MKA Physics -1

7/25/2019 MKA Physics -1

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PROBLEMS IN PHYSICS

MKA1. Two particles are simultaneously thrown

from roofs of two high buildings, as shown

in figure. Their velocities of projection are

2ms-1and 14 ms-1respectively. Horiontal

and vertical separation between points !

and " is 22m and # m respectively.

$alculate minimum separation between

the particles in the process of their

motion.

Ans: 6.00 m

MKA 2.! ball of mass m is thrown at an angle of 4%&to

the horiontal from top of a '% m high tower

!" as shown in figure. !nother identical ball is

thrown with velocity 2& ms-1 horiontally

towards !" from top of a (& m high tower $)

one second after the projection of first ball.

"oth the balls move in same vertical plane. *f

they collide in mid air

+i $alculate distance !.$.

+ii )uring collision the two balls get stuc

together, calculate the distance between !

and the point on the ground, at which the

combined ball stries. iven g / 1& ms -2.

Ans: +i 4& m +ii 1% m

MKA 3. Two inclined planes 0! and 0" having inclination +with horiontal (&& and '&&

respectively, intersect each other at 0 as shown in figure. ! particle is projected from

point with velocity u / 1& ms-1 along a direction perpendicular to plane 0!. *f theparticle stries plane 0" perpendicularly at , calculate

+i velocity with which particle stries the

plane 0",

+ii time of flight,

+iii vertical height h of from 0,

+iv ma3imum height from 0, attained by

the particle, and

+v )istance ..

Ans: +i 1& ms-1 +ii 2 sec.

+iii % m +iv 1'.2%

+v 2& m

(


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MKA 4. ! particle is moving along a vertical circle of radius / 2& m with a constant speed

v / (1.4 ms-1as shown in figure. 5traight line !"$ is horiontal and passes through the

centre of the circle. ! shell is fired from point ! at the instant when particle is at $. *f distance

!" is 2&m and shell collides with the particle at ", calculate

+i smallest possible value of the angle ofprojection,

+ii $orresponding velocity u of projection.+/ (.14 and g / 1& ms-2

Ans: +i /tan-1

+ii 2&

ms-1

MKA 5. ! particle is projected from point 0 on the ground with velocity u / % ms-1 at angle

/tan-1+&.%. *t stries at a point $ on a fi3ed smooth plane !" having inclination of (6&

with horiontal. *f the particle does not rebound, calculate

+i co 7 ordinates of point $ inreference to co-ordinate system

shown in figure.

+ii ma3imum height from the

ground to which the particle

rises. +g / 1& ms-2

Ans: +i +% m, 1.2% m

+ii 4.4% m

MKA 6. Two identical shells are fired from a point on the ground with same mule velocity at angles

of elevation / 4%&and / tan-1( towards top of a cliff, 2& m away from point of firing. *f boththe shells reach the top simultaneously, calculate

+i mule velocity,

+ii height of the cliff, and

+iii time interval between two firings. *f just before striing the top of cliff the two shells

get stuc together, considering elastic collision of combined body with the top,

calculate

+iv ma3imum height reached by the combined body.

Ans: +i 2& ms-1 +ii 1& m

+iii +iv 12 m

+above the ground

MKA 7.! shell of mass m / 6&& gm is fired from ground with a velocity 4& ms-1. !t highest point of its

trajectory, it collides inelastically with a ball of mass 8 / 1.( g, suspended by a fle3ible

thread of length 1.4& m. *f thread deviates through an angle of 12& &, calculate

+i angle of projection of shell,

+ii ma3imum height of combined body from ground, and

+iii distance between point of suspension of ball and point of projection of shell.

Ans: +i '& +ii '2.('2% m +iii #2.%6 m

MKA 8.! circle of radius / 2 m is mared on upper surface of a horiontal board, initially at rest. !particle starts from rest along the circle with a tangential acceleration a / &.2% ms -2. !t the

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21 +2n 1 (&(

=

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+ 1& 2sec 1.62sec =


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same instant board accelerates upwards with acceleration b / 2.% ms -2. *f the co-efficient of

friction between board and particle is / &.1, what distance with the particle travel on theboard without sliding9 +g / 1& ms-2

Ans:

MKA .Two small particles !and " having

masses m / &.% g

each and charge :1/ and : / + ; 1&& $ respectively, areconnected at the ends of a non7 conducting, fle3ible and

ine3tensible string of length r / &.% m. article ! is fi3ed and "

is whirled along a vertical circle with centre at !. *f a vertically

upward electric field of strength < / 1.1 1&%=$-1e3ists inthe space, calculate minimum velocity of particle ", re:uired at

highest point so that it may just complete the circle.

+g /1& ms-2

Ans:

MKA 10.! small sphere of mass m / &.% g carrying a positive charge : / 11& $ is connected with alight, fle3ible and ine3tensible string of length r / '& cm and whirled in a vertical circle. *f a

vertically upwards electric field of strength < / 1&% =$-1 e3ists in the space, calculate

minimum velocity of sphere re:uired at highest point so that it may just complete the circle.

+ g / 1& ms-2

Ans: ' ms-1

MKA 11.! small sphere of mass m / &.' g carrying positive charge : / >& $ is connected with alight, fle3ible and ine3tensible string of length r / (& cm and whirled in a vertical circle. *f a

horiontally rightward electric field of strength < / 1&%

=$-1

e3ists in the space calculateminimum velocity of sphere re:uired at highest point so that it may just complete the circle.

+ g / 1& ms-2

Ans: ( ms-1

MKA 12.! particle of mass m / &.1 g and having positive charge : / 6% $ is suspended from apoint by a thread of length l /1& cm. *n the space a uniform horiontal electric field


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MKA 13. Two blocs ! and " of mass 1 g and 2 g respectively are

connected by a string, passing over a light frictionless pulley.

"oth the blocs are resting on a horiontal floor and the pulley

is held such that string remains just taut.

!t moment t / &, a force ? / 2& t =ewton starts acting on the

pulley along vertically upward direction, as shown in figure.

$alculate+i velocity of ! when " loses contact with the floor,

+ii height raised by the pulley upto that instant, and

+iii wor done by the force ? upto that instant. + g / 1&ms -2

Ans: +i % ms-1 +ii %@' m +iii 16%@' joule

MKA 14. ! uniform fle3ible chain of length 1.%& m rests on a fi3ed

smooth sphere of radius / 2@m such that one end ! ofchain is at top of the sphere while the other end " is hanging

freely. $hain is held stationary by a horiontal thread ! as

shown in figure. $alculate acceleration of chain when thethread is burnt. + g / 1&ms-2

Ans:

MKA 15. *n the arrangement shown in figure 1%. ulley ) and < are small

and frictionless. They do not rotate but threads slip over them

without friction and their masses being 4 g and 11.2%g

respectively while masses of blocs !, " and $ are 2 m, m and

mA respectively. Bhen the system is released from rest,downward accelerations of blocs " and $ relative to ! are found

to be % ms-2and ( ms-2respectively. $alculate

+i accelerations of blocs " and $, relative to the ground, and

+ii mass of each bloc. + g / 1&ms -2

Ans: +i ( ms-2, 1 ms-2+"oth downwards +ii 8ass of ! 7 1> g

8ass of " / # g

8ass of $ / 6 g

MKA 16. *n the arrangement shown in figure. mass of bloc !,

" and $ is 6.% g, ' g and 1 g respectively. the pulleyis solid circular disc of mass &.% g, radius 2& cm and

thicness 1 cm. Thread between bloc ! and pulley is

horiontal and that between pulley and bloc $ is

vertical. The pulley is free to rotate about a3is 0 without

friction and thread does not slip over its curved surface.

=eglecting friction between blocs " and $ and that

between blocs and the floor, calculate resultant

acceleration of bloc $ when the system is released.

+ g / 1&ms-2

Ans:


24 .g 6.%>ms(

+ =

2% ms


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MKA 17.*n the arrangement shown in figure, pulley are small, light

and frictionless, threads are ine3tensible and mass of

blocs !, " and $ is m1/ % g, m2/ 4 g and m(/ 2.% g

respectively. co 7 efficient of friction for both the planes is

/ &.%&. $alculate acceleration of each bloc whensystem is released from rest. + g / 1&ms -2

Ans: a1/ 4 ms-2, a2/ &, a(/ 2 ms

-2

MKA 18. *n the arrangement shown in figure mass of blocs !, " and $ is 1>.% g, > g and 1.% grespectively. "ottom surface of ! is smooth, while co7efficient of friction between " and floor

is &.2 and that between blocs ! and $ is 1@(. 5ystem is released from rest at t / & and

pulleys are light and frictionless. $alculate

+i acceleration of bloc $, and

+ii energy lost due to friction during first &.2 sec.

+g / 1& ms-2

Ans: +i

+ii &.1#

joule

MKA 1. ! bloc resting over a horiontal floor has a symmetric trac

!"$, as shown in figure. 8ass of the bloc is 8 / (.12 g. Cength

!" / Cength "$ / 1 m. ! bloc of mass m / 2 g is put on the

trac at ! and the system is released from rest. =eglecting friction

and impact at ", calculate period of horiontal oscillations

performed by the bloc of mass 8. + g / 1& ms

-2

Ans: 2 sec.

MKA 20.*n the arrangement shown in figure a wedge of mass m(/ (.4% g is placed on a smoothhoriontal surface. ! small and light pulley is connected on its top edge, as shown. ! light,


21&ms


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fle3ible thread passes over the pulley. Two blocs having mass m 1/ 1.( g and m2/ 1.% g

are connected at the ends of the thread. m 1 is on smooth horiontal surface and m2rests on

inclined surface of the wedge. "ase length of wedge is 2m and inclination is (6&. m2is initially

near the top edge of the wedge. *f the whole system is released from rest, calculate

+i velocity of wedge when m2reaches its bottom,

+ii velocity of m2at that instant and tension in the

thread during motion of m2. !ll the surface aresmooth. + g / 1& ms-2

Ans: +i 2 ms-1

+ii

MKA 21.! small, light pulley is attached with a bloc $ of mass 4 g is placed on the top horiontal

surface of $. !nother loc ! of mass 2 g is hanging from a string, attached with " and

passing over the pulley. Taing g / 1& ms-2and neglecting friction, calculate acceleration of

each bloc when the system is released from rest.

*f initial height of lower surface of bloc ! is 12.% cm from bottom of a hole cut in $, calculate

inetic energy of each bloc and loss of potential energy of ! when it hits the bottom of the

hole.

Ans: Dertical acceleration of ! / '.2% ms-2 +)ownward

Horiontal acceleration of ! / 1.2% ms -2+ightward

!cceleration of " / %.&& ms-2+Ceftward

!cceleration of $ / 1.2% ms-2+ightward

E< of ! / 1.'2% F,

E< of " / &.6% F,

E< of $ / &.12% F,

Coss of < / 2.%& F

MKA 22.! board is fi3ed to the floor of an elevator such that the board forms

angle / (6& with horiontal floor of the elevator acceleratingupwards. ! bloc is placed on point ! of the board as shown in figure.

Bhen motion with velocity v1/ 4 ms1is given to the bloc, it comes to

rest after moving a distance l / 1.'m relative to the board. *ts velocity

was v2/ 4 ms1down the board when it returns to point !. $alculate

acceleration a of elevator and coefficient of friction between the boardand the bloc. +g / 1& ms2

Ans: 2.%ms-2, &.2%


11(ms , (.# newton

2


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MKA 23. ! bloc " of mass 1& g is resting over a smooth

horiontal plane. )istance of " from the wall is 4&

cm and it is held at rest by an ine3tensible thread

"). !nother thread is connected to left face of " and

a bloc ! of mass 2 g is suspended as shown in

figure. "loc $ of mass 2 g is resting against the

vertical wall. "locs " and $ are hinged at the endsof a light rigid rod.

!ssuming friction to 2& cm is absent, calculate

acceleration of each bloc when thread ") is burnt.

+g / 1& ms2

Ans: !cceleration of ! / ( ms-2+!cceleration of " / ( ms-2+ !cceleration of $ / 4 ms-2+

MKA 24. Three identical blocs !, " and $, each of mass m / 6 g are

connected with each other by light and ine3tensible strings, as

shown in figure. 5trings pass over light and frictionless pulleys

fi3ed to the edges of trolly of mass 8 / 21 g. *f co efficient offriction between blocs and trolly surfaces is / 4@6, calculatema3imum possible value of angle so that bloc " remainsstationary relative to the trolly. $alculate also, the force ? to be

applied horiontally on the trolley.

$alculate also, the force ? to be applied horiontally on the trolley.

Ans: (6, (1% newton



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MKA 25.Two blocs of mass m1and m2are attached at the ends of an ideal spring of force constant

E and natural length l&. The system rests on a smooth horiontal plane. "locs having mass

m1 and m2 are pulled apart by applying force ?1 and ?2respectively as shown in figure.

$alculate ma3imum elongation of the spring.

Ans:

MKA 26.! uniform solid sphere of radius / 44 cm is cut into two parts

by a plane. )istance of the plane from centre of the sphere is

a / 2'.4 cm as shown in figure. $alculate distance of centre of

mass of heavier part from centre 0.

Ans:

MKA 27. ! vehicle of mass m starts moving along a horiontal circle of radius such that itsspeed varies with distances s covered by the vehicle as v / E, where E is a

constant. $alculate

i tangential and normal force on vehicle as function of s,

ii distance s in terms of time t, and

iii wor done by the resultant force in first t seconds after the beginning of motion.

Ans: +i +ii +iii

MKA 28.! particle of mass m moves along

a horiontal circle of radius such that normal acceleration of particle varies with time as an / Et2,

where E is a constant. $alculate

+i tangential force on particle at time t,+ii total force on particle at time t.

+iii power developed by total force at time t, and

+iv average power developed by total force over first t second.

Ans: +i +ii

+iii +iv

MKA 2. Two identical blocs ! and ", each of mass m / 2 g are connected to the ends of

and ideal spring having force constant E / 1&&& =m1. 5ystem of these blocs and

spring is placed on a rough force. $oefficient of friction between blocs and floor is / &.%."loc " is pressed towards left so that spring gets compressed.

+i $alculate initial minimum compression 3-&of spring such

that bloc ! leaves contact with the wall when system is

released.+ii *f initial compression of spring is 3 / 2 3 &, calculate

velocity of spring isx/ 2 x &,calculate velocity of centre of

mass of the system when blocAjust leaves contact with the

wall. +g / 1& ms72

Ans: +i (cm

+ii

/ &.%1 ms-1

MKA 30.!n ice cube of sie a / 2-& cm is floating in a tan +base area ! / %& cm 3 %& cm partially

filled with water. )ensity of water is 1/ 1&&& g m7(and that of ice is 2 -/ #&& gm7(.$alculate increase in gravitational potential energy when ice melts completely.

Ans: -&.62 F


1 2 2 1

1 2

m ? m ?2

E m m

+ +

1(2cm

(%

5

221 mE smE ,

2

2 21E t4

4 21mE t>

m E 4m + Et +mET1

mEt2

11 1.&%ms2


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MKA 31. ! cubical bloc of wood +density 1 / %&& g m 7( has side a / (& cm. *t is floating in arectangular tan partially filled with water +density 2/ 1&&& g m7(and having base area ! /4% cm 3 '& cm. $alculate wor done to press the bloc so that it is just. *mmersed in water.

Ans: '.6% F

MKA 32.! bloc of mass mis held at rest on a smooth horiontal floor. ! light frictionless, small pulley

is fi3ed at a height of ' m from the floor. ! light ine3tensible string of length 1' m, connected

with ! passes over the pulley and another identical bloc B is hung from the string. *nitial

height of B is % m from the floor as shown in figure. Bhen the system is released from rest, B

starts to move vertically downwards andA sides on the floor towards right.

+i *f at an instant string maes an angle withhoriontal, calculate relation between velocity uof

A and v of B,

+ii $alculate vwhen B stries the floor.

+g / 1& ms72

Ans: +i u /v sec

+ii

MKA 33.! string with one end fi3ed on a rigid wall, passing over a fi3ed frictionless pulley at a distance

of 2 m from the wall, has a point mass M of 2 g attached to it at a distance of 1 m from the

wall. ! mass m of &.% g is attached to the free end. The system is initially held at rest so that

the string is horiontal between wall and pulley and vertical beyond the pulley as shown in

?igure

Bhat will be the speed with which the point mass M will hit

the wall when the system is released9 +g / 1& ms72

Ans:

MKA 34.Two identical buggies each of mass 1%& g move one after the other without friction withsame velocity 4 ms71. ! man of mass m rides the rear buggy. !t a certain moment the man

jumps into the front buggy with a velocity vrelative to his buggy. !s a result of this process

rear buggy stops.

*f the sum of inetic energies of man and front buggy just after collision differs from that just

before collision by 26&& joule, calculate values of m and v.

Ans: %& g, 1' ms-1

MKA 35. Two balls of mass m1/ 1&& gm and m2 / (&& gm are suspended from point A by

two e:ual ine3tensible threads, each of length l / (2@(% m. "all of mass m1 is drawn

aside and held at the same level asA but at a distance from A, as shown in figure.

Bhen ball m1is released, it collides elastically with the stationary ball of mass m2. $alculate


14&v ms41

=

1 1% %% 3 ms (.(#ms'

=

(

2l


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PHY-PP-10

+i velocity u1with which the ball of mass m1collides with

the other ball, and

+ii ma3imum rise of centre of mass of the ball of mass m 2.

+g / 1& ms72

Ans: +i 4 ms-1 +ii &.2& m

MKA 36.Two identical blocsA and B, each of mass m / 1.% g and carrying positive charge q/ (&

$are ept stationary on a smooth horiontal floor. Bhen the blocs are released, due toelectrostatic repulsion. Amoves towards left while B towards right. !fter moving 1&cm, A

comes into contact with a mass less spring of force constant K/ '6%& =m 71 while after

moving 1& cm, Bcollides inelastically with a rigid wall as shown in ?igure. $alculate.

+i velocity ofAwhen it comes in contact with the

spring, and+ii ma3imum compression of the spring.

Ans: +i ' ms-1 +ii 1& cm

MKA 37. Two small blocsA and Bof masses, m1/ &.% g and m2/ 1 g respectively, each carrying

positive charge of q/ 4& $are ept stationary on a smooth horiontal floor. Bhen the blocsare released, due to electrostatic repulsion,A moves towards left while Btowards right. !fter

moving 4& cm,A comes in contact with a mass less spring of force constant K/ 6'&& =m71

while after moving 2& cm, B collides inelastically with a rigid wall as shown in figure. $alculate

+i velocity of Awhen it comes in contact with the

spring and

+ii ma3imum compression of the spring.

Ans: +i 12 ms-1 +ii 1&1 cm

MKA 38. Two identical blocsAand B,each of mass m / '&& gm, each carrying positive charge of

q/ 4& $are ept stationary on a smooth horiontal floor, Bhen the blocs are released, dueto electrostatic repulsion,Amoves towards left while Btowards right. !fter moving 4& cm,A

comes in contact with a mass less spring of force constant K/ '4&& =m71, while after moving

2&cm, Bcollides inelastically with a rigid wall as shown in figure. $alculate.

+i velocity of Awhen it comes in contact with the

spring and

+ii ma3imum compression of the spring.

Ans: +i 1& ms-1 +ii 1&1 cm



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MKA 3.! wooden bloc of mass 6&& gm is suspended by a light rigid rod of length 1 m form A. The

rod is free to rotate in a vertical plane through A,without friction.

! bullet of mass 1& gm is fired from point 0 on the ground with velocity 1&& ms71at angel of

elevation . !t highest point of its trajectory, it stries the wooden bloc. !t that instant blocwas moving in vertical circle with velocity 6 ms71and inclination of rod with vertical was (6asshown in ?ig. The bullet gets embedded into the bloc and the combined body just completes

vertical circle. $alculate

+i velocity of the combined body just after collision.

+ii velocity of bullet just before collision, and

+iii co7ordinates of !, in reference to co7ordinate system as

shown in figure. +g / 1& ms72

Ans: +i ' ms-1 +ii >& ms-1

+iii +4>&.' m, 1>&.> m

MKA 40.! bullet of mass m, moving horiontally with velocity v0/ ( ms7

1stries elastically with a body of mass M / 2 m suspended by

two identical threads of length l/ 1 m each as shown in ?ig.

$alculate.

+i ma3imum deflection angle with vertical of thread, and+ii period of small oscillations of body M. +g / 1& ms72

Ans: +i (6 +ii

MKA 41.! body of mass M/ 2 m rests on a smooth horiontal

plane. ! small bloc of mass mrests over it at left end !

as shown in ?ig. 41. ! sharp impulse is applied on the

bloc, due to which it starts moving to the right with

velocity v&/ ' ms71. !t highest point of its trajectory,

the bloc collides with a particle of the same mass m

moving vertically downwards with velocity v /2 ms71

and gets stuc with it. *f the combined body lands at the

end point A of body of mass M, calculate length l.+=eglect ?riction. +g / 1& ms72


2sec

1&


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PHY-PP-12

Ans: 4& cm

MKA 42. *n the arrangement shown in ?ig. 42, ball and bloc

have the same mass m/ 1 g each, / '&&and lengthl/ 2.%& m. $o7efficient of friction between bloc and

floor is &.%. Bhen the ball is released from the positionshown in the figure, it collides with the bloc and the

bloc stops after moving a distance 2.%& m.

?ind coefficient of restitution for collision between the

ball and the bloc. +g / 1& ms72

Ans: 1

MKA 43. ! bloc Bof mass m/ &.% g is attached with upper end of a

vertical spring of force costant K/ 1&&& =m 71as shown in ?ig. 4(.

!nother identical bloc ! fall from a height h/ 4#.% cm on the

bloc B and gets stuc with it. The combined body starts to

perform vertical oscillations.

$alculate amplitude of these oscillations. +g / 1& ms72

Ans: % cm

MKA 44.Three identical balls each of mass m/ &.% g are

connected with each other as shown in ?ig. 4( and

rest over a smooth horiontal table. !t moment t/ &,

ball " is imparted a velocity v& / # ms71. $alculate

velocity of ! when it collides with bass $.

Ans: ' ms-1

MKA 45. Two small particles, each of mass mcarrying positive

charge qeach are attached to the ends of a non7

conducting light thread of length 2 l. ! third particle of

mass 2 mis attached at mid7point of the thread. The

whole system is placed on a smooth horiontal floor

and the particle of mass 2 mis given a velocity vas

shown in figure. $alculate minimum distance betweenthe two charged particles during the process of



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PHY-PP-13

motion.

Ans:

MKA 46.! pan of mass m/ 1.% g and a bloc of mass M/ ( g are connected with each other by a

fle3ible, light and ine3tensible string, passing over a small, light and frictionless pulley. *nitially

the bloc is resting over a horiontal floor as shown in figure.!t t / &, an inelastic ball of mass m&/ &.%g collides with the pan

with velocity v&/ 1' ms71+vertically downwards. $alculate

+i ma3imum height, upto which the bloc rises,

+ii the time tat which bloc stries wit1h the floor,

iii *f the bloc comes to rest just after striing the floor, calculate

velocity of pan at t/ 2 second. +g / 1& ms72

Ans: +i &.'4 m +ii 1.'& sec

+iii &.4> ms-1+downward

MKA 47. Two identical blocs A and B each of mass 2 g are hanging

stationary by a light ine3tensible fle3ible string, passing over a light

and frictionless pulley, as shown in ?ig. 46. ! shell $, of mass 1 g

moving vertically upwards with velocity # ms71 collides with bloc B

and gets stuc to it. $alculate

+i time after which bloc Bstarts moving downwards,

+ii ma3imum height reached by B, and

+iii loss of mechanical energy up to that instant.

Ans: +i &.# second +ii &.>1 m

+iii (2.4 joule

MKA 48. ! light fle3ible thread passes over a small, frictionless pulley. Two

blocs of mass m / 1 g and M/ ( g are attached with the thread

as shown in ?ig. Heavier bloc rests on a slab. ! shell of mass 1 g,

moving upwards with velocity 1& ms71, collides with the hanging bloc

at time t/ &. $alculate.

+i ma3imum height ascended by Mwhen it is jered into motion, and+ii time tat that instant G

+a *f shell gets stuc the hanging bloc.

+b *f shell collides with the hanging bloc elastically. +g / 1& ms72

Ans: a. +i 1 m a. +ii 2 sec

b. +i &.'2% m b. +ii 2.%& sec.


2

2 2

&

2: l

4 mv l : +


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PHY-PP-14

MKA 4. *n the arrangement shown in ?ig. pulleys are light and

frictionless and thread are fle3ible and ine3tensible. 8ass of

each of the blocsAand B is m/ &.% g. *nitially B is resting

over a slab and ! is hanging.

! shell of e:ual mass m / &.% and moving vertically upwards

with velocity v& / 12 ms71 stries the bloc A and getsembedded into at t / &. $alculate.

+i ma3imum height ascended by Bwhen it is jered into motion

and time tat that instant, and

+ii time twhen ! stries the slab. *nitial height of blocA from

the slab is h/ 1& cm. +g / 1& ms72

Ans: +i 1m, 1.'% second +ii 1.2% second

MKA 50.! ball of mass m/ 1 g is hung vertically by a thread of

length l/ 1.%& m. pper end of the thread is attached to the

ceiling of a trolley of mass 8 / 4 g. initially. Trolley is

stationary and it is free to move along horiontal rails

without friction.

! shell of mass m/ 1 g, moving horiontally with velocity

v&/ ' ms71, collides with the ball and gets stuc with it. !s a

result, thread starts to deflect towards right. $alculate its

ma3imum deflection with the vertical. +g / 1& ms72

Ans: (6

MKA 51.! small steel ball ! is suspended by an ine3tensible thread of length l =1.%m from 0. !nother identical ball is thrown vertically downwards such that its

surface remains just in contact with thread during downward motion and

collide elastically with the suspended ball. *f the suspended ball just.

$ompletes vertical circle after collision. $alculate the velocity of the falling

ball just before collision and its distance from 0 after t/ &.1 second after the

collision. +g / 1& ms72

Ans: 12.% ms-1, 1.(&2 m

MKA 52.! bloc ! of mass m = 5 kg is attached with a spring having force constant k/ 2&&& =m71.

The other end of the spring is fi3ed to a rough plane, inclined at (6with horiontal and

having coefficient of friction / &.2% "loc ! is gently placed on the plane such that thespring has no tension. Then bloc ! is released slowly.



15/52

PHY-PP-15

+i $alculate elongation of the spring when e:uilibrium

is achieved. =ow an ine3tensible thread is connected

with bloc ! and passed below pulley $ and over

pulley ), as shown in figure,. 0ther end of the thread

is connected with another bloc " of mass ( g. "loc

" is resting over a table and thread is loose.*f the

table collapses suddenly and " falls freely through>&@# cm the thread becomes taut, calculate

+ii combined speed of blocs at that instant, and

+iii ma3imum elongation of spring in the process of

motion. +g / 1& ms72

Ans: +i 1 cm +ii &.% ms-1

+iii ' m

MKA 53.! right angled wedge !"$ of mass 8 / 4 g and base

angel / %(is resting over a smooth horiontal plane. !shell of mass m/ &.% g moving horiontally with velocity

v&/ 4& ms71, collides with the wedge, just above point !.

!s a conse:uence, wedge starts to move towards left with

velocity v-/ % ms71. $alculate

+i heat generated during collision.

+ii ma3imum height reached by the shell, and

+iii distance of point ! of wedge from the shell when shell

stries the plane. +g / 1& ms72

Ans: +i 12% joule +ii 4% m

+iii (& m

MKA 54.! uniform chain !A "A of length 2l having mass per unit length ishanging from ceiling of an elevator by two light, ine3tensible

threads !!A and ""A of e:ual length as shown in figure. )istance!" is very small. !t a certain instant, elevator starts ascending



16/52

PHY-PP-16

with constant acceleration a. Two seconds after the beginning of

motion, thread BB is burnt. !ssuring that instant to be t / 0,

calculate tension in thread !!A at time t.

Ans:

MKA 55. ! turn table is free to rotate about a fi3ed vertical a3is and has a smooth groove madeon its upper surface along a radius. The table is rotated about the a3is with constant angular

velocity and a particle of mass m is gently placed in the groove at distance a from the a3is of

rotation. $alculate magnitude of resultant velocity of the particle as a function of its distance 3

from a3is of rotation. $alculate also, tor:ue re:uired to eep the angular velocity constant.

Ans:

MKA 56. ! light ine3tensible string is passed through a hole made in a

smooth horiontal table top. Two masses m1/ ( g and m2/ '.2

g are connected at the ends of the strings as shown in ?ig. %'.

*nitially, m2 is held at rest and m1 is rotated along a horiontal

circle of radius r&/ 2& cm with angular velocity &/ 1> rad sec71.$alculate.

+i acceleration of m2when it is released from rest, and velocity of

m1when radius of its circular path becomes (& cm.

Ans: +i

+ii

MKA 57. ! uniform solid sphere of mass 1 g and radius 1& cm is

ept stationary on a rough inclined plane by fi3ing a highly

dense particle at B. *nclination of plane is (6 with horiontaland AB is the diameter of the sphere which is parallel to the

plane, as shown in ?ig. %6 $alculate.+i mass of the particle fi3ed at B, and

+ii minimum re:uired coefficient of friction between sphere

and plane to eep sphere in e:uilibrium.

Ans: +i ( g +ii &.6%

MKA 58.! ball of radius R/ 2& cm has mass m/ &.6% g and moment of inertia +about its diameter

I/ &.&12% g m2. The ball rolls without sliding over a rough horiontal floor with velocity v&/

1& ms71towards a smooth vertical wall. *f coefficient of restitution between the wall and the

ball is e/ &.6 calculate velocity v of the ball long after the collision. +g / 1& ms72

Ans: 2 ms-1

MKA 5.!" is a horiontal diameter of a ball of mass m / &.4 g and radius

/ &.1& m. !t time t / &, a sharp impulse is applied a " at angle of

4%& with the horiontal, as shown in figure. 5o that the ball

immediately starts to move with velocity v&/ 1& ms71.

+i $alculate the impulse. *f coefficient of inetic friction between the

floor and the ball is / &.1, calculate,+ii velocity of ball when it stops sliding.

+iii time t at that instant.

+iv horiontal distance traveled by the ball upto that instant,

+v angular displacement of the ball about horiontal diameter

perpendicular to !", upto that instant, and+vi energy lost due to friction.


2 2(l+g a +g a t4

+ + +

2 2 2 2 223 a ,2m 3 3 a

2((1ms2(

1'.6'ms


17/52

PHY-PP-17

Ans:+i 4

+ii Iero

+iii 1& second +iv %& m+Ceftward

+v 12%& radians +clocwise +vi 6& joule

MKA 60.! solid ball of diameter d / 11 cm is rotating about its one of the horiontal diameters with

angular velocity &/ 12& rad@sec. *t is released from a height so that it falls h / 1.> m freelyand then collides with the horiontal floor. $o7efficient restitution is e / %@' and co7efficient offriction between the ball and the ground is / &.2. $alculate fraction of energy lost duringcollision and the distance between the points where the ball stries the floor for the first and

second time. + g / 1& ms72

Ans: &.4(2, 2.2 m

MKA 61.! steel ball of radius / 2& cm and mass m / 2 g is rotating about a horiontal diameter

with angular velocity &/ %& rad@sec. This rotating ball is dropped on to a rough horiontalfloor and falls freely through a height h / 1.2% m. The coefficient of restitution is e / 1.& and

coefficient of friction between the ball and the floor is / &.(. $alculate+i distance between points of first and second impact of the ball with the floor, and

+ii loss of energy due to friction.

Ans: +i ( m +ii (>.% joule

MKA 62. ! uniform rod of length l and mass 8 is suspended on two

vertical ine3tensible strings as shown in figure. $alculate tension

T in left string at the instant, when right string snaps.

Ans:

MKA 63.! triangular prism of mass 8 / 1.12 g having base

angle (6& is placed on a smooth horiontal floor. !

solid cylinder of radius / 2& cm and mass m / 4

g is placed over the inclined surface of the prism. *f

sufficient friction e3ists between the cylinder surface

and the prism, so that cylinder does not slip,

calculate also, force of friction e3isting between the

cylinder and the prism. + g / 1& ms72

Ans: (.6% ms2, 12 newton

!ngular acceleration of cylinder / (& radian@sec2

+clocwise

MKA 64.! solid metallic cylinder of mass m / 1 g and radius

/ 2& cm is free to roll +without sliding over the

inclined surface of a wooden wedge of mass 8 / &.2>

g. 5urface of wedge in inclined at (6& with the

horiontal and the wedge lies on a smooth horiontal

floor. Bhen the system is released from rest, calculate

+i acceleration of the wedge,+ii angular acceleration of the cylinder, and


12gms

8gT

4=


18/52

PHY-PP-18

+iii force of interaction between cylinder and the

wedge. +g / 1& ms72

Ans: +i (.6% ms-2 +ii (& rad sec-2

+iii =ormal reaction / %.6% =

?riction / (.&& =

*nteraction force /

MKA 65. ! uniform rod of mass m / (& g and length l /

&.>& m is free to rotate about a horiontal a3is 0

passing through its centre. ! particle of mass 8 /

11.2 g falls vertically through a height h / ('@24% m

and collides elastically with the rod at a distance l@4

from 0. !t the instant of collision the rod was

stationary and was at angle / (6&with horiontalas shown in figure. $alculate

+i angular velocity of the rod just after collision, and

+ii velocity +direction and magnitude of particle

after collision. +g / 1& ms72

Ans: +i ( rad@sec +ii #@6 ms-1

+horiontally rightward

MKA 66.! homogeneous rod !" of length C and mass 8 is hinged at the centre 0 in such a way that

it can rotate freely in the vertical plane. The rod is initially in horiontal position. !n insect 5 of

the same mass 8 falls vertically with speed D on point $, midway between the points 0 and". *mmediately after falling, the insect starts to move towards " such that the rod rotates with

a constant angular velocity .

+i calculate angular velocity in terms of Dand C,

+ii if insect reaches the end " when the rod

has turned through an angle of #&&,

calculate v in terms of C.

Ans: +i +ii

MKA 67.! s:uare frame is formed by four rods, each of length l / '& cm. 8ass of two rods !" and

"$ is m / 2%@1> g each while that of rods !) and $) is 2m each. The frame is free to rotate

about a fi3ed horiontal a3is passing through its geometric centre 0 shown in figure. ! spring

is placed on the rod !" at a distance a / 1% cm from ". The spring is held vertical and a bloc

is placed on upper end of the spring so that rod !" is horiontal.


2 2%.6% ( '.4#=+ =

12D

6C

62gC

12


19/52

PHY-PP-1

+i $alculate mass 8 of the bloc,

+ii *f the spring is initially compressed by connecting a thread

between its ends and energy stored in it is 6'.% joule, calculate

velocity with which bloc bounces up when the thread is burnt.

$alculate also ma3imum angular velocity of frame during its

rotational motion assuming that the bloc does not collide with theframe in subse:uent motion. + g / 1& ms72

Ans: +i 2%@# g +ii 6.2 ms-1

+iii (.%( rad@sec

MKA 68.! heavy plan of mass 1&2.% g is placed over two cylindrical rollers of radii / 1& cm and

r / % cm. 8ass of rollers is 4& g and 2& g respectively. lan is pulled towards right by

applying a horiontal force ? / 2% = as shown in figure. )uring first second of motion the

plan gets displaced by 1& cm.

*f plan remains horiontal and slipping does not 1tae place, calculate magnitude and

direction of force of friction acting between

+i plan an bigger roller,

+ii plan and smaller roller,

+iii bigger roller and floor, and

+iv smaller roller and floor. +g / 1& ms72

Ans: +i (= +ii 1.%& =

+iii 1.&& = +iv &.%& =

MKA 6.! semi circular of radius / '2.% cm is cut in a bloc.8ass of bloc, having trac, is 8 / 1 g and rests over a

smooth horiontal floor. ! cylinder of radius r / 1& cm and

mass m / &.% g is hanging by a thread such that a3es of

cylinder and trac are in same level and surface of

cylinder is in contact with the trac as shown in figure.

Bhen the thread is burnt, cylinder starts to move down the

trac. 5ufficient friction e3ists between surface of

cylinder and trac, so that cylinder does not slip.

$alculate velocity of a3is of cylinder when it reaches

bottom of the trac. +g / 1& ms72

Ans: 2 ms-1

MKA 70.! trolley initially at rest with a solid cylinder placed on its

bed such that cylinder a3is maes angle with directionof motion of trolley as shown in figure, starts to move

forward with constant acceleration a. *f initial distance of

mid point of cylinder a3is from rear edge of trolley bed is

d, calculate the distance s which the trolley goes before

the cylinder rolls off the edge of its horiontal bed.

!ssume dimensions of cylinder to be very small in

comparison to other dimensions. =eglect slipping.

$alculate also, frictional force acting on the cylinder.

Ans:


2 2 2( 1dcos ec , ma sin # cos2 (

+


20/52

PHY-PP-20

MKA 71.! uniform circular disc of mass 8 and radius is free to rotate

about a vertical a3is 0 passing through its rim. !n insect of

mass m is at point ! such that line 0! is the diameter of the

disc as shown in figure. The insect describes a complete circle

relative to disc and returns to the starting point !. $alculate the

angle moved by the disc relative to the ground.

Ans:

MKA 72.! unifo1rm rod !" of mass m / 2 g and length l / 1&&

cm is placed on a sharp support 0 such that !0 / a / 4&

cm and 0" / b / '& cm. ! spring of force constant E /

'&& =m1 is attached to end " as shown in figure. To

eep the rod horiontal, its end ! is tied with a thread

such that the spring is elongated by y / 1 cm. $alculatereaction of support 0 on the rod when the thread is

burnt. +g 1& ms72

Ans: 2& newton

MKA 73. *n the system shown in figure, blocs ! and " have mass m1 / 2 g and m2 / g

respectively. ulley having moment of inertia * / &.11 g m2can rotate with out friction

about a fi3ed a3is. *nner and outer radii of pulley are a 1& cm and b / 1% cm

respectively. " is hanging with the thread wrapped around the pulley, while ! lies on a rough

inclined plane. $oefficient of friction being /. $alculate+i tension in each thread, and

+ii acceleration of each bloc. + g / 1& ms72

Ans: +i Tension in thread connected with ! is 16 =

+ii Tension in thread connected with " is 2' =

+iii !cceleration of ! / 2 ms -2+up the plane

+iv !cceleration of " / ( ms-2 +vertically downward

MKA 74. *n the arrangement shown in figure, mass of blocs

! and " is m1/ &.% g and m2/ 1& g, respectively

and mass of spool is 8 / > g. *nner and outer radii

of the spool are a / 1& cm and b / 1% cm

respectively. *ts moment of inertia about its own a3is

is *&/ &.1& g m2. *f friction be sufficient of prevent

sliding, calculate acceleration of blocs ! and ".

+ g / 1& ms72

Ans: ( ms-2+upward, &.% ms-2+downward


(81

(8 >m

+

2'

6

(

1&


21/52

PHY-PP-21

MKA 75. ! pulley of radius b / 2& cm is fi3ed with a shaft of radius a / 1& cm. 8oment of

inertia of shaftpulley system is * / g m2and the system is free to rotate about a3is 0of the shaft without friction. ! bloc " of mass m2/ > g is resting over and ideal spring

of force constant. E / 2&4> =m1. Cower end of the spring is fi3ed to the floor and the spring

is vertical. Thread connected between shaft and bloc " is just taut.

!nother thread is connected between pulley and bloc ! of mass m1/ 4 g. *nitially this thread is loose. Bhen bloc ! is released, first it

falls freely through a height h / 4&%@1&24 m, then the thread

becomes taut and bloc " is jered into motion. $alculate

+i initial compression of the spring,

+ii velocity of bloc " when it is jered into motion,

+iii loss of energy during that jer, and

+iv ma3imum elongation of spring +from its natural length in the

process of motion. +g / 1& ms72

Ans: +i 12%@(2 cm

+ii >& cm sec-1+iii joule or '.>2 joule

+iv

MKA 76.! wheel of radius / 1& cm and moment of inertia * / &.&% gm2is rotating about a fi3edhoriontal a3is 0 with angular velocity &/ 1& rad@sec. ! uniform rigid rod of mass m / ( gand length l / %& cm is hinged at one end ! such that it can rotate about end ! in a vertical

plane.

=


22/52

PHY-PP-22

MKA 78.! uniform rod of length l / 6% cm is hinged t one of its end and is free to rotate in vertical

plane. *t is released from rest when the rod is horiontal. Bhen rod becomes vertical, it is

broen at mid point and lower part now moves freely. $alculate distance of the centre of lower

part from hinge, when it again becomes vertical for the first time. + g / 1& ms72

Ans: 2.%2 m

MKA 7. ! man can jump over b / 4 m wide trench on earth. *f mean density of an imaginaryplanet is twice that of the earth, calculate its ma3imum possible radius so that he may

escape from it by jumping. iven radius of earth, e/ '.4 1&'m.Ans:

MKA 80. ! thin uniform rod of length 2 a has mass per unit length. $alculate magnitude ofgravitational field strength an potential as a function of distance r from centre of the rod

along the straight line

+i perpendicular to the rod and passing through the centre,

+ii coinciding with the rodAs a3is +at points lying outside the rod.

Ans: +i +ii

MKA 81.! particle of mass m is placed on centre of curvature of a fi3ed, uniform

semicircular ring and mass 8 as shown in figure. $alculate

+i interaction force between the ring and the particle, and

+ii wor re:uired to displace the particle from centre of curvature to

infinity.

Ans: +i +ii

MKA 82.! system consists of a thin ring of radius and a very long uniform wire oriented along a3is

of the ring with one of its ends coinciding with the centre of the ring. *f mass of ring be 8 andmass of wire be per unit length, calculate interaction force between the ring and the wire.

Ans:

MKA 83. *nside a fi3ed sphere of radius and uniform density , there is aspherical cavity of radius @2 such that surface of the cavity passes

through the centre of the sphere as shown in figure. ! particle of mass m

is released from rest at centre " of the cavity. $alculate velocity with

which particle stries the centre ! of the sphere. =eglect earthAs gravity.

Ans:

MKA 84.*n a vertical cylindrical vessel of base area ! / >& cm2water is filled to a height h / (& cm. *f

density and "ul 8odulus of water be / 1&&& g m(and " / 2 1=m2, calculate elasticdeformation energy of water in the vessel. + g / 1& ms72

Ans:

MKA 85.! ring of radius / 4 m is

made of a highly dense material. 8ass of the ring is m 1/ %.4 1g. )istributed uniformlyover its circumference. ! highly dense particle of mass m2/ ' 1& >g is placed on the a3is

of the ring at a distance 3&/ ( m from the centre. =eglecting all other force, 5 e3cept mutualgravitational interaction of the two, calculate


'.4 m

2 2 2

2 2

a r a2 a, 2 log rr r a

+ + +

e2 2

2 a r a

, log r a+r a

+

2

28m

8m

8

2 2JK

(

2 2 (K g !h -'/1.>31& joule'"


23/52


24/52

PHY-PP-24

MKA 0.! cylindrical tan having crosssectional area ! / &.% m2is filled with two li:uids of density1/ #&& g m(and 2/ '&& g m(, to a height h / '& cm each as shown in figure. ! smallhole having area a / % cm2 is made in right vertical wall at a height y / 2& cm from the

bottom. $alculate

+i velocity of efflu3,

+iihoriontal force ? to eep the cylinder in statice:uilibrium, if it is placed on a smooth horiontal

plane, and

+iii minimum and ma3imum values of ? to eep the

cylinder in static e:uilibrium, if coefficient of friction

between the cylinder and the plane is / &.&1+g / 1& ms72

Ans: +i 4 ms-1 +ii 6.2 =

+iii Iero, %2.2 =

MKA 1. $urved surface of a vessel has shape of a truncated cone having semi verte3 angle/ (6&. Top and bottom radii of the vessel are r1/ ( cm and r2/ 12 cm respectively andheight is h / 12 cm. The vessel is full of water +density / 1&&& g m( and is placed on asmooth horiontal plane in vacuum. $alculate

+i mass of the li:uid in the vessel,

+ii force on the bottom of the vessel,

+iii resultant force on curves walls.

! hole having area 5 / 1.% cm2 is made in curved wall near

the bottom. $alculate

+iv velocity of efflu3,+v horiontal range of water jet, and

+vi horiontal force re:uired to eep the vessel in static

e:uilibrium. =eglect atmospheric pressure.

Ans: +i &.6%'. g +ii 16.2>. = +iii #.62 = +vertically

upward

+iv

+v 2(.&4 cm +vi &.2>> =

MKA 2.! cylindrical tan of base area ! has a small orifice of area a at the bottom. !t time t / &, a

tap starts to supply water into the tan at a constant rate m (s1. $alculate relation between

height h of water in the tan and time t.

Ans: t /

MKA 3. ! steel rod of length l1 / (& cm and two

identical brass rods of length l2/ 2& cm each,

support a light horiontal platform as shown in

figure. $ross sectional area of each of the three

rods is ! / 1 cm2. $alculate stress in each rod

when a vertically downward force ? / %&&&

= is applied on the platform.

iven, LoungAs modulus of elastically for steel,


124ms

e

2gh! 2h log

g g

+


25/52


26/52

PHY-PP-26

&


27/52

PHY-PP-27

MKA 100. Two identical blocs ! and " of mass m / ( g are attached with ends of an ideal spring

of force constant E / 2&&& =m-1and rest over a smooth horiontal floor. !nother identical

bloc !moving with velocity D&/ &.'ms-1as shown in figure stries of bloc ! and gets

stuc to it. $alculate for subse:uent motion

+i velocity of centre of mass of the system.

+ii fre:uency of oscillations of the system,

+iii oscillation energy of the system, and+iv ma3imum compression of the spring.

Ans: +i &.2 ms-1 +ii

+iii &. joule

+iv

MKA 101. Two bloc ! and " of masses m 1/ ( g and m2/ ' g respectively connected with each

other by a spring of force constant E / 2&& =m -1as shown in ?ig. 1&1. "locs are pulled away

from each other by 3o/ ( cm and then released. Bhen spring is in its natural length and

blocs are moving towards each other, another bloc of mass m / ( g moving with velocity v &/ &.4ms-1+towards right collides with ! and gets stuc to it. =eglecting friction, calculate

+i velocities v1 and v2 of the blocs ! and " respectively just before collision and their

angular fre:uency,

+ii velocity of centre of mass of the system, after collision,

+iii amplitude of oscillations of combined body, and

+iv loss of energy during collision.

Ans: +i &.2 ms-1, &.1 ms-1, 1& rad@sec

+ii &.1 ms-1+towards right

+iii

+iv &.&( joule

MKA 102. *n the arrangement shown in pulleys are small and

light and spring are ideal, E1, E2, E(and E4are force

constants of the springs. $alculate period of small

vertical oscillations of bloc of mass m.

Ans:


2

% 1&H

( 1&mm

24cm

1 2 ( 4

1 1 1 14 m

E E E E

+ + +


28/52

PHY-PP-28

MKA 103. !" and $) are two ideal springs having force

constant E1and E2respectively. Cower ends of these

springs are attached to the ground so that the

springs remain vertical. ! light rod of length (a is

attached with upper ends " and $ of springs. !

particle of mass m is fi3ed with the rod at a distance

a from end " and in e:uilibrium, the rod ishoriontal. $alculate period of small vertical

oscillations of the system.

Ans:

MKA 104. ?ig shows a particle of mass m / 1&& gm,

attached with four identical springs, each of length

l / 1& cm. *nitial tension in each spring if ?&/ 2%

newton. =eglecting gravity, calculate period of

small oscillations of the particle along a line

perpendicular to the plane of the figure.

Ans: &.&2sec

MKA 105. *n the arrangement shown in ?ig. 1&%, body Bis a solid cylinder radius / 1& cm with

mass 8 / 4 g. *t can rotate without friction about a fi3ed horiontal a3is ",! bloc ! of mass

m / 2 g suspended by an ine3tensible thread is wrapped around the cylinder. ! horiontal

light spring of force constant E / 1&& =m-1 fi3ed at one end eeps the system in static

e:uilibrium. $alculate

+i initial elongation in the spring, and

+ii period of small vertical oscillations of the bloc.

+g / 1& ms-2

Ans: !i 2& cm +ii &.4 second

MKA 106.! solid uniform sphere of radius r rolls without sliding along the inner surface of a fi3ed

spherical shell of radius and performs small oscillations. $alculate period of these

oscillations.

Ans:

MKA 107. 0ne end of each of two identical

springs, natural length # cm and force constant E / 4% =m -1is attached with a small

particle of mass m / (& gm. 0ther end of right spring if fi3ed with a wall and other end of left

spring is attached with a fi3ed bloc having a positive charge : / 1 $ as shown in figure.The particle rests over a smooth horiontal plane and springs are non-deformed.$alculate deformation of springs when a positive

charge : / 1 $ is given to the particle ande:uilibrium is attained.$alculate also, fre:uency of small longitudinaloscillations of the particle.

Ans: 1 cm,


1 2

1 2

m+E 4E2

( E E

+

6+ r2

%g

(&H


29/52

PHY-PP-2

MKA 108.! non-conducting piston of mass m and area 5 divides a non-conducting, closed cylinder

into two parts as shown in ?ig. 1&>. iston is connected with left wall of cylinder by a spring of

force constant E. Ceft part is evacuated and right part contains an ideal gas at pressure .

!diabatic constant of the gas is and in e:uilibrium length of each part is l.

$alculate angular fre:uency of small

oscillations of the piston.Ans:

MKA 10.! rectangular tan having base 1% cm 3 2& cm is filled

with water +density / 1&&& g m-( upto 2& cm height.0ne end of an ideal spring of natural length h &/ 2& cm

and force constant E / 2>& =m

-1

is fi3ed to the bottom ofa tan so that spring, remains vertical. This system is in

an elevator moving downwards with acceleration a / 2

ms-2. ! cubical bloc of side l / 1& cm and mass m / 2

g is gently placed over the spring and released

gradually, as shown in ?ig. 1.

+i $alculate compression of the spring in e:uilibrium

position.

+ii *f bloc is slightly pushed down from e:uilibrium

position and released, calculate fre:uency of its vertical

oscillations.

Ans: +i 4 cm

+ii

MKA 110. "oth the limbs of a -tube are vertical. 0ne end of a light spring of

force constant E / 6> =m-1is fi3ed with top of left limb and a piston of

mass m / %& gm is attached with lower end of the spring as shown in

?ig. 11&. $ross-sectional area of tube is 5 / 1 cm2. Bater +density / 1&&& g ms-( is poured into right limb till elongation of spring

reduces to a / ' mm.

+i $alculate difference h between level of water in right limb and

level of lower face of the piston

+ii *f mass of whole in the tube is 8 / 1%& gm, calculate angular

fre:uency of small oscillations. +=eglect !tmospheric pressure.

Ans: +i (2 mm +ii 2& rad@sec

MKA 1"! bus is traveling along a straight road with velocity v/ '.4ms-1. ! boy is sitting a distance NaA

way from line of motion of the bus. He throws a stone with velocity u / 1& ms -1at the instant

when a glass window of the bus is infront of the him. *f the stone stries this glass window at

highest point of its trajectory and height of the window above the point of projection is #7 1.>

m, calculate.

+i time of flight of the stone,

+ii distance NaA, and

+iii inclination of plane of trajectory of stone with the road. +g/ 1& ms-2

Ans: +i &.' second +ii 2.>> m

+iii (6


5 El

ml

+

1% 2 sec


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MKA 2". ! particle of mass m/1g is moving along 3-a3is with constant velocity of magnitude

v&/2 ms-1. Bhen it passes through origin, it e3perience a constant force F/ =ewton

inclined at angle /tan-1+2 with 3-a3is so that the particle now moves in negative :uadrant of3-y plane. =eglecting gravity, calculate e:uation to the trajectory of the particle.O

Ans: 432; 43y ; y2; 1'y /&

MKA 3". *n the arrangement shown in ?ig., pulleys are light,

small and smooth. 8ass of blocs !, " and $ is m1/

14 g, m2/ 11 g and 8 / %2 g respectively. The

bloc ! can slide freely along a vertical rail, fi3ed to left

vertical face of bloc $. !ssuming all the surfaces to

be smooth, calculate magnitude of resultant

acceleration of each of the blocs !, " and $.

+g / 1& ms-2

Ans:

MKA 4".! shell of mass m /1 g is fired from a point 0 on the ground with velocity u /' ms-1at angle

/ '&with the horiontal. !t highest point of trajectory, the shell just comes into contact to ahoriontal plan of mass 8 / 2 g which is resting over a horiontal platform as shown in

figure. $oefficient of friction between shell and plan is 2/ &.% and that between plan and

platform is 1/ &.1. *n the figure, 3-a3is is horiontal a3isthrough 0 and is in the line of trajectory of the shell and

y-a3is is vertical a3is through 0. $alculate co-ordinates f

the point where the shell finally comes to rest and

displacement of plan upto that instant.

+g / 1& ms-2

Ans:

MKA 5". ! particle is projected form ground withvelocity u / 1& ms -1at an angle with horiontal.!t highest point of its trajectory, it comes intocontact with lowest point of a vertical circular

trac of radius / 1 m as shown in figure andit starts to move along inner surface of thetrac. Height of lowest point of the trac fromground is h / (.1& m. =eglecting frictionbetween particle and the trac, calculatema3imum height reached by the particle abovethe ground +g / 1& ms-2

Ans: 4.>#2 m

MKA 6".! rod !" of length a / #& cm can rotate freely in a horiontal plane about a vertical a3is00A, passing through its one end ! as shown in figure, ! particle is suspended from other and

rod by a light, ine3tensible thread of the length l/ %& cm. The thread is capable of withstanding a ma3imum tension e:ual to 1.2% times the weight of the particle. *f rod starts to


%

2 2 22ms , 1& ms ,1ms

P+1 &.# (m,1.(%mQ,&.2%m+


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rotate and its angular velocity increases slowly and if height 0! is e:ual to h / 12& cm,calculate.+i speed of the particle when it stries theground, and+ii distance of the point at which particle striesthe ground from foot 0 of the a3is of rotation.

+g / 1& ms-2

Ans: +i % ms71

+ii

MKA 7".! small bloc of mass m / 1 g is attached with one end of aspring of force constant E / 11& =m -1. 0ther end of the spring

is fi3ed to a rough plane having inclination /sin71+(@% with the

horiontal and having coefficient of friction / &.2. The springis ept in its natural length by an ine3tensible thread tiedbetween its ends as shown in figure. *f the thread is burnt,calculate elongation of spring when the bloc attains statice:uilibrium position.

Ans:

MKA 8".! stationary light, smooth pulley can rotate without friction about afi3ed horiontal a3is. ! light rope passes over the pulley. 0ne end ofthe rope supports a ladder with a man and the other end supports acounterweight of mass 8, 8ass of the man is m. *nitially, centre ofmass of the counterweight is at a height h from that of the man asshown in ?ig. >R. *f the man starts to climb up the ladder slowly,calculate wor done by him to reach his centre of mass level withthat of the counterweight.

Ans:

MKA ".! steel ball of mass m is suspended by an ine3tensible thread of length l/ 1 mG Thepoint of suspension is at height h / ' m from the ground. The bass is drawn aside andan impulse is given to it so that it passes through the e:uilibrium position with velocity/ %ms 71.

!nother ball of mass m@2 is projected from the ground with velocity u at angle withhoriontal such that its plane of trajectory passes through the point of suspension and isnormal to .!t highest point of trajectory, second ball collides with the first ball. *f at the instant of collision,first ball was passing through its e3actly lowest position and first ball, subse:uently,completes vertical circle, calculate.

+i ma3imum possible value of , and+ii corresponding value of u. +g / 1& ms72

Ans: +i %( +ii 12.% ms71


12& 2 cm

'4cm

11

m8gh

+8 m

&v&D


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MKA 10".! ball of mass m / 1 g is suspended from point N0A of a toy cart of mass 8 / ( g by anine3tensible thread of length l/ 1.%'2% m. The ball is first raised to point ! and then the ball isreleased from rest. oint ! is in the same level as 0 and distance 0! is such that the ball fallsfreely through a height h / 1.2%m as shown in figure, then thread becomes taut. =eglectingfriction, calculate+i velocity of cart just after the thread becomes taut, and

+ii loss of energy when thread becomes taut.Ans: +i +ii

MKA 11".! wedge of mass 8 / (.' g and having base angle / (.6 is resting over a smoothhoriontal surface. ! ball of mass m / 1 g is thrown vertically downwards such that it striesthe wedge with velocity v&/ 11 ms

-1at height h / 4> cm from base of the wedge as shown in

figure. $oefficient of restitution between ball and wedge is e / &.% !ssuming all the surfacesto be smooth, calculate

+i velocity of wedge just after collision,+ii vertical component of velocity of ball just aftercollision,+iii time of flight of ball from the instant of collision withwedge to the instant when ball stries the floor, and+iv distance between ball and right edge of the wedgeball stries the floor. +g / 1& ms-2

Ans: +i 2 ms71 +ii 1.4 ms71+downwards+iii &.2 second +iv 1.2& m

MKA 12". ! uniform rod of length 4land mass m is free to rotate about a horiontal a3is passingthrough a point distant lfrom its one end. Bhen the rod is horiontal, its angular velocity is asshown figure. $alculate+i reaction of a3is at this instant,+ii acceleration of centre of mass of the rod at this instant,+iii reaction of a3is and acceleration of centre of mass of the rod when rod becomes verticalfor the first time, and+iv minimum value of so that centre of rod can complete circular motion.

Ans: +i+ii+iii

+iv

MKA 13".! uniform solid cylinder of mass m / 2 g and radius / 1& cm is gently placed on a rough

having inclination / (6with horiontal and having coefficient of friction / 1@>, such thata3is of the cylinder is normal to line of greatest slope of the plane. $alculate after one second.+i Einetic energy of the cylinder, and+ii loss of energy against friction. +g / 1& ms-2

Ans: +i 26 F +ii ( F


1% ms6

(%& F4#

224mg 6l

1

6 4g

+

22(g +l

6

+

2 21( 'gmg ml , l6 6 + + 'g61


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PHY-PP-33

MKA 14".! solid ball of radius / 1& cm and mass m / &.> g rolls down, from rest, along a rough

plane inclined at angle / (6with horiontal. Dertical component of displacement of ball is h1/ 1.6% m as shown in the figure. "ase of inclined plane is at height h 2 / (6 cm from the

ground. *f coefficient of friction and restitution between the ball and ground are / &.&2% ande / &.% respectively, calculate distance, from 0, of the point at which the ball stries theground.

+i for the first time, and+ii for the second time,$alculate also, loss of energy againstfriction during first collision with theground. Bhat amount of energy is lost duringfirst collision with the ground 9

+ / 1& ms-2

Ans: +i 4' cm+ii 212 cm, >>.% ml, 4.>>>% F

MKA 15". Two identical balls, each of mass m / 1 g are attached at ends of anideal spring of natural length l&/ 1& cm and having force constant E /1(6%& =m-1. The system is placed on a smooth horiontal table. !sharp impulse is applied at one of these balls in a direction normal toa3is of the spring. *f the impulse is horiontal and is e:ual to $/ ' =-s,calculate+i energy supplied by the impulse, and+ii an e3pression from ma3imum elongation 3& of the spring duringsubse:uent motion.

Ans: +i 1> F

+ii F2+4

MKA 16". !n earth satellite is * revolving in a circular orbit of radius NaA with velocity v &. ! gun

is in the satellite and is aimed directly towards the earth. ! bullet is fired from the gunwith mule velocity . =eglecting resistance offered by cosmic dust, calculate.+i minimum and ma3imum distance of bullet from earth during its subse:uent motion, and+ii period of revolution.

Ans: +i +ii

MKA 17". Two cylindrical tans having cross- sectional area ! and 2! are ept on ahoriontal floor. ?irst tan is filled with water to a height h while the other is empty . *f two tanare connected by a pipe of cross-sectional area a+aSS! at the bottom at t/ &, calculatetime twhen level of water in two tans becomes same.$alculate also, loss of gravitations potential energy during the process

!ssume density of water and gravitational acceleration to be uniform and e:ual to and g

respectively.


2 2& & & & & &3 4l 3 2mE3 +l 23 + = +

&

2

v

2a,2a

(&

1' a

( (v


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PHY-PP-34

Ans:

MKA 18". *n the arrangement shown ?ig. 1>R, !" is a uniform rod oflength l/ #& cm and mass 8 / 2 g. The rod is free to rotateabout a horiontal a3is passing through end !. ! threadpasses over a light, smooth and small pulley. 0ne end of the

thread is attached with end " of the rod and the other endcarries a bloc of mass m / 1 g. To eep the system ine:uilibrium, one end of an ideal spring of force constant E /6%&& =m-1is attached with mid point of the rod and the otherend is fi3ed such that in e:uilibrium, the spring is vertical andthe rod is horiontal. *f in e:uilibrium, the spring is verticaland the rod is horiontal. *f in e:uilibrium, part of the threadbetween end " and pulley is vertical, calculate fre:uency ofsmall oscillations of the system.

Ans: radian, 6%#.(6%joules

MKA 1". ?igure shows a solid, uniform cylinder of radius andmass 8, which is free to rotate about a fi3ed horiontal a3is0 and passes through centre of the cylinder. 0ne end of anideal spring of force constant E is fi3ed and the other end ishinged to the cylinder at !. )istance 0! is e:ual to @2. !nine3tensible thread is wrapped round the cylinder andpasses over a smooth, small pulley. ! bloc of e:ual mass8 and having cross-sectional area ! is suspended fromfree end of the thread. The bloc is partially immersed in a

non-viscous li:uid of density .

*f in e:uilibrium, spring is horiontal and line 0! is vertical,

calculate fre:uency of small oscillations of the system.

Ans:

MKA 111.! small sphere is charged uniformly and placed at point ! +u, v%so that at point " +>, 6electric field strength is &/ +%4'; 62j=$-1and potential is ;#&& volt.$alculate+i magnitude of charge.+ii co-ordinates of point !, and+iii if di-electric strength of air is ( 3 1&'Dm-1, minimum possible radius of the sphere

Ans: +i 1 $ +ii +2, -1+iii

MKA 112. Two long wires are placed on a smooth horiontal table. Bires have e:ual but opposite

charges. 8agnitude of linear charge density on each wire is .$alculate +for unit length ofwires wor re:uired to increase the separation between the wires from a to 2a.

Ans:

MKA 113. Two long wires have uniform

charge density per unit length each. The wires are non-coplanar and mutuallyperpendicular. 5hortest distance between them is d. $alculate interaction force betweenthem.


22! 2h 1, !gh(a g (

1% % 4H,

2 4&%

1 +E !pg

2 '8

+

((31& mor%.4>cm

2

e

&

log 22


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PHY-PP-35

Ans:

MKA 114. Two short electric dipoles having dipole moment p1 and p2 are placed co-a3iallyand uni-directionally, at a distance r apart. $alculate nature and magnitude of force betweenthem.

Ans: !ttraction,

MKA 115. ! small cor ball ! of mass m is suspended by a thread oflength l. !nother ball " is fi3ed at a distance l from point ofsuspension and distance l@2 from thread when it is vertical, asshown in ?ig. 11%. "alls ! and " have charge +;: each. "all ! isheld by an e3ternal force such that the thread remains vertical.Bhen ball ! is released from rest, thread deflects through a

ma3imum angle of / (&, calculate m in terms of otherparameters.

Ans:

MKA 116.! positively charged sphere of mass m / % g is attached by aspring of force constant E / 1&4=m -1. The sphere is tied with athread so that spring is in its natural length. !nother identical,negatively charged sphere is fi3ed with floor, vertically below thepositively charged sphere as shown in figure. *f initial separationbetween sphere is r&/ %& cm and magnitude of charge on each

sphere is q / 1&&$, calculate ma3imum elongation of springwhen the thread is burnt. +g / 1& ms-2

Ans: 1& cm

MKA 117.! non-conducting hollow sphere having inner and outer radii a and b respectively is madeof a material having di-electric constant E and has uniformly distributed charge over its entire

solid volume. Dolume density of charge is . $alculate potential at a distance rfrom its centrewhen+i r b,+ii r S a,+iii a S r S b.

Ans: +i D1/+iiDi/

+iii D /

MKA 118. )istance between centers of two spheres ! and ",each of radius is ras shown in ?ig. 11>. 5phere " hasa spherical cavity of radius @2 such that distance ofcentre of cavity is +r 7 @2 from the centre of sphere !and @2 from the centre of sphere ". )i-electric constantof material of each sphere is E /1 and material of each

sphere has a uniform charge density per unit volume.$alculate interaction energy of the two spheres.

Ans:


2

&2

1 2

4

&

'p p1 .4 r

2

2 (@ 2&

: +1 2 ( .

2 gl +2 (

( (

&

+b a

( r

( ( 2 2 (

& &

+b a b a a +b a

( b ( E 2 ab

+

( ( 2 2 (

& &

+b a b r a +b r

( b ( E 2 rb +

2 '

&

+6r 4

# r +2r


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PHY-PP-36

MKA 11. ! non-conducting sphere of radius / % cm has itsenter at origin 0 of co- ordinate system, shown in ?ig.11#. t has a spherical cavity of radius r/ 1 cm, whosecentre is at +&, ( cm.5olid material of sphere has uniform

positive charge density / coul m-(. $alculate potential at point(+4 cm, &.

Ans: (%.1' volt

MKA 120.! solid non-conducting hemisphere of radius has a uniformly distributed positive charge

of density per unit volume. ! negatively charge particle having charge qis transferred fromcentre of its base to infinity. $alculate wor performed in the process. )i-electric constant ofmaterial of hemisphere is unity.

Ans:MKA 121. Two circular rings ! and ", each of radius a / (& cm, are

placed coa3ially with their a3es vertical as shown in fugure.)istance between centres of these rings is h/ 4&cm. Cower ring

! has a positive charge of 1&$ while uppering " has annegative charge of 2& $ . ! particle of mass m / 1&& gmcarrying a positive of : / 1& c is released from rest at thecentre of the ring !.+i $alculate initial acceleration of the particle.+ii $alculate velocity of particle when it reaches at the centre ofupper ring ". +g / 1& ms-2

Ans: +i 46.' ms-2 +ii > ms-1

MKA 122. Two circular rings ! and ", each of radius a / (& cmare placed coa3ially with their a3es horiontal in auniform electric field & / 1&% =$-1 directed verticallyupwards as shown in figure. )istance between centresof these rings ! and " is h / 4& cm. ing ! has a

positive charge q1 / 1&$ while ring " has a negativecharge of magnitude :2/ 2&$, ! particle of mass m /1&& gm and carrying a positive charge q / 1&$ isreleased from rest at the centre of the ring !. $alculateits velocity when it has moved a distance of 4& cm.

Ans:

MKA 123.! particle having charge : / >.>%$ is placed on the a3is of a circular ring of radius / (&cm. )istance of the particle from centre of the rings is a / 4& cm. $alculate electrical flu3passing through the ring.

Ans:


'1&

2

&

:

4

1' 2 m s

% 1 2

2 2&

:a 1 11& =$ m

2 a a

=

+


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PHY-PP-38

+i *f this beam stries a plane surface at an angle / (&with normal to the surface, howmany -particles strie the surface in t/ 4 second 9+ii How many -particles are there in length l/ 2& cm of the bearn9+iii $alculate power of the source used to accelerate these -particles from rest. +8ass of -particles / '.'> 3 1&-26g.

Ans: +i 2.% 3 1&12 +ii '.2% 3 1&4-

+iii >.(% mB.

MKA 131.

1%

6r1%


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PHY-PP-3

MKA 134. $alculate e:uivalent resistance between ! and" of the circuit shown in figure.

Ans: '.6%

MKA 135. Temperature coefficients of resistance of two wires, ! and " 1/ ( 3 1&-(@$ and 2/ ' 31&-(@$ respectively and that of their series combination is s / % 3 1&-(@$. $alculatetemperature coefficient of resistance of a circuit segment consisting of these two wires whenthey are connected in parallel.

Ans: 4 3 1&-(@$

MKA 136. *f a battery of emf > volt negligible internalresistance is connected between terminal and of the circuit shown in figure, calculate current

through 2.% resistance and hence calculatee:uivalent resistance of the circuit.

Ans: Iero, 4

MKA137. =ine identical capacitors, each of capacitance

$ / 1%?are connected as shown in figure.

$alculate e:uivalent capacitance betweenterminals 1 and 4.

Ans: 11 ?

MKA 138. ! voltmeter of resistance vand an ammeter of resistance !are connected in seriesacross a battery of emf & and of negligible internal resistance. Bhen a resistance isconnected in parallel to voltmeter, reading of ammeter increases to three times while that ofvoltmeter reduces to one-third. $alculate !and vin terms of .

Ans: !/


D

>., >

(=


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PHY-PP-40

MKA 13. *n the circuit shown in ?ig. 1(#, 1, / 2,(/ (, 4/ 2, %/ 2,


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MKA 143. *n the circuit shown in figure, emf of eachbattery is < / 2& volts and capacitance is $1/

% ?, $2/ (? an $(/ '?. $alculate chargeon capacitor $(when switch 5 is closed andsteady state is reached. $alculate also, heat

generated in the circuit.

Ans: Iero, 2%& F

MKA 144.! parallel plate capacitor is filled by a di-electric whose di-electric constant varies withpotential difference D according to law E / aD, where a / 2volt -1. !n air capacitor having samedimensions charged to a potential difference of D &/ 2> volt is connected in parallel to theuncharged capacitor filled with above mentioned di-electric . $alculate ration of charge on

capacitor filled by aforesaid di-electric to charge on air capacitor in steady state.

Ans: 6

MKA 145. 5witch 5 of circuit shown in ?ig. 14% is in position 1 for along time. !t instant t / &, it thrown from position 1 to 2.$alculate thermal power 1+t and 2+t generated acrossresistance 1and 2 respectively.

Ans:

MKA 146. ! capacitor of capacitance $1/ &.1 ? is chargedby a battery of e.m.f.


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PHY-PP-43

MKA 14. *n steady state, calculate energy,stored in capacitors shown in figure. andthe rate which battery supplies energy.

Ans: >&.% F, 1& B

MKA 150. *n the circuit shown in figure, $ is a parallel plateair capacitor having plate f area ! / %& cm2each anda distance )/ 1 mm apart 1, 2and (are resistors

having resistance ( , 2and 1respectively. Twoidentical sources each of emf D and negligibleinternal resistance are connected as shown in figure*f dielectric strength of air is


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PHY-PP-44

MKA 152. *n the circuit shown in figure, emf andinternal resistance of battery are ' D and &.%

respectively, $alculate charge on eachcapacitor in steady state.

Ans: &, 2 $, &

MKA 153.*n the circuit shown in figure,

1/ >, 2/ %, $1/ '?, $2/ ?,


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PHY-PP-45

MKA 155. *n the circuit shown in figure,


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PHY-PP-47

!B$

MKA 160. $onductor of length lhas shape of a semi-cylinderof radius R +S S l. $ross section of the conductor isshown if ?ig. 1'&. Thicness of the conductor is t

+SS R and conductivity of its material varies withangle only, according to the law / & cos . *f abattery of emf D and of negligible internal resistance isconnected across its end faces, calculate magneticinduction at mid point " of the a3is of the semi-cylinder.

Ans:

MKA 161.


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PHY-PP-48

Ans:

MKA 162. ! non-conducting thin spherical

shell of radius has uniform surface charge density . The shell rotates about a diameterwith constant angular velocity $alculate magnetic induction " at the centre of the shell.

Ans:

MKA 163.! s:uare loop of side a / ' cm. carries a current I / (& amp. $alculatemagneticinduction " at point , lying on a3is

MKA Physics -1

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