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This DPP is to be discussed (29-04-2014)

CT-3 to be discussed (29-04-2014)

DPP No. # 06

Total Marks : 165 Max. Time : 180 min.Single choice Objective (��1� negative marking) Q.1 to Q.15 (3 marks 3 min.) [45, 45]Multiple choice objective ('�1' negative marking) Q.16 to Q.21 (4 marks 4 min.) [24, 24]Subjective Questions ('�1' negative marking) Q.22 to 30 (4 marks 5 min.) [36, 45]Assertion and Reason (no negative marking) Q. 31 to 33 (3 marks 3 min.) [9 , 9]Comprehension ('�1' negative marking) Q.34 to Q.42 (3 marks 3 min.) [27, 27] Match the Following Q.43 to Q.45 (no negative marking) (2 × 4 or 5) (8 marks 10 min.) [24, 30]

1. The system shown in figure consists of a circular ring of radius r lying in xyplane centerd at origin and two semi circular rings of radius r lying in xy and yzplane with their centres at origin. Each segment of the system is made of a

uniform wire of mass per unit length = r

m

. The moment of inertia of the

system about the axis AB(along x axis) as shown in figure is:

(A) 2

mr 2

(B) 25

mr2 (C) 23

mr2 (D) 4 mr2

2. A uniform thin hemispherical shell is kept at rest and in equilibrium on aninclined plane of angle of inclination = 30º as shown inf figure. If the

surface of the inclined plane is sufficiently rough to prevent sliding then

the angle made by the plane of hemisphere with inclined plane is :

(A) value of µ is needed (B) 30º (C) 45º (D) 60º

3. A massless stick of length L is hinged at one end and a mass m attached to its other end. The stick is freeto rotate in vertical plane about a fixed horizontal axis passing through frictionless hinge. The stick is heldin a horizontal position. At what distance x from the hinge should a second mass M = m be attached tothe stick, so that stick falls as fast as possible when released from rest

(A) L2 (B) L3 (C) L)12( (D) L)13(

TARGET : JEE (ADVANCED) 2014

TEST SYLLABUS

TEST : PART TEST-3 (Test Date : 30-04-2014)Test Syllabus : EMI, AC, Wave Optics, RBD, SHM, Properties of Matter.

Course: VIJETA (JPAD)

& VIJAY (JRAD)

Date : 27-04-2014

DPP No. : 6

PHYSICS

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4. A uniform rod of mass m and length L is hinged at one end and free to rotatein horizontal plane. All the surfaces are smooth. A particle of same mass mcollides with the rod perpendicular to the length of rod with a speed V

0. The

coefficient of restitution for the collision is e = 21

. If hinge reaction during the

collision is zero then the value of x is :

(A) No such value of x is possible (B) x = 2L

(C) x = 3L2

(D) x = L

5. A block of dimensions × t × h and uniform density w rests on a rough floor. Wind blowing with speed V

and of density a falls perpendicularly on one face of dimension × h of the block as shown in figure.

Assuming that air is stopped when it strikes the wall and there is sufficient friction on the ground so thatthe block does not slide, the minimum speed V so that the block topples is :

(A) t.h

g2/1

a

w

(B) t.

h

g2/1

w

a

(C) t.

hg

2/1

(D) None of these

6. An oscillation is superposition of three harmonic oscillations and decribed by the equation x = A sin2 1t whereA changes with time according to A = A0(1 + cos22t) with A0 to be constant. The frequencies of pure harmonicoscillations forming this oscillation are :

(A) 1,2 , || 21 (B) 1, || 21 , 1 + 2

(C) 1,2, || 12 (D) 1 , 2, 1 + 2

7. The natural frequency of the system shown in figure is:{The pulleys are smooth and massless.}

(A) Mk21

(B)

Mk22

(C)Mk1

(D)

Mk41

8. A uniform disc of mass m is attached to a spring of spring constant k asshown in figure and there is sufficient friction to prevent slipping of disc. Timeperiod of small oscillations of disc is:

(A) km

2 (B) k3m2

2

(C) k2m3

(D) k3

m2

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9. A particle is executing simple harmonic motion in a conservative force field. The total energy of simple harmonicmotion is given by E = ax2 + bv2 where �x� is the displacement from mean position x = 0 and v is the velocity of

the particle at x then choose the INCORRECT statements.{Potential energy at mean position is assumed to bezero}

(A) amplitude of S.H.M is aE

(B) Maximum velocity of the particle during S.H.M is bE

(C) Time peried of motion is a

b2

(D) displacement of the particle is proportional to the velocity of the particle.

10. Two particle of mass m each are fixed to a massless rod of length 2 . The rod ishinged at one end about a smooth hinge and it performs oscillations of small angle invertical plane. The length of the equivalent simple pendulum is:

(A) 23

(B) 3

10(C)

35

(D) None of these

11. A uniform rod of length and cross�sectional area A rotates in horizontal plane about a vertical axis passing

through one end of the rod and perpendicular to length of rod. The extension produced in the rod due to centrifu-gal force is ( is weight of rod per unit length & is angular speed of rotation of rod, Y is young's modulus ofelasticity of rod)

(A) AY

2(B)

AY3

32(C)

AY

32(D) 32

AY3

12. A thin uniform elastic rod of natural length L, density and Young'smodulus Y is immersed completely in vertical position in a fluid of density2 by applying a vertically downward force at its top end such that the topend of rod coincides with the free surface of fluid. The rod is in equilibrium,then pick up the correct option.(g is acceleration due to gravity, neglectatmospheric pressure)

(A) Net compression in rod is Y4gL2

(B) Net compression in rod is

Y2gL3 2

(C) Net compression in rod is Y2gL2

(D) Net compression in rod is

YgL2

13. A uniform metal rod (fixed at both ends) of 2 mm2 cross-section is cooled from 40 ºC to 20 ºC. The co-

efficient of the linear expansion of the rod is 12 106 per degree & it�s young modulus of elasticity is

1011 N/m2. The energy stored per unit volume of the rod is:(A) 2880 J/m3 (B) 1500 J/m3 (C) 5760 J/m3 (D) 1440 J/m3

14. In the given figure, two elastic rods A & B are rigidly joined to end supports.A small block of mass �

m

� is moving with velocity v between the rods. All

collisions are assumed to be elastic & the surface is given to be smooth.The time period of oscillations of small mass �

m

� will be: (A = area of

cross section, Y = young�s modulus, L = length of each rod)

(A) vL2

+ 2 YALm

(B) vL2

+ 2 YALm2

(C) vL2

+ YALm

(D) vL2

15. Two forces F1 and F2 act on a thin uniform elastic rod placed in space.Force F1 acts at right end of rod and F2 acts exactly at centre of rod asshown (both forces act parallel to length of the rod).

F2

F1C

(i) F1 causes extension of rod while F2 causes compression of rod.(ii) F1 causes extension of rod and F2 also causes extension of rod.(iii) F1 causes extension of rod while F2 does not change total length of rod.The correct order of True / False in above statements is(A) T F F (B) F T F (C) F F T (D) F F F

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16. A uniform disc of mass m and radius R is free to rotate about its fixed horizontalaxis without friction. There is sufficient friction between the inextensible light stringand disc to prevent slipping of string over disc. At the shown instant extension in

light spring is Kmg3

, where m is mass of block, g is acceleration due to gravity and

K is spring constant. Then select the correct alternative(s).

(A) Acceleration of block just after it is released is 3g4

(B) Tension in the string continuously increases till extension in the spring reaches maximum value.

(C) Acceleration of the block just after release g43

(D) Angular acceleration of disc just after release is R3g4

17. A uniform solid cylinder of mass m, radius R is at rest on an extremely rough horizontalsurface. Now a horizontal force F = kt (where k = positive constant and t = time), isapplied at the highest point on the cylinder. Assume that the cylinder performs purerolling . Then select the correct alternative(s).

(A) The friction force acting on the cylinder varies with time as

(B) Velocity of the highest point after time t will be m3kt2 2

(C) speed of the particle which is at highest point of the cyllinder after time t will be m3kt4 2

(D) If the coefficient of friction between the ground and the cylinder is , the cylinder will start sliding at

kmg3

t

.

18. A particle constrained to move along x-axis given a velocity u along the positive x-axis. The acceleration ' a

' of the

particle varies as a = bx, where b is a positive constant and x is the x co-ordinate of the position of the particle. Then select the correct alternative(s):

(A) The maximum displacement of the particle from the starting point is b

u

(B) The particle will oscillate about the origin(C) Velocity is maximum at the origin(D) Given data is insufficient to determine the exact motion of the particle.

‘

19. A particle is performing SHM along x-axis. Its acceleration as function of time t is given by a = A sin t (A and

are positive constants and t is time). At t =

3, the particle is at origin moving with velocity �v�. Then select the

correct alternative(s).

(A) v =

2A

(B) Amplitude of SHM is 2

A

(C) At any time t displacement of particle from mean position is,x =

2

3tsin

A2

(D) The equilibrium position is x = 22

A3

20. Figure shows roughly how the force F between two adjacent atoms in a solidvaries with inter atomic separation r. Which of the following statements arecorrect ? (A) OQ is the equilibrium separation.(B) Hooke's law is obeyed near P.(C) The potential energy of the atoms is the gradient of the graph at all points.(D) The energy to separate the atoms completely is obtained from the magnitude of the area enclosed below theaxis of r.

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21. A 20 gm particle is subjected to two simple harmonic motionsx1 = 2 sin 10 t,

x2 = 4 sin (10 t + 3

). where x1 & x2 are in metre & t is in sec.

(A) The displacement of the particle at t = 0 will be 32 m.

(B) Maximum speed of the particle will be 720 m/s.

(C) Magnitude of maximum acceleration of the particle will be 200 7 m/s2.(D) Energy of the resultant simple harmonic motion will be 28 J.

22. A solid cylinder of mass M radius R is resting on a horizontal platform (which is parallel to x�y plane) with its

axis fixed along the y-axis and free to rotate about its axis. The platform is given a motion in X-direction given byX = A cos t. There is sufficient friction present in the surface of contact that can prevent the slipping betweenthe cylinder and platform. maximum torque acting on the cylinder during its motion in N-m is Find the value of3. [Take M = 4kg, R = 1m, A= 2m, = 1 rad/s]

23. A uniform thin, rod AB of length L and mass m is undergoing rotation about fixed axis passing through endA and perpendicular to the rod, such that end A remains stationary as shown. The kinetic energy of

section AP of rod is equal to kinetic energy of section BP of rod at an instant. Then the value of 3

APAB

is

(AB and AP are lengths of respective parts of rod)m,L

A P B

24. A uniform thin rod of mass 'm' and length '3' is released from rest fromhorizontal position as shown. When it passes through the vertical line AD,it gets broken at point 'C' due to some reason. Find the angle (in radian)rotated by rod BC till its centre of mass passes through the horizontal linePQ from the instant of breakage. (g = 10 m/s2) (hinge is smooth and thereis no thrust on one part of rod due to the other part at the time of breaking.)

25. A solid sphere (radius = R) rolls without slipping in a cylindrical trough

(radius = 5R). The time period of small of oscillations is kgR)3k(

22

.

Find the value of k (axis of cylinder is fixed and horizontal). 26. A small block is kept on a platform executing SHM in the horizontal plane,

described by x = A sint. The time period of SHM is T and the coefficient offriction between the block and the platform is . The condition that the block

does not slip on the platform at any instant is 2

2

gT

Ax then write the value

of �x�.

27. Two particles P1 and P

2 are performing SHM along the same line about the same mean position. Initially they are

at their positive extreme position. If the time period of each particle is 12 sec and the difference of their amplitudesis 12 cm then find the minimum time after which the separation between the particles become 6 cm.

28. Two opposite forces F1 = 120N and F2 = 80N act on an heavy elastic plank ofmodulus of elasticity y = 2×1011 N/m2 and length L = 1m placed over a smoothhorizontal surface. The cross-sectional area of plank isA = 0.5m2. If the change in the length of plank is x × 10�9m, then find x ?

F1F2

29. A ring of radius r made of wire of density is rotated about a stationary vertical axispassing though its centre and perpendicular to the plane of the ring as shown in figure.Determine the angular velocity (in rad/s) of ring at which the ring breaks. The wire

breaks at tensile stress . Ignore gravity. (Take

= 4 m2/s2 and r = 1m)

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30. The length of an elastic string is 5 metre when the longitudinal tension is 4 N and 6 metre when the tensionis 5 N. If the length of the string (in metre) is "2X" when the longitudinal tension is 9 N (assume Hooke�slaw is valid) then the value of X will be :

31. STATEMENT-1 : Three identical planks of uniform mass density and length L are kept on each other asshown.

maximum length of AB so that all the planks remain in equilibrium is 4L7

.

STATEMENT-2 : For equilibrium, combined centre of mass of plank-2 and plank-3 must have horizontaldistance lesser than L from A.(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1(C) Statement-1 is True, Statement-2 is False(D) Statement-1 is False, Statement-2 is True.

32. STATEMENT�1 : For a particle performing SHM, its speed decreases as it goes away from the meanposition.STATEMENT�2 : In SHM, the acceleration is always opposite to the velocity of the particle.(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1(C) Statement-1 is True, Statement-2 is False(D) Statement-1 is False, Statement-2 is True.

33. Statement-1 : Elastic forces can be conservative even beyond the proportionality limit.Statement-2 : Materials can be stretched even beyond the proportionality limit.(Note: The point until which Hooke's law is valid is called the proportionality limit)(A) Statement-1 is true, statement-2 is true and statement-2 is the correct explanation for statement-1(B) Statement-1 is true, Statement-2 is true and Statement-2 is not the correct explanation for statement-1(C) Statement-1 is true, but Statement-2 is false(D) Statement-1 is false, but Statement-2 is true.

COMPREHENSION - 1 A block of mass m is attached to an unstretched ideal spring of force constant k and

held at rest. A bullet of mass m/2 is vertically fired to it with speed u = 4 3 mgk

as

shown. The moment bullet strikes the block it is released and it is seen that bullet

comes out of the block instantaneously with speed 2u

. Due to this the block starts

oscillatory motion.

34. Velocity of block just after bullet comes out of block is,

(A) 4u

(B) 2u

(C) 2u (D) 3u

35. Amplitude of oscillations of block is :

(A) 2mg

k(B)

mgk

(C) 32mgk

(D) 2mg

k

36. Maximum speed of block during oscillations is :

(A) 2g k

m(B)

kgm

(C) 32g k

m(D) 2 kg

m

COMPREHENSION - 2A rod of mass 'm and length L is attached to a L shaped plank at 'A'. rod can movefreely about A. A string is tied between rod and plank as shown in figure. Whole systemis moving with a constant acceleration g in x-direction

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37. Tension in the string is:

(A) Zero (B) 2mg (C) 2

mg(D) mg

38. Force exerted by hinge on the rod is :

(A) mg (B)2

mg(C)

25mg

(D)4mg5

39. If string is cut at any instant then the angular acceleration of rod (with respect to the plank) at that instant is

(A) 3g2

(B)

g6(C)

3g2

(D) 2g3

COMPREHENSION - 3A uniform rod of mass M and length L, area of cross section A isplaced on a smooth horizontal surface. Forces acting on the rod areshown in the digram

40. Ratio of elongation in section PQ of rod and section QR of rod is(A) 1 : 1 (B) 3 : 5 (C) 5 : 7 (D) 1 : 2

41. Ratio of elastic potential energy stored in section PQ and section QR of the rod is(A) 19 :37 (B) 21 : 39 (C) 23 : 41 (D) 17 : 35

42. Total elastic potential energy stored in the rod is :

(A) AY6

LF7 2

(B) AY6

LF11 2

(C) AY6

LF5 2

(D) AY2

LF3 2

43. In column-I some situations are shown and in column-II information about their resulting motion is given. Selectthe correct answer using the codes given below the columns. :

Column�I Column�II

(P) A uniform solid sphere of mass (1) friction will be in the direction of acceleration1 kg and radius 1 m,

s = 0.05 of centre of mass of body.

(Q) A uniform body of mass 1 kg, r = m21

, (2) friction wil be opposite to direction of

R = 1 m, I (about axis passting acceleration of centre of mass of body.through centre and perpendicularto plane of paper) = 2 kg m2,

s = 0.3

(R) A uniform solid cylinder released on a (3) Body rotates clockwise.fixed incline plane m = 2 kg, R = 1 m,

s =

52

.

(S) A uniform body of mass 1 kg (4) Body rotates anticlockwise.

r = m21

, R = 1 m,

I(about axis passting throughcentre and perpendicular to plainof paper) = 2kgm2

s = 0.5

(String tightly wound on inner radius

is pulled).

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Codes :P Q R S

(A) 1 2 4 3(B) 2 1 3 4(C) 4 1 2 3(D) 4 2 3 1

44. A particle of mass m = 1 kg executes SHM about mean position O with angular frequency = 1.0 rad/s and totalenergy 2J. x is positive if measured towards right from O. At t = 0, particle is at O and moves towards right. Thenselect the correct answer using the codes given below the columns. :

Column-I Column-II

(P) speed of particle is 2 m/s at (1) x = + 1m

(Q) Kinetic energy of the particle is 1J at (2) x = � 1m

(R) At t = /6 s particle is at (3) x = + 2 m

(S) Kinetic energy is 1.5 J at (4) x = � 2 m

Codes :P Q R S

(A) 1 2 4 3(B) 2 1 3 4(C) 4 3 1 2(D) 4 2 3 1

45. Match the column :In a spring block system on a horizontal smooth surface. K = spring constant, A = amplitude, m = mass of theblock. In column I some changes are given and column II respective effect is written. Then select the correctanswer using the codes given below the columns. :

Column I Column II

(P) If mass of the block is doubled (1) time period increases(keeping K, A unchanged)

(Q) If the amplitude of oscillation is doubled (2) time period decreases(keeping K, m unchanged)

(R) If force constant is doubled (3) energy of oscillation increases(keeping m, A unchanged)

(S) If another spring of same force constant is attached parallel to the previous one (4) energy of oscillation remains constant

(keeping m, A unchanged)Codes :

P Q R S(A) 1 3 2 3(B) 2 1 3 4(C) 3 4 1 2(D) 4 2 3 1