZvcgvÎv, T2 = ?
1g I 2q cv‡Îi M¨v‡mi eM©g~j Mo eM©‡eM h_vµ‡g c1 I c2 n‡j,
c1 = c2 n‡Z n‡e|
3RT1
M1 =
3RT2
M2
ev, T1
M1 =
T2
M2
ev, T2 = M2
M1 T1 =
0.0320.002 273 = 4368K
2q cv‡Îi M¨v‡mi ZvcgvÎv evov‡Z n‡e (4368 – 273) K = 4095 K
myZivs cvÎ ywU GKB mv‡_ Lvwj n‡Z n‡j wØZxq cv‡Îi ZvcgvÎv
4095K evov‡Z n‡e|
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