ALP Solutions RBD Physics Hindi
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Transcript of ALP Solutions RBD Physics Hindi
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RESONANCE RIGID BODY DYNAMICS - 1
TOPIC : n
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RESONANCE RIGID BODY DYNAMICS - 2
4.
;gk f?kjuh o /kkxs esa fQlyu ugha gSAvr% (a = r) ..........(i)fcUnq nzO;eku ds fy, :
mg T = ma ...........(ii)pdrh ds fy, cy vk?kw.kZ lehdj.k
Tr = .
Tr = .2
mr2
T = 2mr
=
2
mg...........(iii)
mg 2mg
= ma
mg = 2mg3
a = 3g2
.
5.
R
RV
2V 2V
2R
x
x = gR4
v2g
2R2v2
= gRv16 2
6.F sin
F) Fcos
mg
f
Rr
(F sin + N = mg)F cos js[kh; pky dks c
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RESONANCE RIGID BODY DYNAMICS - 3
8. vkosx = laosx esa ifjorZu
P. 2
=
12m 2
. (AB ds dsUnz ds ifjr%)
= m
P6
= 2
ds fy, ; 2
= t t =
2 = p62m
t = p12m
9.
G
g sina
o`kkdkj ikbi fojke esa gSA blfy, g sin = a
10. fcUnq P ds lkis{k dks.kh; laosx laj{k.k ls
MV 2L
=
12)L2()M2( 2
w 2V
= 3wL2
w = L4V3
Ans. (C)
11. ?kw.kZu tkZ = 221 = mK2
js[kh; tkZ = 2mv21
K = ?kw.kZu f=kT;k
dqy tkZ = 221
+ 2mv
21
v = R
dqy tkZ dk Hkkx ?kw.kZu tkZ ls lacaf/kr
= 22
2
mv21
mv21
21
= 2222
22
RmmKmK
=
22
2
KRK
cgq p;ukRed iz'u12.* ( A, B, C )
(A)
= A
L
i.e dtdL
= A
L
;g lEcU/k n'kkZrk gS fd dtdL
, A
o L
nksauks ds gh yEcor~ gSA blfy, (A) fodYi lgh gSA
(C) L
. L
= L2 ;gk
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RESONANCE RIGID BODY DYNAMICS - 4
le; ds lkFk vodyu djus ij ge kIr djrs gSA
L
.
dtdL
+dtdL
. L
= 2L dtdL
2 L
.
dtdL
= 2L dtdL
.....(1)
ysfdu pwafd L
dtdL
L
dtdL
= 0
vr% lHkh lehdj.k (1) lsdtdL
= 0
;k L
dk ifjek.kLo:i gS vFkkZr~ L le; ds lkFk ifjofrZr ugha gksrk gSA(B) blfy, ge nks fcUnqvksa ds fo"k; esa fuf'pr gS :
(1)
or dtdL
L
vkSj
(2) | L | ;k L le; ds lkFk ifjofrZr ugha gksrs gS] vr% ;g og fLFkfr gSA tgk L dh fn'kk ifjofrZr gksrh gSA ysfdu
ifjek.k fu;r jgrk gS ,oa
lHkh fcUnqvksa ij L
ds yEcor~ gSA
bls ge fy[k ldrs gSA
;fn L
= ( a cos ) i
+ ( a sin ) j
t gk a = /kukRed vpj gSA
rc
= (a sin ) i ( a cos ) j
blfy,, L
.
= 0 vkSj L
vr% A
,d vpj jkf'k gS vkSj ;g ges'kk ds yEcor~ gSA vr% A
dks fy[k ldrs gSA
A
= AA k
ge ns[k ldrs gS L
. A
= 0
vFkkZr~ L
A
yEcor~ gSA
vr% ge dg ldrs gS fd L
o A
ds vuqfn'k ?kVd 'kwU; gS ;k L
dk A
ds vuqfn'k ?kVd ges'kk fu;r gSA
vr% ge fu"d"kZ fudkyrs gS fd
, A
o L
rhuksa ,d nwljs ds yEcor~ gSA
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RESONANCE RIGID BODY DYNAMICS - 5
13.
SA B
fu;rfu;r ugha gksxk
dks.kh; Roj.k B fcUnq ds ckn 'kwU; gks tk,xk] D;ksafd f = 0 ?k"kZ.k cy gksus ds dkj.k cy vk?kw.kZ 'kwU; gSA ijUrq nzO;eku dsUnzij cy ds dkj.k js[kh; Roj.k c
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RESONANCE RIGID BODY DYNAMICS - 6
Hkkx - II
1.CM
x
m
12m 2
CM
= CM + mx2
x = m
CM =
m12
m 2
x = m34.0607
2. (i) xdx
A
B
AB = 2dmx AB =
0
3dxax=
4a 4
.
(ii)x
dm
dx
xcm
=
Q
0
0
2
dx ax
dxax
=
32
2 /3
cm
dx
ABA
B
AB = cm + m2
32
cm = AB 9
m4 2
m =
0
dx ax =
2a 2
cm
= 9a2
4a 44
36a 4
cm
Ans.
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RESONANCE RIGID BODY DYNAMICS - 7
3. (a)
2 kg nzO;eku d s fy,T1 2g sin 45 = 2a ......(i)
4 kg nzO;eku d s fy,4g sin45 T2 = 4a ............(ii)
f?kjuh d s fy,
r(T2 T1) = = (a/r) ............(iii) ( =
2mr 2
)
lehdj.k (i), (ii) o (iii) ls
a =
2r24
sing)24(
a =
01.05.024
52/110)24(.
a = 0.248 = (0.25 m/s2).
(b) m1 = 4kg m2 = 2kg = 0.2 (2 kg CykWd o ur ry d s vuqfn'k) = 0.5 kg-m2 r = 0.1 mm1gsin T2 = m1a .........(i)T1 (m2gsin + m2gcos) = m2a .........(ii)
r(T1 T2) = . =
r
a .........(iii)
lehdj.k (i), (ii) o (iii) ls
m1g sin (m2g sin + m2gsin) + 2ra
= m1a + m2a
eku j[kus ij
4g sin45 (2g sin45 + 0.2 2g sin45) + 01.05.0
a = 6a
27.80 (13.69 + 6.95) = 56a
= a = 567
= (0.125 m/s2).
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RESONANCE RIGID BODY DYNAMICS - 8
4. N2 sin = f (i)N1 + N2 cos = mg (ii)fcUnq A ds lkis{k cyk?kwZ.k
( N2 cos )
tan
b + N2 sin b = mg 2
a cos )
N2 =
b2sincosmga
N2 cos =
b2sincosmga 2
lehdj.k ....(ii) ls
N1 = mg N2 cos = mg b2sincosmga 2
N1 = mg b2)sincosab2( 2
N2 sin = N1 = 12NsinN
=
b2)sincosab2(mg
b2sincosamg
2
2
=
sincosab2sincosa2
2
5. mg 2/b = , = 6mb2
+ m
2
2b
I = 6mb2
+ 2
mb2 =
2mb2
311
= 3mb2 2
vr% 2mgb
= 3mb2 2
= b22g3
Accn of corner C = 22 bb = 2g3
{ksfrt fn'kk esa Roj.k 'kwU; gS vr%
C Hkkx dk Roj.k = 22 bb = 2g3
O fcUnq dk osx 'kwU; gSA blfy, Nx = 0
mg Ny = m 2b
= m 2b
b22g3
=
4mg3
Ny = 4mg
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RESONANCE RIGID BODY DYNAMICS - 9
6.
L
21kxsin30 kxsin30
30 30 BA
mg
(i)
dkVus ds igys 2k xsin30 = mgkx = mg (T = kx = mg)dkVus ds ckn(ii) COM ds lkis{k cy vk?kw.kZ
(Tsin30) x 2
=
4mg
= .12
m 2
=
g3 nf{k.kko`kZ
(b) A fcUnq dk Roj.kma
x = T cos30
ax =
m23mg
=
2g3
= aAC
mg T sin30 = may
mg 2mg
= may
aAy =
2g
+ 2
= 2g
+ 2
g3 = (g) ()
aA = g ji
23
(c) aBx = g23
aBy = 22g
= 2g ( )
aB = g j2i
23
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RESONANCE RIGID BODY DYNAMICS - 10
(ii)kxsin30
30
kxsin30
30
1 2
L/3 L/3 L/3
mg
dkVus ds igysmg = 2kx sin30 = kx = TT = mg.
dkVus ds ckn
(a) COM ds lkis{k cy vk?kw.kZ
(T sin30)
6
= .
(mg)
21
6
= .12
m 2
=
g (cw).
(b) (T cos30) = max
mg23
= max
ax =
2g3
aAx = )i(2g3
mg 2ma
= may
ay = 2g ( j )
aAy = (ay 2 ) =
2g
2
g
= 0
ig
23
aA
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RESONANCE RIGID BODY DYNAMICS - 11
(c) ig23
acx
j 22
gacy
= g j
ac = a
cx i + acy j =
jgig
23
.
7.
A fcUnq ds lkis{k dks.kh; laosx :Li = m1vs (us : VDdj ds ckn xsan dk vafre osx)
L = 3m 22 + m1us
Li = L
(m1vs = 3.m 22
+ m1us)
2 5 = 32.18
b + (2 us)
10 = 1032
+ 2us
............ (i) izR;koLFkku xq.kkad
e = s
s
v
u
0.8 = s
s
v
u
54
=
5
u2.1 s
4 = 56
us
us =
5
206............ (ii)
lehdj.k (ii) dks lehdj.k (i) esa j[kus ij
10 = 1032
+ 2
5
20610 = 10
32 + 5
4012
100 = 32 + 24 80
= 1445
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RESONANCE RIGID BODY DYNAMICS - 12
dks lehdj.k (ii) esa j[kus ij
us =
5
206
us = 5
2014456
us = 514
280270
= 51410
=
71
blfy, fn'kk () us
71
8.
ekuk rk{kf.kd ?kw.kZu v{k ds funsZ'kkad P(x,y).rc P dk osx C ds lkis{k 'kwU; gSA 0ivCP
b
t ( k ) 0iv]jyi)tvx[( x = vtand yt = VvkSj yt = VmijksDr lehdj.k ls t dks gVkus ij
1x.v
y
or xy =
2v
vr% P dk fcUnqiFk vfrijoy; gksxk
(b) C fcUnq ds funsZ'kkad =
0,Nt21 2
0ivCP
0itwjyi)tw21
x(k 2
x = 2tw21
y = w
tt dks gVkus ij
x = 2
2
yw
w21
x = w2
2y2
lehdj.k ijoy; dh gSA
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RESONANCE RIGID BODY DYNAMICS - 13
9. a = Rmg sin 30 0 T = ma .........(1)
vkSj 2mg
T = ma .........(2)
= I
= 221
TR
MR
= MR2T
.........(3)T ds fy, lehdj.k (1), (2) rFkk (3) dks gy djus ij
T = 21
mM 2mg M
eku j[kus ij ge izkIr djsaxs
T =
21
(0.5)(2)2.8)(2)(0.5)(9
= 1.63 N
T = 1.63 Nlehdj.k (iii) ls Me dk dks.kh; eanu
= MR2T
= )2.0)(2((2)(1.63)
= 8.15 rad/s2
CykWd dk js[kh; eanua = R = (0.2) (8.15) = 1.63 m/s2
og {k.k tc Me dk dks.kh; osx
0 = 10 rad/sCykWd dk js[kh; osx
v 0 = 0 R = (10) (0.2) = 2 m/s
vc CykWd }kjk r; nwjh tc rd ;g fojke esa ugha vk tk;s
s = 2av 02 [ v 2 = v02 2as dk mi;ksx djus ij v = 0 ]
= )63.1(2(2)2
m
;k s = 1.22 m (a) 1.633 N (b) 1.224 m
10.
fcUnq A ds lkis{k laosx laj{k.k ls
xO
3R10
mvR + mv
10R3R = 2mR2
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RESONANCE RIGID BODY DYNAMICS - 14
v ' = 2v7.1
R2
v7.1'
tkZ laj{k.k ls
2mv21
+ 212 = mg 0.3 R + 2
1mv'2 + 2
1'2
mv2 = mg 0.3 R + mv'2
v2 = g 0.3 R + 22
v27.1
vU;wure
= gR3.07.1
2
11. CykWd m dks iwjk ,d pDdj yxkus ds fy, U;wure osx gR5 gksuk pkfg,A
;kaf=kd tkZ laj{k.k ls
212 = Mg 2
R =
MgR
fcUnq P ds lkis{k VDdj rFkk VDdj ds ckn dks.kh; laosx laj{k.k ls = m.R gR5
MgR = mR gR5
MgR = m2R2 5gR
= 3ML2
j[kus ij
m
M = 15
Ans. : 15m
M
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RESONANCE RIGID BODY DYNAMICS - 15
12. (i)
L = fLizax dks fcuk [kksyh voLFkk esa yEckbZ
(ii)
(a) (i) rFkk (ii) esa tkZ laj{k.k ls
x2
mgI21kx
21I
21 2
122
............. (i)
I = Icm
+ ,
2
x2
I = 12
m 2 + m
2
x2
............. (ii)
Lyxx 22
............. (iii)lehdj.k (ii) rFkk (iii) dks (i) esa j[kus ij
22
x2
m12
m
21
2 + 2/1
22 LyxK21
=
22
x2
m12
m
21
12 + mg
x
x = 150 mm, y = 20 mm, = 450 mm, K = 300 N/mm = 3 kg, = 4 rad/sec
lehdj.k esa eku j[kus ij
1 = 8632
rad/sec
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RESONANCE RIGID BODY DYNAMICS - 16
(b) 180 ?kqekus ij
;g izkjfEHkd fLFkfr dh rjg gS] vr% 2 = 2 = 4 rad / sec.
13. O fcUnq ds lkis{k cy vk?kw.kZ
N
= dtMd
= tb2
ekuk M
rFkk N
ds chp dks.k = 45 at t = t0
rc 21
= NMN.M
=
040
22
20
bt2tba)bta(
=
0422
30
2
bt2.tbatb2
= 4
022
20
tbabt
gy d jus ij, t0 = ba
( t0 +_ .kkRed ugha gks ldrk)
vr% bab2tb2N 0
14. = dks.kh; Roj.k
= dks.kh; Roj.kr[rs ds fy,
F = m11 ....... (i)xksys ds fy, cy vk?kw.kZ C fcUnq ds lkis{k
fr = Ic = 5
2m2r
2 ....... (ii)ekuk 2 xksys ds nzO;eku dsUnz dk dks.kh; Roj.k A fcUnq ij gSA
(1 = 2 + r) ....... (iii)lehdj.k (i), (ii) rFkk (iii) ls
1 =
21 m7
2m
FrFkk 2 =
17
2
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RESONANCE RIGID BODY DYNAMICS - 17
15.
csyu ds fy,Mg + T1 2T = Ma ......(i)
?kw.kZu v{k ds lkis{k cy vk?kw.kZ
2TR + T1(2R) = =
Ra
. ....(ii) (a = R)
A Hkkj ds fy,mg T1 = ma
f?kjuh o Mksjh ds e/; dksbZ fQlyu ugha gSAa1 = a + (2R) = (3a) ....(iii)
lehdj.k (i), (ii) rFkk (iii) ls
2
1
Rm9M
g)m3M(3a
16. A Hkkx dh /kjkry ls Vdjkrs le; osx = gh2 ;fn VDdj ds Bhd ckn nzO;eku dsUnz dk osx v gks NM+ }kjk izkIr dks.kh;osx nf{k.kkorZ fn'kk esa fp=kkuqlkj gSA izR;koLFkrk xq.kkad dh lehdj.k dkmi;ksx djus ijikl vkus dk osx = nwj tkus dk osx (A fcUnq ij)
gh2 = v + 2L cos .............(1)
D;ksafd A fcUnq ij vkosfxr cy dk;Zjr gSA blfy, A fcUnq ds lkis{k VDdj ds igys rFkk VDdj ds ckn dks.kh; laosx laj{k.kls
gh2 M 2L
cos = cm
Mv 2L
cos .............(2)
= ( gh2 v) cosL2
dk eku j[kus ij
lehdj.k (1) ls
gh2 M. 2L
cos =12
ML2 ( gh2 v) cosL
2 Mv
2L
cos
cos6L
v + 2vcosL
=
cos6gh2L
2gh2 Lcos
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RESONANCE RIGID BODY DYNAMICS - 18
v
cos6cos31 2
=
cos6)cos31( 2
gh2
v =
cos31cos1 2
gh2
nzO;eku dsUnz ij tkrs le; vf/kdre pkbZ ij gksxk tc mldk osx 'kwU; gks tk,xkAO = v2 2g H
H = g2v 2
=
2
2
2
cos31cos31
h.
[ Ans.: H = 1 31 3
2
2
2
cos
cos
h; h =49144
]
17. NC + NB = 250NB x = 250 3
NB = x750
f1 = x750
f2 =
25x
750
?k"kZ.k ds fo:) fd;k x;k dk;Z
W = dx)( 21 = dx5.73.0x15005.4
3
= 450 n 2
3 + 7.5 (4.5 3)
= 450 0.41 + 7.5 1.5
21
mv2 = 400 1.5 195.75
v2 = (600 195.75) 5.22
=161.7 2 = 323.4
v = 18.52 m/sec.
18. cgqr NksVk gSA 0{kSfrt fn'kk esa cy larqfyr djus ij
N1 = N2P fcUnq ds lkis{k cy vk?kw.kZ larqfyr djus ijFor to be very small we can directly write
T.b + N2b 2Wb
N2a = 0
T
w/2
A
N2
PN1
T N1
AN2
b
a
;fn f[kM+dh dk Roj.k A gks rks y fn'kk esa cy
w N1 N2 T = gwA
... (ii)
CykWd ds fy,
T 2W
= g2WA
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RESONANCE RIGID BODY DYNAMICS - 19
T =
g2WA
2W
.... (iii)
lehdj.k (iii) dks lehdj.k (i) esa j[kus ij
2W
b + g2W
Ab + N1b = N1a + 2Wb
g2WAb
= N1 (a b)
N1 =
)ba(g2WAb
..... (iv)
N1 rFkk T dk eku lehdj.k (ii) esa j[kus ij
W 2
)ba(g2
WAb
2W
g2WA
= gWA
2W
)ba(gWAb
= g2WA3
1 )ba(gAb2
= g
A3
g (a b) = (2b + 3a 3b)A
A = )ba3(g)ba(
Ans.
19. VDdj ds ckn] ekuk COM, V osx ls xfr djrk gS] vkSj fudk; COM ds lkis{k dks.kh; osx ls ?kqekuk izkjEHk dj nsrkgSA js[kh; laosx laj{k.k yxkus ij
mv0 = 3mv v = 3v0
COM ds lkis{k dks.kh; laosx laj{k.k yxkus ij
mv0. 32a
= =
3
3ma2
.
= ma2
= a32
v0
(a) vk/kk pDdj iwjk djus esa yxk le;
t =
=
0va32
(b) bl vUrjky esa d.k B vk/kk pDdj iwjk djrk gSA bldh fLFkfr esa COM fn[kkbZ xbZ fLFkfr+ Disp. due to Angular motion.
x-fn'kk esa B dk foLFkkiu = COM ds js[kh; xfr ds dkj.k foLFkkiu + dks.kh; xfr ds dkj.k foLFkkiu
xB = 3v0
.t + MN
= 3v0
.
0va32
+ 3a2
. cos30 = 32
a + a
Y fn'kk esa foLFkkiu
YB = 3a2
cos60 = 3a
d qy foLFkkiu = 2B2B yx
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RESONANCE RIGID BODY DYNAMICS - 20
20.
f
A f = mg
A fcUnq ds lkis{k cy vk?kw.kZ
R( mg) = 2
mR2
R
g2
R1025.02
=
R5
fu;r dks.kh; pky ij
=
Rv
)n2(2Rv
2
n =
.R.541818
R4v
2
2
n =
.R.201818
=
3107520
1818
n =
7520
101818 3 =
202010186 3
n =
2
4186
n = 636
=
216
fu;r dks.kh; osx ij vkus ls igys pdrh }kjk yxk;s x;s pDdjksa dh la[;k n =
216.
21.a2
f2
m
lrg ds dkj.k IysV ij ?k"kZ.k f1 = 7.5 0.2 10 = 1525 15 f2 = 1.5 a1f2 = 6a210 = 1.5 a1 + 6a2 ....(i)f2 . r = mr . f2 = ma2 ...(ii)f2 = ma1 ma2
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RESONANCE RIGID BODY DYNAMICS - 21
a2 + r = a1 a1 a2 = a2 a2 = a1 a2 a1 = 2a2
10 3a1 = 1.5 a1
a1 = 45100
= 920
a2 = 2a1
= 1820
v1 = a1t = 920
43
= 35
(IysV)
v2 = a2t = 1820
43
= 65
(ikbi)
2 = s/rad42.10
21601000
x65
r
v2 (ikbi)
22. 4.8 Ma2ekuk fd iVy xy ry esa gSArc yEcor~ v{kksa ds es;k ls I
x + Iy = Iz
ysfdu Ix= Iy ( lfEer ls )
vkSj Iz= 1.6 Ma2 (fn;k gqvk gS )
Ix= 2
Iz = 0.8 Ma2
vr% lekUrj v{kksa dh es; ls IAB = Ix + M(2a)2
= 0.8 Ma2 + 4Ma2= 4.8 Ma2
A By
x
23. nh xbZ 'krZ d s vUrxZr d soy ;gh lEHko gS fd ?k"kZ.k ij d h vksj o Roj.k uhps d h vksj gS t Slk uhps n'kkZ;k x;kgSA
xfr d h lehd j.k gS &
a = m
fsinmg =
m
f30sinmg = 2
g
m
f......(1)
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RESONANCE RIGID BODY DYNAMICS - 22
=
=
fR =
mRf2
......(2)?kw.kZu (fcuk fQlys) d s fy,
a = R or g/2 f/m = 2f/m
m
f3 = g/2 ;k f = mg/6
(1) f fmax
mg cos 30 23 mg
(2) vU; lEHkkouk,sa t ks lEHko ugha gS fuEu gS &(a) ?k"kZ.k uhps d h vksj gSA bl fLFkfr esa a o n'kkZ;s vuqlkj gSA
(b) pd rh ij ?k"kZ.k ij d h vksj o js[kh; Roj.k ij d h vksj
bl fLFkfr esa ?kw.kZu lEHko ugha gSA
24. t = 0 ls t = t0 le; d s e/;] ;gk vkxs d h vksj fQlyu gks jgh gSA blfy, ?k"kZ.k f cka;h vksj gS ,oa vf/kd re gSA vFkkZr~ mg gSA t > t0, le; d s fy,] ?k"kZ.k f 'kwU; gSA D;ksafd 'kq) ?kw.kZu xfr kjEHk gks xbZ gSA vFkkZr~ ;gk lEidZ fcUnqvksa d se/; d ksbZ fQlyu (lkis{k xfr) ugha gSA
blfy, t < t0 d s fy,
js[kh; Roj.k , = m
f = g (f = mg)
vkSj d ks.kh; Roj.k, = I
= 2mR2IRf
= Rg2
vc V js[kh; osx gS vkSj d ks.kh; osx gSA rc t = t0 pd rh d kV = V0 at0 = V0 gt0 ......(1)
-
RESONANCE RIGID BODY DYNAMICS - 23
vkSj = t0 = Rgt2 0
......(2)'kq) ?kw.kZu d s fy,
V = RvFkkZr~ V0 2to = 2to
t0 = g3V0
lehd j.k (1), esa j[kus ij] ge kIr d jrs gS
V = V0 g
g3V0
V = 32
V0?k"kZ.k cy }kjk fd ;k x;k d k;Z
t t0 d s fy,] fd lh le; t ij pd rh d k js[kh; osx V = V0 gt gS vkSj d ks.kh; osx = at = Rgt2gSA d k;Z t kZ
es; ls] t le; esa ?k"kZ.k }kjk fd ;k x;k d k;Z = t le; ij pd rh d h xfrt t kZ le; (t = 0) ij pd rh d hxfrt t kZ
W = 21
mV2 + 21
I2 21
mV02
= 21
m [V0 gt]2 + 21
2mR21
2
2gt2
2
1mV02
= 21
[mV02 + m2g2t2 2mV0gt + 2m2g2t2 mV02]
;k W = 2gtm
[3gt 2V0]t > t
o ij ls fy,] ?k"kZ.k cy 'kwU; gSA ?k"kZ.k cy }kjk fd ;k x;k d k;Z 'kwU; gSA vr% t kZ lajfpr jgsxhA
blfy,, t le; es ?k"kZ.k }kjk fd ;k x;k d k;Z t0 rd fd ;k x;k d qy d k;Z gSA (D;ksafd bld s ckn ?k"kZ.k cy }kjkfd ;k x;k d k;Z 'kwU; gSA)
W = 2gtm 0
[3gt0 2Vo]t0 = V0/3g, j[kus ij ge izkIr d jrs gS
W = 6mV0
[V0 2V0] W = 6mV20
25. ekuk fcuk ewM+h njh dk nzO;eku M rc
M =
2R
M
2
2
R = 4
M
;kaf=kd tkZ laj{k.k fu;e ls
MgR M g 2R
= 21
4M
v 2 + 21
I 2 R/2R
MM
v
;k MgR
4M
g
2R
= 8Mv2
+ 21
4421 2RM
2
R/2v
;k 87
MgR = 163Mv2
v = 3Rg 14
-
RESONANCE RIGID BODY DYNAMICS - 24
26. tc F vf/kdre gS rc ?kw.kZu lkE;koLFkk lehdj.kF.R. = (N1 + N2) R .............(1)
{kfrt fn'kk esa lkE;koLFkk ds fy,f1 = N2 = N1 ............(2)
/oZ fn'kk ds fy,F + N1 = mgF = [(mg F) + (mg F)]
21
)Fmg(21)Fmg(
21putting
ij j[kus21
F
21
211
= 43
mg
F = 83
mg = 83
w [ Ans.: 3w/8 ]27. tSls gh dhV (ePNj) nqljs fljs dh vksj xfr djsxk nzO;eku dsUnz mlh fLFkrh ij jgsxk ftlls (straw) NM ck;h rjQ
LFkkukUrfjr gksxhA
AB ds lkis{k cyk?kwZ.k lUrqfyr gSA
2mg
3
= (m + mA)g
6
4m = m + mAmA = 3m
28. 2mg 2L mg 2
L T 4
L = 0
T = 2mg Ans.
(b) NP = 6mg Ans. (c) 2mg
2L
mg 2L
= (2m 4
2+ m
4
2+
12m2 )x
= mg 2
=
x22
12m
4m3
=
2mg
=
12m10 2
= 10
g6 =
5g3
Ans.
-
RESONANCE RIGID BODY DYNAMICS - 25
(d) 22
12m10
21
= 2mg 2 mg 2
=2
2
12m10
21
= 2mg
, 2 = 10g12
5g6
= 5g6
, v = 2
=
5g6
2 Ans.
29. fudk; ?kw.kZu djus ds fy, LorU=k gS ysfdu LFkkukfUrfjr xfr ds fy, LorU=k ugh gSA lEiw.kZ fudk; ( NM A + NM B +nzO;eku m ) dk fcUnq P ds lkis{k cyk?wk.kZ 'kwU; gSAvr% VDdj ls iwoZ fudk; dk dks.kh; laosx = VDdj ds ckn fudk; dk dks.kh; laosx (fcUnq P ds lkis{k ).ekuk VDdj dsrqjUr ckn fudk; dk dks.kh; osx gS rc
A
P
Bv
m
L i = L f
mv (2l) = ;gka , = fcUnq P ds lkis{k fudk; dk tMRo vk?kw.kZ
= m (2) 2 + m A ( 2 / 3 ) + m B
22
)212
fn;k gS : = 0.6 m, m = 0.05 kg, mA = 0.01 kg vkSj mB = 0.02 kgekuks dk izfrLFkkiu djus ij
I = 0.09 kgm 2Therefore, from Eq. (1) vr% lehdj.k (1) ls
= I2mv
= 09.0)6.0)()(05.0)(2( v
= 0.67 v ........(2)vc VDdj ds ckn ;kfU=kd tkZ laj{k.k ls
- 0
vr% ?kw.kZu xfrt tkZ eas deh = xq:Roh; fLFkfrt tkZ esa of`)
;k 21
I = mg (2) + mA g
2
+ m B g
2
-
RESONANCE RIGID BODY DYNAMICS - 26
;k 2 = Immg A )m 34( B
=
09.0
)02.0301.005.04(6.0)8.9(
= 17.64 (rad /s) 2 = 4.2 rad/s .........(3)lehdj.k (2) vkSj (3) dh rqyuk djus ij
v = sm /0.674.2
;k v = 6.3 m/s
30. Q fcUnq ds lkis{k cyk?kw.kZ lUrqfyr gSA vr%
Pc = mg b
P =
c
bmg
N = mgdksbZ fQlyu ugh gks blfy;s f = P mg = PP
max = ( mg)
Cmax
= Pbmg
= mgbmg
=
b
31. (i) lhekUr voLFkk esa] vfHkyEc izfrf;k 'O' ls xqtjrh gSA rFkk ?ku O ds lkis{k yq