alp solutions rbd physics hindi

RESONANCE RIGID BODY DYNAMICS - 1

TOPIC : n


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


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


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


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


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.


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).


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


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


(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


(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


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


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


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


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


(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


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


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


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


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


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)


=

=

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)


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


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.


(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


;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

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