11 physics test papers

7/22/2019 11 Physics Test Papers

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PHYSICS(Class 11)

Index

Chapters page

1. Physical World & Measurement 012. Kinematics 093. Laws of Motion 304. Work, Energy & Power 435. Motion of System of particles & Rigid Body 556. Gravitation 657. Properties of Bulk Matter 788. Thermodynamics 1289. Behaviour of Perfect Gas & Kinetic Theory of gases 19710. Oscillations & Waves 216


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Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

&

&

1. = + 2 .

1

2. .

() 3.0/2= /2

() 6.67 10112/2= 132

1

3. . 1

4. 1.56

1016. ( 3 108/)

2

5. () () () ,

.

2

6. 1= 100 3 2= 200 4 .

2

7. , ,

2

8. 19 1010 /2. 2.

.. .

3

9.

.

3

10.

2

mF

r

=

3.5 0.1m kg kg=

20 / 1 / 12.5 0.5m s m s r m m = =

(1) (2) %

.

3

Page 1 of 259


4/261



( & )

&

1 /x

a m st

= =

2/

xb m s

t= =

2 () 3.0/23

2

3 10

1

3600h

/2 3.104/2

() 6.67 10112/2 13211 1 3 2

11 3 2 3

8 1 3 2

6.67 10

6.67 10 10 (10 )

6.67 10

kg m s

cm

g cm s

=

=

=

3 . () ().

4 distancetimevelocity

=

16

8

1.56 10time

3 10

=

5 2 2; (1)F MLT A LT = = -1

2

2 -2

-1 2 4 -2

2

F=MA M=FA (2)From equation 1) L = AT

Dimensions of energy = ML T

FA A T T

FAT

=

6 1 100 3R =

2 200 4R =

255.2 10 sect onds=

Page 2 of 259


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1 2netR R R= +

( ) ( )

( )

100 3 200 4

300 7 .

net

net

R

R ohms

= +

=

7 force = mass acceleration

velocityforce = mass

time

time forcemass

velocity

=

1

FTmass

V

mass FTV

=

=

[ ] 2 1 2

2 2

F MLTy ML T

L L

= = =

1 1 1

2 1

2 2 2

1, 1, 2

a b c

a b c

M L Tn n

M L T

= = =

=

[ ]

1 1 2

10

2

1210

2

1 1 119 10

1 1 1

1000 10019 10 1

1 1

kg m S n

g cm S

g cmn

g cm

=

=

10

2

119 10 1000 1

100n

=

a b cp g

(1)a b cR p g =

[ ]1 3 2b ca

LT L ML LT =

[ ] [ ] [ ]3 21

3 1

a b c b coM LT L M T

a b c

+ =

+ =

11

2 19 10n =

Page 3 of 259


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&

&

1. ..? 1

2. .. ? 1

3. ? 1

4. 824.7

35.7211 . .

2

5.

. ,

6.31010?

2

6. ? 2

7. ? 2

8. ( )

2

P aV b RT

V

+ =

2

9. , , , ,

. 2 5 2/EL m G

2

10. ()

() =

() =

() : 24m

P=D l

, 1%, 1.5% 0.5%. %

?

3

Page 5 of 259


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( & )

&

1 .. 1 1010

1.. 1.46 1011.

2 1

1.

3 .

4 6824.7 10r km=

435.72

35.72radian

60 60 180

=

=

?l =

6 35.72824.7 10

60 60 180

l r

l

=

=

52

tx c=

1082 6.3 102 3 10 /

7 60xc m s

t

= = =

6 (1) , .

(2)

.

51.429 10l km=

Page 6 of 259


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(1) ( )

( )

Q Heat Energy

m mass=

2 22 2oML T

M L TM

=

(2) ( )2 2Q ML T

Sm Q M K

=

( ) 2 2 1oS M L T K =

22aP a PV

V= =

2

22

3

2

Fa V

A

MLTa L

L

=

=

2 6

2

5 2

MLT La

L

a ML T

=

=

3

o o

Also b V

V M L T

=

=

2 2E ML T =

[ ]

2 1

1 3 2

L ML T

m M

G M L T

=

=

=

[ ]

22 2 2 12

5 2 25 1 3 2

3 6 4

3 6 4

Dimensions of

1

ML T ML TEL

m G M M L T

M L T

M L T

=

= =

,

Page 7 of 259


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10 () () 2

1 2

2

MLTF ML TA L

= = =

2 2

1 2

3

1 2 1 2

ML TML T

L

ML T ML T

= =

()

[ ] [ ]1 2 1 3

1 2 4 o

ML T M LT L T

ML T ML T

=

=

,

()2

4m

D l

=

m D l= +2 +

P m D l

P%=1%+2(1.5)%+0.5%

P

P %=4.5%

P

% 4.5%l

l

=

Page 8 of 259


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1. s t= 1

2. .

1

3. 1

4. 2

5. .

2

6. 60/

40/. .

2

. v at = + . 3

.

( )3 2

6 15 40x t t t m= +

() .

() .

() .

3

. . (1) 0 4

(2) 0 (3) 0 12 .

.

5

Page 9 of 259


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()

1 .

2 .

3 ( )Va Vb V say= =

0Vab V a V b V V = = =

4 (1) .

(2) , .

5

6 1 2

1 2

2 2 60 4048 / .

60 40av km hr

= = =

+ +

7 t

tan

tan acceleration(a)

v

t

But

=

=

va

t

v at

=

=

8 3 26 15 40x t t t= +

( )

( ) 2

23 12 15

6 12

ms

ms

dxt t

dt

dva t

dt

= =

= =

v at = +

Page 10 of 259


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1. 400 .

1

2. 2.5 .

.

1

3. 14. 2

5. . .

2

6. 32+ 7 9,

.

2

7. . () 5

9 .

2

8.

54/. .

126/ .

100/.

3

9. 2 (2 1)a

Snth u n= + .

3

10. 44.1

. 9.8/ 2

3

Page 12 of 259


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1: ,

.

2: 0

2.5 + 2.5 5.0.

3: .

4:

(1)

(2) .

5: (1) .

(2) ()

.

6: 23 7 9x t t= +

2

6 7 m/s

6 /

dx tdt

da m s

dt

= = +

= =

7: 21

2s ut at = +

9 5.

4 .

210 4 ( ) (4)

21

4 9.8 162

2 9.8

g

=

=

=

8: 54/ 15/ 126/ 35/

100 / .b m s =

19.6 / m s =

Page 13 of 259


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35 15 20/. ...

100 20 80 / ..

80/.

9:1

Snth Sn Sn

=

2

2

1

1

2

1( 1) ( 1)

2

n

n

S un an

S u n a n

= +

= +

2 21 1( 1) ( 1)

2 2Snth un an u n a n= +

( )

2 21 1 1

2 2 2

1 Snth = u-

2

= u+ 2 12

un an un u an a na

a na

an

+ + +

+

.

10:

0

9.8/2

( )

( )

2

102

2 1

9.844.1 0 2 1

2

55

aSnth u n

n

n

= +

= +

= =

2

2

1

2

12

h ut at

h un gn

= +

= +

210 5 9.8 (5)2

4.9 25

h

h

= +

=

h = 122.5m

Page 14 of 259


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1.

1

2. 1

3. .

, , , .

1

4.

2

5.

. (1) = 3 (2) = 6.

2

6. 4.9/. 245.

2

7. 126/.

200. ( )

2

8. 2

1

2s ut at = +

3

9. () ()

() 500/

1500/ .

3

10. () = + () 2 2= 2 3

Page 15 of 259


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1:

.

2: ( )

3:

4:

5: 3t x=

2

3

( 3)

x t

x t

= +

= +

() 2( 3)dx

tdt

= = +

3 sec 2(3 3) 12 / t m s= = + =

() 6 sec 2(6 3) 18 / t m s= = + =

6: 4.9 / ( )u m s upward =

245h m=

( ) = = 9.8/2()2

2

1

2

1245 4.9 (9.8)

2

s ut at

t t

= +

= +

24.9 4.9 245t t =

7.6 5.6t s or s=

u at = +

4.9 (9.8)(76) = +

t = 7.6s

69.6 / m s=

Page 16 of 259


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7: 126 / 35 / u km hr m s= =

0 200s m = = 2 2

2 2

29

2

v s

vas

=

=

( ) ( ) ( ) ( )2 2 2 2

0 126 0 35

2 200 2 200a

= =

= 3.06/2()

= +

0 35

3.06

V vt

a

= =

8: () = t

=

= +

( )

1

2

1

2

S OA OC AD BD

AD DBS ut AD

AD

= +

= +

( )

( )

2

2

1

2

1

2

DBS ut AD

AD

DBS ut t

AD

= +

= +

( ) ( )21

tan2

BDS ut t a a

AD

= + = =

9: () V AB

. .

() ()

V AB V A V B A V A

B V B

V AB

=

()

( ) AVAB VA V B A

VAB VA VB B

=

= +

V AB

t = 11.4s

21

2S ut at = +

Page 17 of 259


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() = 500/ . =

1500/ ( )

gJ g J

g gJ J

V V V

V V V

=

= +

= 1500 + 500 = 1000/

( )

10:

()dv

adt

=

dv adt =

dv adt = (1)V at k = +

0when t u

K u

= =

=

()2 2

2v as =

dv

adt

=

dv dx

adt dx

=

dva

dx=

adx vdv=

dxv

dt

=

2 2

( )2 2

x

o v

o

a dx dv

va x x

=

=

( )( )2 2

s displacement2

o

vas x x

= = =

V at u= +

2 2 2v as =

Page 18 of 259


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1. 1

2. 30 10/.

1

3. 1 1

4. 2 3 4 2

5.

2

6. 45

2

. 3

4

2

. .

.

3

. . 3

10. ()

() .

.

3

Page 19 of 259


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()

1: .. .

2: 45.

3: , 2 = , 2WT

=

4: 1

22 2R=(A +B +2ABcos) 1

22 24 (2 3 2 2 3cos )

16 4 9 12cos

= + +

= + +

13cos cos 0.25

12

75 32'o

= =

=

5: , ( )2 2 2 2sin 2 sin 2

Horizontal Range2

u u SinR

g g g

= =

212sin 2 sin = ( )

2 2

Maximum Height2

u Sinhm

g

=

2122sin cos sin tan 4 75.96

o

ie = = =

6: 2sin 2u

Rg

=

1 2

2 2

11

45 ; 45

sin 2 sin 2(45 )u uR

g g

= + =

+= =

2

1

2

1

2 2

22

2

sin(90 2 )

2(1)

2 2(45 )

(90 2 ) =

uR

g

u CosR

g

u Sin u SinR

g g

u Sin

g

+=

=

= =

2

2

2u CosR

g

= , R1= R2

Page 20 of 259


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7: 2 24 3

16 9

OB

OB

= +

= +

25 5

5

OB

Range km

= =

=

8: (1) then (1)l

PQ lt

= =

W=l

(2)l lr r

= =

t andl V W t = =

(1) W t

tV =

v rwr =

(2) 2

d ddv dw dr a r w w

dt t t r r

= = = = = =

9:

, .

21

2, ,

S ut at

s x u u t t

= +

= = =

0

(1)

and a

x ut

xtu

=

=

=

21

2

, 0, ,

S ut at

s y u t t a g

= +

= = = =

21

2y gt =

21 (2)2

y gt=

(1) (2)

( )22

2

1 1

2 2

xxy g gu u

= =

2

ar

=

Page 21 of 259


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10: ()

.

.

()

.

Page 22 of 259


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1. Give an example of a body moving with uniform speed but having a variable

velocity and an acceleration which remains constant in magnitude but changes in

direction

1

2. 1

. A and B

. .A B

1

. 10 120

.

10

2

. .

2

.

2

.

2

. A

2 B

. 2A B+

.

2

.

2 3 4 3 5A i J k and B i J k= + + = +

2

10. () A and B

A and B

1

2AB

()

Page 23 of 259


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(b) Triangular law of vector addition states that if two vectors can be represented both in

magnitude and direction by the sides of a triangle taken in order then their resultant is

given by the third side of the triangle taken in opposite order.

Proofin ADC2 2 2

2 2 2

( ) ( ) ( )

( ) ( ) ( )

AC AD DC

AC AB BD DC

= +

= + +

2 2 2 2

2 2 2

( ) ( ) ( ) 2( )( ) ( )

( ) ( ) ( cos )2 2( )( cos ) ( sin )

AC AB BD AB BD DC

AC P Q P Q Q

= + + +

= + + +

2 2 2 2 2( ) (sin cos ) cos

2 cos sin

BDAC P Q

BC

CDPQ

BC

= + + =

+ =

( )2 2 2 2

2

( ) 2 cos sin cos

2 cos

R P Q PQ Q

R P Q P

= + + +

= + +

Page 26 of 259


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1.

,

1

2.

30 60,

1

3. 1

4. .

2

5. . ,

.

2

6. 30/,

45. , 10/2.

2

7. 600

. .

2

. ,

. .

3

. ' ' 30

60

.

3

10. () ()

()

3

Page 27 of 259


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Ans1: Four times the initial horizontal range.

Ans2: Three times the initial vertical height.

Ans3: 6 radian per hour.

Ans4: Acceleration must be perpendicular to the direction of motion and is called centripetal

acceleration.

Ans5: When a stone is moving around a circular path, its velocity acts tangent to the circle.

When the string breaks, the centripetal force will not act. Due to inertia, the stone

continues to move along the tangent to circular path, and flies off tangentially to the

circular path.

Ans6:( )

22 30 sin 2sin245

10

uR

g

= = =

( )2

450sin2

30

1sin22

2 30 150 15 75o o o o

or or

=

=

= =

Ans7: 600revolutions minv =

600revolutions sec.

60

6002 2

60

20 /

v

w v

w rad s

=

= =

=

Ans8: When an object moves in a circular path with constant speed then the motion is called

uniform circular motion

Time period The time taken by the object to complete one revolution

Frequency The total number of revolutions in one second is called the frequency.

Angular velocity It is defined as the time rate of change of angular displacement.

2 12W v v

T T

= = =

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Ans9: (1) When ( )2

30 sin 2 30o oARg

= =

23

2A

R

g

=

When ( )2

60 sin 2 60o o

BR

g

= =

2 3

2

: 1:1

B

A B

Rg

R R

=

=

(2) When2

230 sin 30o oA

Hg

= =

21

2 4AH g

=

When2

260 sin 60

2

o o

BH

g

= =

( )2

34

: 1: 3

B

A B

Hg

H H

=

=

Ans10: (1) P.E. Will be maximum at the highest point

(P.E.) highest point = mgH

( )

( )

2 2

2 2

sin.

2

1. sin

2

H

H

P E mgg

P E m

=

=

(2) K.E will be minimum at the highest point

( ) ( )21

. .2

HHK E m v=

(Vertical component of velocity is zero)

( )

2 21

. . cos2HK E m =

(3) Total mechanical energy

( ) ( )

( )

2 2 2 2

2 2 2

. . . .

1 1cos sin

2 2

1cos sin

2

H HK E P E

mu mu

mu

+

+

+

21

2m u

Page 29 of 259


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1. ,

1

2. 10..

1

3. .

,

,

1

4. .

2

5. 0.05

6/ .

.

2

6. .

,

.

2

. .

2

. 0. ,

() 10/.

() 5/2.

() 5/2. ./2.

3

. () () 0.1 .

30/ 20/.

.

3

10. .

.

() ( ) .

()

() .

3

Page 30 of 259


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1 , .

2 10.

3 . ,

.

4

.

5 0.05(6) 0.3 /

,

0.05(6) 0.3 /

0.3 0.3 0.6 /.

6 ,

.

1 1 2 2

m m +

1 1

1 1 2 2 2

2

0 m

m mm

+ = =

,

. .

, 0, ./2

() , .

0 . 66

Page 31 of 259


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()

5/2

( ) 0(. + 5) 336

() 5/2

( + ) 0 (. + 5) 1036

,

( ) ( ) 0

()

( )m v 2 1

P P

() 0.1 30/ 20 /m s =

2 1

P P m mv =

( )m v

(20 30) 5

10 () .

, (3 ) (3 )

. .

()

.

..

2 + (3) (3 )

.

()

, 4 () (4 )

.

Page 32 of 259


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1: = ( ) = ( ) = 0.

2:

.

3:

.

4:

.

5:

.

..

.

6:

.

. ,

7: = 40, = 600 ( )

() ()

() = 6/2

= ( + ) = 40 (10 + 6) = 640 ( )

() = 4/2

= ( ) = 40 (10 6) = 240 ( )

() = 5/ () = 0

= = 40 10 = 400 ( )

() = ; = ( ) = ( )

= ( )

8:

Page 34 of 259


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(1)

Fs R Fs sR =

(2)FssR

=

(1) & (2)

10 30

. . 0.5

10 30 0.5

(sin cos )

om kg k

a g k

= = =

=

9.8(sin30 0.5cos30 )

9.8( .5 0.5 0.866)

o oa

a

=

=

9: . .

.. 0 constantFext then P= =

1 12 1 1 1 1m F t m m v= =

2 21 2 2 2 2m F t m m v= =

12 21F F=

( )1 1 1 1 2 2 2 2

1 1 2 2 1 1 2 2

m m v m m v

m m v m m

=

+ = +

.

.

tan Fs

R =

tan FssR

= =

a = 0.657m/s2

Page 35 of 259


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10: = 0.40

= 0/

= 8.0

8.0

0.40

Fam

= =

(1) = 510 ( 5)x ut= =

(2) = 25

2

2

1

210 25 ( 20)(25)

x ut at

x

= +

= +

(3) = 30

2

1

2

1

1

2

110 30 ( 20)(30)

2

x ut at

x

= +

= +

(4) = 30

10 ( 20)(30)

590 / .

u at

m s

= +

= +

=

30 1002

590 70x ut= =

= 1+ 2

2

20 /a m s=

x = -50m

18700x m=

x = -6000m

x = -50000m

241300x m=

Page 36 of 259


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1. 1

2. 1

3. . . 1

4. .

2

5. 16 12 200

.

2

6. 3000.

33000. 9.8/2.

2

7. 2

8.

() ,() .

() .

3

9. , .

.

3

10. 20 54/.

, 0.3/2 .

()

()

()

3

Page 37 of 259


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1

.

2

3

.

4 (1) .

(2) .

2 21 2 1 2

2 cosF F F F F = + +

( )2 21 22 2

90

( 16) (12)

oF F F

F

= + =

= +

20

20

200

F N

Fa

m

=

= =

( )F ma=

( )

ma T mg

T m a g

=

= +

33000 3000( 9.8)

33000 3000 9.8

3000

T N a

a

= = +

=

(1) .

(2) .

()

.

() .

20.1 / a m s=

21.2 /a m s=

Page 38 of 259


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()

.

.

.. d p

dt

( )

d pF k

dt

P m

=

=

dF km

dt

=

F kma=

( .. 1)

10 20 20 1000 54 / 15 / m tonnes kg u km hr m s= = = = 2

0.3 / 0a m s = =

() 20000 ( 0.3)F =

() u at = +

0 15

0.3

u at

ut

a

=

= =

()2 2

2v as = 2 2

(0) (15) 2( 0.3)s =

F kma=

F = -6000N

t = 50s

S = 375m

Page 39 of 259


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1. 1

2. 1

3. 1

4.

() .

() .

2

5. 1, 2

. .

.

2

6. 0.020 100.

0/,

2

7. 30 54/.

106.

3

. ,

.

3

. () . ..

()

5

Page 40 of 259


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1: .

2: FsR

=

.

3: .

4: (1) 0 0 ( )

(2) .

5: 1 2 1 2

1+ 2 0

11+ 2 2 0

11 2 2

1 1

2 2

a m

a m=

6: 0

(+)

g

0 g

g

0.02 80

100

b bg

g

b bg

g

m

m

m

m

=

= =

7: (1)

.

(2)

(3)( )

22 15tan

30 9.8rg

= =

tan 0.7653 =

0.016 / g m s =

37.4o=

Page 41 of 259


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8:

(3)

3Fa

m=

(1)

13

FF ma m

m= =

(2)

2 ( + )

2 2 2F m a m= =3

F

m

(3)

3F m a m= =3

F

m

9: () .

I=Ft

.Fav

() .

d pF

dt

or d p Fdt

=

=

1

2

0At t P P and att t P P

= =

= =

2

1

2 1

P t

vP

dp Fdt

P P Ft

=

=

( ) ImpulseFt I=

3 3FF =

22

3FF =

1 3FF =

2 1 P P I =

Page 42 of 259


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1. .

1

2. . . .. 1

3. .. 1

4. .

..

2

5. .

2

6. 2

7. .. 2

8. ,

3

9. 2 7

0.1.

(1) 10

(2) 10

(3) 10.

3

10. 3

Page 43 of 259


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47/261


48/261



A 10.

.

= F

.

B :

1)F k x=

A, extF =-F

, 1) extF =-F

extF =+k x 2)

) = extF .dx

= ( =00)

= ( 2)

,

, dw= kx dx

.. = =

2 2 2

20 0 1

2 2 2 2

x

x xK K Kx

= =

Page 46 of 259


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1. , 1

2.

1

3. 15 15

1

4. .

2

5. 2

6. 0 5

2 + 32. .

2

. 3

. .

2

.

3

.

.

.

3

10. 20% ..

3

Page 47 of 259


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51/261



7: 1 1 2 2

3 ? 0m kg v m v= = = =

( )

1

1 2 1 2 2

11 2

3

2

m m v m v

As m m

=

+

=

+

( ) ( )

( )

2 2

2

2

3 2 0

3 3

3

m m

m

m

+

=

+

+ ( )23 m = 3

2 2

2

2

3 9 3

4 9 3

4 6

m m

m

m

+ =

=

=

8:

0

. (1)

PE mgh

KE

T E mgh

=

=

=

( )1 2

1 1

( )

1

(0) 22

1

2

PE mg h x

KE M V gx

KE

=

= =

= 2M gx

KE mgx

TE mgh mgx

=

= mgx+

(2)TE mgh=

( ) (( )

2

2

2

2

. . 0

1

2

0 2

P E

KE M

g h

=

=

=

( )1

22

. (3)

KE M hg

KE mgh

ThusT E mgh

=

=

=

(1), (2) (3) .

2 6 4m =

Page 49 of 259


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9:

1 2

1 2

0

(1)

eu

eu

=

=

1 2

1 2(2)

mu m m

u

= +

+ =

(1) (2)

( )

( )

2

2

2 1

1 (3)2

u e

ue

= +

= +

(1) (2)

( )1 1 (4)2ue =

(4) (3)

10:2

. .2

PK E

m=

' 20%

206' 5 5100

P P of P

PPP P P P

= +

= + = + =

2

1

2

. '2

36 36. '

25 2 25

PK E

m

PK E E

m

=

= =

% . '

100%E E

E

' 361 100 1 100

25

E

E

= =

11100 44%

25= =

( )

( )1

2

1

1

e

e

+

=

Page 50 of 259


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1. If two bodies stick together after collision will the collision be elastic or inelastic? 1

2. , 1

3. .

, .

1

4.

, ,i j k

2 3 ,F i j k N = + +

, ,i j k ,

4

2

5. 1 2.

2

6. 2

7. 0.4 4/

0.6 2/.

, ..

2

8.

3

9. () . .

() ,

5

Page 51 of 259


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Ans1: Inelastic collision.

Ans2: Potential energy of an air bubble decreases because work is done by upthrust on the

bubble.

Ans3: The loss in potential energy appears as kinetic energy of the molecules of the cid.

Ans4: 2 3 ,F i j k N = + +

( ) ( )

4

.

2 3 . 4

S k

W F S

W i j k k

=

=

= + +

Ans5: Velocity of approach 1 12gh =

(Ball drops form height h1)

Velocity of separation 22gh=

(Ball rebounds to height h2)

Coefficient of restitution

2 2

1 1

2

2

ghe

gh

= =

Ans6: It states that work done by force acting on a body is equal to the change produced in its

kinetic energy.

If F

force is applied to move an object through a distance dS

Then .dw F dS =

W = 12 J

2h

e h=

Page 52 of 259


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.

.

F ma

dw ma d S

d v

dw m d S dt

=

=

=

dsdw m d

dt

dw m d

Integrating

=

=

2

2

w

o u

u

dW W m d

VW m

= =

=

Hence W = Kf Ki Where Kf and Ki are final and initial kinetic energy.

Ans7: m1= 0.4kg, u1= 4m/s, m2= 0.6kg u2= 2m/s.

Total K.E. be fore collision

( ) ( ) ( ) ( )

2 2

1 1 2 2

2 2

1 1

2 2

1 10.4 4 0.6 2

2 2

4.4

Ki m u m u

Ki

Ki J

= +

= +

=

Since collision is perfectly inelastic

1 1 2 2

1 2

2.8 / m u m u

m sm m

+

= =+

Total K.E. after collision

( )

( ) ( )

2

1 2

2

1

2

10.4 0.6 2.8

2

Kf m m

Kf

= +

= +

3.92Kf J=

Loss in K.E. ( )K =Ki Kf = 4.4 3.92 = 0.48J

2 2

2 2

m mu

W

=

Page 53 of 259


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57/261



. .

.

.

.

.

d tt

dt=

.

. +

+ .

.

. ()

()

() w

()

1

2 2G Me Ms r

Page 55 of 259


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( )

1 , .

2 .

3 .

4 sinA B AB x =

A

B

0

0

o

A B

=

=

5 L I w=

.

( ) (1) Argular Momentumd L d Id w

I w Idt dt dt

= = = =

(2)

d wwhere Torque

dt

I

= =

=

(1) (2)

M

r

2

2

Md rF

dt=

( )Md d r Md

F V cmdt dt dt

= =

0F =

( )

( )

0

0 0

MdV cm

dt

dor V cm as M

dt

=

=

( ) constantV cm =

d L

dt =

Page 56 of 259


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60/261



1. 1

2. 20 kg

1

4m

1

. 1

.

3 4 5 6 6 ?w i j x and r i j k

= + = +

2

. 2 2 2ow w = 2

.

2

.

2

. 0 1

10

.

. () .

() , . .K E

.

Page 58 of 259


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( )

1 .

2 22

5I MR=

22 1

205 4

I

=

3 (1)

(2)

(3)

4 w r =

( ) ( )

3 4 5 6 6

18 13 2

i j k i j k

i j k

= + +

= +

5dw

dt =

d

dw d

dt d

dw d

d dt

=

=

dww

d

d wdw

=

=

2

2

o

o

w

o w

w

w

d wdw

w

=

=

I = 0.5kgm2

Page 59 of 259


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2 2

2 2

ow w =

.

22

5MR

I M=2 2

5K M=

2

2 22

5

R

K R=

1 1 2 2

1 2

cm

m r m r r

m m

+=

+

1 1 2 2

1 1 2 2

0

0

cmr

m r m r

m r m r

=

+ =

=

1 1 2 2m r m r =

40 20 0.22

R cm m= = =

2 2 2

1 10rotations/sec.

. 1 (0.2) 0.04

M kg

M I MR kgm

= =

= = =

2 2 10 20 /

0.04 20

w rad s

L IW

L

= = =

=

=

2 22ow w =

2

5K R=

1 2

2 1

=m r

m r

2

2.51 /L kgm s=

Page 60 of 259


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() , .

.

()

( )' ' constant

''

IW I W IW

IWW

I

= =

=

..

2

2

1. ' '

2

1. '

2 '

K E I W

IWK E I

I

=

=

2 21 I W

K.E=2 I

As I'


64/261



1. 0.5 12/.

1

2. 1

3. 1

4. .

2

5. ow w at = + 2

6. 2 5 4F i j k = +

( )3 3 3r i j k m= + +

2

7.

27

5MR .

3

8. 1, 2, 3 4 , ,

1. .

3

. ()

() . 500 10

() ()

()

5

Page 62 of 259


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( )

1 v rw=

12

0.5

24 / .

vw

r

w rad s

= =

=

2

.

3 L r p=

sinor L rp =

r and p

4

.

.

5 dw

dt =

dw dt =

o

w t

w o

o

dw dt

w w t

=

=

6 r F =

( )

i j k

= 2 -3 4

3 2 3

= 17i-6 j-13k Nm

ow w t= +

Page 63 of 259


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7 . .

.

, , + ()2

1(. ) 27

5MR

1

2 2

I=I -M(OC)2

7I= MR -MR

5

8 1 1 11 ( , ) (0,0)m kg x y= =

2 2 2

3 3 3

4 4 4

2 ( , ) (1,0)

3 ( , ) (1,1)4 ( , ) (0,1)

m kg x y

m kg x ym kg x y

= =

= =

= =

1 1 2 2 3 3 4 4

1 2 3 4

0.5

m x m x m x m xXcm

m m m m

Xcm m

+ + +=

+ + +

=

1 1 2 2 3 3 4 4

1 2 3 4

0.7

m y m y m y m yYcm

m m m m

Ycm m

+ + +=

+ + +

=

(0.5, 0.7)

9 ()

.

() ()21

4Id MR=

2 21500 10 12500

4Id gmcm= =

() 2'Id Id MR= +

2 2

2 2

2

1

' 4

5 5' 500 10

4 4

' 62500

Id MR MR

Id MR

Id gmcm

= +

= +

=

()21

2I MR=

( )2

2

1500 10

2

25,000

I

I gmcm

=

=

225

I MR=

Page 64 of 259


67/261



1. 1

2. 1

1

1

3. .

1

4. 5.2 .

2

5.

12 21=F F

2

6. 2

7.

2

8. 400 2

6.4106

61024

. 2

.

3

10. (1) (2) (3)

.

3

Page 65 of 259


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01. .

02.

.

03. 21

F r

1 3

1/9

04. = 1 = 5.2

2 3

=Re

J JT R

Te

=(5.2) 3/2 1

= 11.86

05.12

F

= 1 2212

Gm m

rr

(1)

12F

21F

12r

21

r

1 221 122

=Gm m

F rr

(2)

& (1) (2)

1 1

aa

a

=

1 212 21

3

=Gm m

F rr

Page 66 of 259


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1 221 123

=Gm m

F rr

12 21

=r r

1 2

21 213=

Gm mF r

r

06. (1)

(=24)

(2)

..

07.

=GM

r

=2 r

=2 r

GM

r

=3

2r

GM

2 =2 3

4 r

GM

24

= R (constant)GM

2 1 1 2F F=

2 3T r

Page 67 of 259


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08. = 61024 = 400

= 6.4106

= 2= 12.8106

= 6.67104 2/2

=6.67 10 11 6 1

=2

Gmm

r

( )

11 24

6

6.67 10 6 10 400=

2 2 12.8 10

Gmm

r

. . =2

= -2K.E2

GMm

r

= 2 6.25 109 = 12.5109

. . = . + .

. . = 6.25 109 12.50 109

. . = 6.25 109

09. 2

= 1= 2+=3

= 1 22

Gm m

r

=

( )

( )

3 3

2

4 42

3 3

3

G p R pR

R

10.

dx

r

U F

=

=2

rGMm

dx

x

96.25 10KE Joules=

F =2 2 4128

27GP R

O3R

2R

O

R

Page 68 of 259


71/261


72/261



1.

1

2. . 1

3. 1

4. 2

5.

.

2

. 0

41 .

2

.

30 .2.

2

.

2

2

.

, 2 .

2

10.

5

Page 70 of 259


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01. 1 22

Gm m

r

1 2(2)

1 2 2

( )( ) 2

2 2

2 2F = =

4

G m m Gm

r r

.. .

02.

w mg o = =

03.

.

04. ( )1 /d R

1 R

R

0

0 .. .

05.

2

GMm

R

2

=F GMm

Rm

m

2

GMI

R=

M

RO

A

m

Page 71 of 259


74/261


75/261


76/261



1. 1

2.

1

3. 1

4. 2

. 2

.

2

. 400, 1

100

2

. .

3

. 4%

400

3

10.

3

Page 74 of 259


77/261


78/261



07. = 6400 = 6400 103

= 1600

= = 1 = 100 = 1009.8

() =

=

2R

R h

+

= 1009.8

26400

1600 6400

+

= 649.8 = 64

08.

=2 r

=Gm

r

=

34

3R P=

=3

3

24.3

R

G R P

32

4T

GP

=

09. =

2R

R h

+

g = 4% =4

100

g

= 6400

3T

GP

=

32 2

2v

r r rT

GmGm

r

= = =

Page 76 of 259


79/261


80/261



1. . ? 1

2. 0.1%

100

?

1

3. , :

1) 2) 3) .

2

4. A 51052.

1% : 1) 2) .

2

5. = = 0.74A,

= 0.5A; 1.9A ?

2

6. A

? ?

2

7. C A

?

2

8. . ? 2

9. :

1) B 2) B 3) .

3

10. 1, 1 2

2. ?

2

Page 78 of 259


81/261


82/261



1) ;

3

3

3

991

100 30.03

100

lV

V l

= = =

2) B , =5

7 2Stress 5 10 1.67 10 |Volumetric Strain 0.03

Normal XX N m= =

A 05. , = 0.74A.

= 0.74A

1) A = 0.5 A( ), .

2) A = 1.9A( ), .

3) A = , .

A 06. A

.

A 07. = Normal Stress

Longitudinal Strain

=F

AlL

F = F

A = A

=

=

= (G)

A =2 2

o x x+=

, D = F. A

Page 80 of 259


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( )( )

,

F=

work Dore = .2

= .2

FLNow y

Al

YAl

L

YAl x

L

YAx xl x given

L

=

=

A 08. .

A 09. 1) E B A

. .

B.

2) B A

,

: .

3) E ,

.

A 10. = .

A = .

C = (1)

C = (2)

, =Normal Stress

Longitudinal Strain

Y=T A

l l

=

=

A = A

Work Done =2

2

YAx

L

Page 81 of 259


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= C

=

( )

( )

1

1

2

2

for first case.

Y= for second case.

T l

Y XA l l

T lX

A l l

=

,

,

( ) ( )

( ) ( )

( )

2

1 2

1

1 2 2 1

1 2 1 2 2

2 1 2 1 1 2

1

T Tl lX X

A l l A l l

T l l T l l

T l T l T l l T

l T T T l T l

=

=

=

=

l= .

l = 2 1 1 2

2 1

T l T l

T T

Page 82 of 259


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1. 1

2. 1

3. .

2

4. 2

5.

)

2)

3)

2

6.

3

7.

. ,

) ,

) , .

) .

3

8. . 2

9. 2.010+11 /2.

2.81010, 109

2

3

Page 83 of 259


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(

1.A. H

.. A H ,

= E

E =

2.A. D

. : , I

B

. : C I, G.

3.A. B = B A;

,

.

4.A.

, .

E ( ) =Stress

Strain

:

1) = Normal StressLongitudinal Strain

2) B =Normal Stress

Volunetric Strain

3) =Stress

Strain

Tangential

Shearing

Page 84 of 259


87/261



5.A. 1) , H

H

.

2) B B A

.

3) , A

B .

6.A. I ,

, .

, .

=

= 2

I = >

.

7.A. 1) =Normal force

Area

= F

=2

cos cos=

cos

F F

A A

2) =StressTangential

Area

= F

A = A/

=sin

cos

F

A

V

o d r

Page 85 of 259


88/261


89/261



A = A

=

9 2F F

=10 N|m ; force constant = K =A l

= ? ; ?F

l=

, = =F

A

=F l

XA y

=

9 10

1110 2.8 102 10

x x

x

10 9 112.8 10

2

xl

+

=

122.8 10

2

xl

=

121.4 10l x m = (A0= 1010)

A 2 l = A = l l= l2(1)

=F F

=A A

=2

x

F

( )Using equation i

=Fx

(F ) = = F

;

=K

l

= l

= 2.010112.81010

= 22.8101110

= 5.6104

0 =0.014A

K = 56Nm-1

Page 87 of 259


90/261



1. 1

2. 1

3. 1

. 2

.

, .

2

. 0.0 /

2

2

. 1

2

3

. 1000 .

.

0.0 /

.

Page 88 of 259


91/261



( )

A01. ,

.

A02.

,

.

A03.

A04. A

,

.

:

1)

2) .

A05. ,

, .

. .

A06. , = = 0.06/

= 2 = 0.02

2= 5 = 0.05

, = 24 12

= 24(0.02)2

Page 89 of 259


92/261



= 321042

= 2422

= 24(0.05)2

= 2001042

= 20010432104

= 1681042

=

= 0.06168104

= 0.003168

A07. C 1 2 2.

:

1=3 3

1 2 2

4 4; v =

3 3r r

=

1& 2=

1= 21 2

4 4; P =

S S

r r

=34

3r

=4S

r

A , B

11+ 22=

3 3 3

1 2

1 2

4 4 4 4 4 4

3 3 3

S S Sx r x r x r

r r r + =

16 12+1622= 162

r =2 2

1 2r r+

Page 90 of 259


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A08. = 4

= 2 = 2103

= = 0.07/

= 1000

3 34 41000

3 3r x r =

3

42 102 10

10

xr x m

= =

= 42

10004242

= 4 100022

= 4 ( ) ( )( )2 2

4 3221000 2 10 2 10

2x x x

= 4 8 622

1000 4 10 4 104

x x x

= 8 5 622

10 107

=6 23168

107

x m

=

= 0.07 63168

107

x

= 3168108

A09. . A ,

,

:

R = 10r

Page 91 of 259


94/261



= A C

A = A D.

C.

, = f

a

A , D.

D :

= A

=f

Aa

A A,

..

. A

.

Page 92 of 259


95/261



1. 1

2. 2

3. 1000 ,

2

4. . ,

2

5.

= 76

= 13.6/

2

6.

2

7. .

,

2

8.

. .

,

2

9. 20% .

= 10003.

2

10. 5 :

1)

2) .

= 13.63 = 7.23

2

Page 93 of 259


96/261



(

A 01. A ,

.

A 02.

,

.

A 03. =

= B

, = D

1000 = 1000 D = 34

10003

r P

B = 34

3

R P

( )34

Volume of sphere = radius3

, 1000 = B

3 3

3 3

33

4 41000

3 3

1000

1000

r P R P

r R

Rr

=

=

=

:

1

10

.

10

Rr =

Page 94 of 259


97/261



A 04.

,

.

A 05. =

= 4

V

, = 1=

F = 2=

4

V

; 1= 76

:

2 76 of Hg13.6

hP cm

= +

A, B ,

1 1 2 2

76 7613.6 4

PV PV

h VXV X

=

= +

76 4 7613.6

304 7613.6

304 7613.6

228 13.6

hX

h

h

X h

= +

= +

=

=

A 06.

: =force

,Arer

A ..

.

.

h = 3100.8 cm

Page 95 of 259


98/261



A 07. = .

CD =

D

= D

= A

A = A .

= a l

, CD = ( )Al P AlP =

F = M a

accelerationa =

F = i)AlP a

A ,

CD

=force

Area

=

= ; = D; =

, F = A

F = A 2)

E 1) 2) C D :

AlP a = hPg A

A 08.

.

.

A 09. =

= 20

V100

alh

g=

C l D

a

h

Page 96 of 259


99/261



20Vw=V-

100

100 20

10080

100

0.8

V

V VVw

VVw

Vw V

=

=

=

= D

B , FB=

=

FB= 0.8 1)

= D

= A

= D

=

= 2)

A. ,

B =

F 1) & 2)

W

W

0.8V P g=PVg

P=P 0.8

, D = =1000

= 1000 0.8

, D = 800 3.

A 10. 1) =

B =

= 5 5 5

= 125 3.

B = D

= 125 7.2

= 900 1)

P = 800Kg|m

Page 97 of 259


100/261



= B D

= (5 )

= 5 5 (5 )

= D

= 5 5 (5 ) 13.6 2)

F, :

B = D

B g

= D g

F 1) & 2)

900 5 5 (5-h) 13.6

900 5 5(5-h) 13.6

900=125 13.6 (5-h)

900=1700 (5-h)

9005

1700

5 2.65

5 2.65

g g

h

h

h

=

=

=

=

=

2) , =

D = (5 )

= 5 5 1 = 25

1 = D 3

= 5 5 (5 ) 13.6 = 25 13.6 (5 )

A. ,

B = + .

900 = 25 + (25 13.6 (5))

900 = 25 + 1700 (5)

900 = 25 + 8500 1700

900 8500 = 1700 + 25

7600 1450

7600

1450

x

x

=

=

h = 2.35cm

2.54 cm = x

Page 98 of 259


101/261



1. . 1

2. ,

.

1

3. . 1

4. 0.1

7.2 102/

1.013 105/2

2

5. 2

6.

.

1

7. .

,

2

8. 1 1000 .

72 103/

2

. 4 ,

0.07 /

2

10. 1 64

72 103/2

2

11. 1

12.

. ,

( )

( )

3 3 3

2 2 24

P c a bT

a b c

=

+

Page 99 of 259


102/261



A 01. ,

,

.

A 02.

.

A 03. :

2 cosTh

rPg

=

=

=

= A

=

= D

= A

, C ,

.

A 04. ; = 0.1

= 0.1 103 (1 =103)

, = 7.2 102.

:

2 1

2TP P

R =

2=

1 = A

Page 100 of 259


103/261



( )23 2

2 1 3

3 2

2 1

5 3

5 2

2 7.2 10P P 1.44 10 /

0.1 10

, P = P +1.44 10 N/m

= 1.013 10 +1.44 10

= 1.027 10 N/m .

N m

Now

= =

A 05. , C

:

=

=

= A

=

= D

= A .

2TCosh

rPg

=

, ,

C . , .

A 06. ,

, .

A 07. =

= A = 42

E (E1) =

2 A

(2 )

E1= 2 (4 2)

E1= 8 2 1)

= 2

Page 101 of 259


104/261


105/261



A = A A

( )5 622 22

4 10 4 10 47 7

X

=

, D = A

( )

( )

( )

3 5 6 4

3 5 6

3 5 1

22 2272 10 4 10 4 10 eq 4)

7 7

2272 4 10 10 10

7

2272 4 10 10 1 10

7

X from

=

=

=

D 822 1

72 4 10 1

7 10

=

822 972 4 10

7 10

=

A 09. D = 4 = 4 102

= 2 102

A = 2 4 2

(2, 2 )

A = 2 4 (2 102)2

4

4 2

2 4 4 10

8 4 10 m

=

=

, = A

4

4

8 4 1022

0.07 8 4 107

T

=

=

E 422

0.07 2 4 4 107

=

4 2

4

7 8 22 410 10

7

7 10

X

J

=

=

Work Done = 8.14 10-

J

Page 103 of 259


106/261



A 10. = = 1 = 103

=

B = 64

3 3

3 3

33

4 464

3 3

64

64

R r

R r

Rr

=

=

=

310 1)

4 4

Rr

= =

A = 4 2

( )2

34 10 2) =

A = 24 64r

( )

2

23

4 644

104 64 3)

16

R

=

=

= A A

( ) ( )2

32

31064 4 4 10 equation 2 &34

from

=

D = A

( )6

3 6

3 6

1072 10 64 4 4 10 equation 4

4

6472 4 10 10 4

4

from

=

=

( )9

9

2272 4 10 16 4

7

72 4 22 12 10

7

=

=

Work Done = 2.7 10- J

Page 104 of 259


107/261



A 11. , C = =2TCos

rPg

.. (A )

C

.

A 12. , = 1= +4T

a

, 314

3V a=

, =2

4TP P

b= +

324

,3

b V b=

C = 34T

P Pc

= +

C, 334

.3

V c=

, , B :

1 1 2 2 3 3PV PV PV + =

3 3 3

2 3 2 23 3

2 3 2 23 3

4 4 4 4 4 4

3 3 3

4 16 4 16 4 16

3 3 3 3 3 3

4 16 4 16 4 160

3 3 3 3 3 3

T T TP a P b P c

a b c

T a P b T b T cP a P c

T a P b T b T cP a P c

+ + + = +

+ + + = +

+ + + =

4

3

( ) ( )

( ) ( )

( ) ( )

3 2 3 2 3 2

3 3 3 2 2 2

3 3 3 2 2 2

2 2 2 3 3 3

4 4 4 4 0

4 0

4

or

4T

Pa Ta b Tb Pc Tc

P a b c T a b c

P a b c T a b c

a b c P c a b

+ + + =

+ + + =

+ =

+ =

( )

( )

3 3 3

2 2 24

P c a bT

a b c

=

+

Page 105 of 259


108/261



1.

2

2. .

1

3. 1

4.

2

5. 1

5 . ,

2

6. 2

7. 5 5 1

1.55 2.

2

2

8. . , .

2

9. 15 5 0.06 0.02

.

15 ,

:

1)

2) .

3

10. 1 103 1 104 3

.

, .

:

1 103 2; 10 2;

1 103 3

3

Page 106 of 259


109/261



(

A 01.

. , = 0. ,

.

A 02. ,

1:4.

A 03.

f= A

f=viscous force

A=Area

velocity gradient

=co-efficient of viscosity

dv

dx

dv

dx

=

2

2

f

1[ ]

1 [ ]

dvA

dx

N

LT

N MLT

=

=

=

-2

2 1

TML

L T

=

1 -1TML =

Page 107 of 259


110/261



A 04. .

, ,

.

A 05. =

=

= 34

3r

= 34

3R

, ,

= 8 .

3 3

3 3

4 48

3 3

8

R r

R r

=

=

( )

( )

2

=2X1 mm 1 (Given)

=2 mm

=0.2cm 1 10

R r

r mm

cm mm

=

=

=

, = ( )2

T

2 rN = P- g 1)

9

= ( )2

T

2V = 2)

9

RP g

= C

= D

= D

=

D 42) 1)

2

2

2

2

T

T

T T

V R

N r

RV N

r

=

=

Page 108 of 259


111/261



= 5

( )

( )

2

2

0.25

0.1

0.04 =5

0.01

TV =

A 06.

. ,

.

A 07. = 5

B = 5

A = A = B

A = 5 5

A = 25 2

C = 21.55 |Ns m =

= = 1 = 103

= = 2 102

, =

22 3

3

1

dv 2 102 10

dx 10

=2 10 =20 m.

+

= =

, F=dv

Adx

24

3

-4

2 101.55 25 10

10

=1.55 25 10 20

F

=

VT= 20 cm |s

F = 0.0775 N

Page 109 of 259


112/261



A 08. , =Mass

Density

=

D =

= V=m

d

D = 1

= 1 1 1)m

m dd

=

( ), :

F =

2) =

( )

1

1

1

=mg-m g

mgd=mg- Using equation 1)

d

d=mg 1-

d

2)

, F = 11 d

mgd

A 09. F, :

4Pr

V =8 l

=

=

P

l=

= C

,

.

= 0.06

= 0.02

Page 110 of 259


113/261


114/261


115/261


116/261



( )

A01.

.

A02. () =C

SDV

n

= D

D = D

C= C

= C

I / 2000, 3000,

.

A03. B .

A04. , ,

=4Pr

8 l

=

=

= 2 .

=

= .

, 1

,

,

A05. A. :

F = 6 r v F :

F =

Page 114 of 259


117/261



1) =

2) =

3) =

A06. A , , = 1 H

= 0.01 H

= 0.01 (13.6103) 9.8

, 1= 0.35

A , 2= ?

2= 0.65

D , = 103 3

A. B ,

1+2 2

1 2 2

1 1

2 2sV P sV = +

( )2 22 1 2 11

2P P s V V =

= 0.0113.61039.8 ( ) ( )( )2 231

10 0.65 0.35

2

x

= ( )1 31

13.6 10 9.8 10 .4225 0.12252

x x x o

= ( )31

1332.8 10 0.32

x x

= 31332.8 0.15 10x

= 1332.8 150

= 1182.8 31182.8

Hg9.8 13.6 10 m ofx x

A07.

.

3

:

P2= 0.00887 m 0f Hg

Page 115 of 259


118/261



1)

2) =

3) (F) .

= D

=

= D .

= =

= 34

3r sg

F=

= 34

3r sog

=

A.

F = 6 rv

,

F+ F=

= 3 34 463 3

r sog rv r sg + =

=( )22

9

r s So g

1)

.

2)

A08. C A B

1, 2= A B

Page 116 of 259


119/261



1, 2= A B

1, 2= D A B

A = 11

A = 111

B = 112.

I ,

A = B

111= 222

I ,

1= 2 =

11 = 22

1 1= 22

. 1

a

I ,

.

H 1 D1= 21 = 21 = 2

D2 = ?

1 = 2

2= 10

D1= D

1=

1=

= 2, D=2

=2

,4 2

d Dr =

F

11= 22

22 2

1 21 2

4 4

D Dxv v

=

a v = constant

Page 117 of 259


120/261


121/261



F A = 11

F B =22

/ A = =

/ B =

A B = 1

2(1)

I A B I

A B = 2 1(2)

I / A B = 2 22 11 1

(3)2 2

mv mv

F, :

= I . E. / + I . E/

F 1, 2, & 3

1 2 = (2 1) +2 2

2 1

1 1

2 2mv mv

1 + 1+2 2

1 2 2 2

1 1

2 2mv p v mv mgh= + +

D

2 21 21 1 2 2

1 1

2 2

p v p vgh v v gh

m v m+ + + +

2 21 21 1 2 2

1 1

2 2

p pgh v v gh

s s+ + = + +

1 m

sv

= D

H, 4)

p

s=

=

21

2v =

H ), B .

21

2

pgh v

s

+ + = Constant

Page 119 of 259


122/261



1. B ? 1

2. .

?

1

3. .

, ?

1

4. . ? 1

5. 2 102 4 102

.

?

2

6. A 20 4 .

2/ 10 / 2. C

?

2

7. 2.5 105 /2.

, 2.0 10 5

/2. C ?

2

8. A . C

C

2:5. = 0.8

2

9. , .

, ?

2

10. , 5

/2 1 .

? D = 800/3?

3

11. 1) 8 1033/.

40 1042, C

.

2)

3 104. D 1000 /3.

3. 3.6 10 4?

3

Page 120 of 259


123/261



( )

A 01. A B ,

() .

A 02. ,

:

4

Q= 1)8

PR

l

1

( )

( )

( )

4

1

4

1

2

8 2

16 Using equation 1)

2 8 l

P RQ

l

PRQ

=

=

A 03. , 2 ,V gh=

, .

A 04. D ,

. A , (..

) .

A 05. D = 2 102

D = 4 102

A. ;

1 1 2 2 p p Q Qa v = a v A)a v a v=

, = C

, =

Q1= 8 Q

Page 121 of 259


124/261



, 2 r=radiusQa r=

2

2

2

2

1)4

)4

Q

Q

Q Q

P P

da

a d

a d

=

=

=

, A)

2 2

2 2

P Q Q2

4 4

V =d V

P P Q Q

P

QP

Q P

d V d V

d

dV

V d

=

=

( )

22

P

2

Q

2P

Q

P Q

V 4 10

V 2 10

V2

V

V =4V

=

=

.. .

A 06. A. ,

1 1 2 2

1 12

2

1)

a v a v

a vv

a

=

=

, 1= 1 = 2/

2= 2 = ? 2, 1= C A 2 & 1.

2

2 2 2; r =2cma r=

( )2 2 -2

2 0.02 =2 10 ma m=

Page 122 of 259


125/261



( )

2

1 1

2 2

1 1

-2

-1

0.1 r =10cm

=10 10 m

=10 m

=0.1m

a r

a m

=

=

, 1)

( )

( )

2

2 2

2 4

0.1 2

0.02

0.01 2

2 10

v

v

=

=

A. B , , ,

2, 1= 1 & 2

= D

( )

2 2

1 1 2 2

2 2

2 1 2 1

7 2

1

2

1

3 3

1 1

2 2

1

2

10 /

50 /

2 /

10 /

P sv P sv

P P s v v

P N m

v m s

v m s

s Kg m

+ = +

=

=

=

=

=

, ( ) ( )2 27 3

2

110 10 50 2

2P =

[ ]7 3

7 5

110 10 2500 42

10 12.48 10

=

=

A 07. 1=2.5105/2

1

2= 2.0105/2

2 50 /v m s=

6 2

2 8.752 10 / P N m=

Page 123 of 259


126/261



2

1= 0 ( )

2= ?

D = = 1000 3

A. B ,

( )

( )

2 2

1 1 2 2

2

1 2 2

1 22

2

5 5

2

2

52

2

2

1 1

2 2

1

2

2

2 2.5 10 2 10

1000

2 10 0.5

1000

100

P sv P sv

P P sv

P P

v s

v

v

v

+ = +

= +

=

=

=

=

A 08.4

11

18

P RQ

l

=

1= 1

1=

=

1= 1C

1= C

2= D

1

1

vQ

t=

=

1= C 1

2 10 /v m s=

Page 124 of 259


127/261


128/261


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