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Section -1 ( straight objective type )

1- A straight smooth tunnel is dug in the earth which can beassumed as a uniformly dense sphere of mass M and radius R.The distance of the tunnel from earth center is X. A particle ofmass m is placed in the tunnel an allowed to move. The particle ,during its motion presses the tunnel wall with force which is

(a) zero(b) constant at all positions in the tunnel and directly proportional to X(c) directly proportional to distance from the center of earth.(d) constant at all positions in the tunnel and inversely proportional to X

2-In the arrangement shown, a uniformly dense rod of mass MR is hingedat one end and its other end is tied to a masslessstring which is passing over a fixed pulley as shown andother end of the string carry a block of mass m.

In the equilibrium , the rod makes angle from the horizontal and thestring makes angle from the horizontal. Read following statements :(i) m g sin = MR g (ii) 2m sin( + ) = M cos (a) only (i) is correct (b) only (ii) is correct(c) both (i) and (ii) are correct (d) none of (i) and (ii) is correct.

3- A uniformly dense cylinder is projected horizontally with some speed without rotationalvelocity on a horizontal uniformly rough ground at time t =0.The rolling of the cylinder getsestablished at time t = T. If the rotational kinetic energy of the cylinder at times t = T/2 , 2T and3 T are RE1 , RE2 and RE3 respectively then

(a) RE1 =RE2 = RE3 (b) RE1 =2RE2 > RE3(c) 4 RE1 = RE2 = RE3 (d) 2 RE1 =RE2 > RE3

4- When a particle of mass m moves on the equator from east to west with velocity v relative tothe earth surface, it presses the earth with a force F1 and when it moves west to east with samerelative speed v, it presses the earth with force F2 . If angular speed of earth is we , F2 F1 =

(a) 2m v we (b) 4m v we (c) -2m v we (d) -4 m v we

5- A particle is taken from surface of the earth to height H above the earth surface by spendingenergy E1 against gravity and then it is projected by imparting additional energy E2 so that theparticle starts rotating in circular path around the earth. Then

(a) E1 > E2 for any value of H (b) E1 > E2 only if H > Re / 2(c) E1 > E2 only if H < Re / 2 (d) E1 = E2 for any value of H

6- A particle is rotating in circular path around the center of earth with some speed v . The orbit ofthe particle is at height 1.5 Re from the earth surface. The particle is stopped in the orbit and thenprojected towards earth center with same speed v with which it was rotating. The particle will hit

the earth surface with speed

(a) v (b) v2 (c) 4 v (d) 2 v

7- A projectile of mass 0.1 kg , is projected horizontally from the highest point of a tower, withspeed 30 m/s. Angular momentum of the projectile about the point of projection, 4 secondafter the projection is

(a) 120 kg-m2 s-1 (b) 720 kg-m2 s-1 (c) 240 kg-m2 s-1 (d) zero

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8- A uniform rod of length L and mass M hinged at one end , is released from the horizontal

position. The rod begins to rotate about the hinged end. Read the following two statements:

(i)Just after its release the resultant acceleration of the COM of the rod will be lesser than g

(ii) When the rod becomes vertical, the resultant acceleration of the COM of the rod will be

directed upward.

(a) only (i) is correct (b) only (ii) is correct

(c) both (i) & (ii) are correct (d) none of (i) and (ii) is correct.

9- A uniform rod of length d and mass m is allowed to fallfreely for some time under gravity. When velocity of the rodis v then suddenly, one end of the rod is caught by aman and the man did not allow this end to move at all.Just after one end of the rod is caught, its angularmomentum about the COM of the rod will be

(a) zero (b) m v d/2 (c) m v d /12 (d) m v d /8

SECTION-2 ( Assertion & Reason type )

In this section four questions are given and each question is followed by two

statements. In the answer sheet, mark(a) ifboth the statements are true and statement 2 is correct explanation of thestatement 1(b) ifboth the statements are true but statement 2 is not correct explanation of thestatement 1(c) if the statement 1 is true but the statement 2 false(d) if the statement 1 is false but the statement 2 is true.

10-Statement 1 :The angular momentum, about the origin ,of any particle moving under theinfluence of theonly force F = - (2 x) i + ( -2y)j will remain constant.

becauseStatement 2 : Theforce F = - (2 x) i + ( -2y)j is a central force and under central forces ,angular momentum remains constant.

11- Statement 1 : A disc with center C is rolling without slipon a fixed surface. The speed of point P will be greater than thevelocity of point Q.

becauseStatement 2 : In rolling motion without slip, speed of any point is proportional to the distancefrom the center.

12- Statement 1 : Gravitational potential on the earths center is lesser than that of itssurface.

becauseStatement 2 : Gravitational potential of the point will be more if positive external work isrequired against the gravitational field to reach at the point..

13- Statement 1: A rod of non uniform density , having its center of mass not at the center, willfall without rotation if released freely from horizontal orientation.because

Statement 2 : The gravitational force will not provide any torque on the rod about its center ofmass as the gravitational force has its line of action passing from thecenter of mass it self.

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SECTION-3 ( Linked Comprehension Type )This section contains two paragraphs P14-16 and P17-19 .Each paragraph is followed by 3questions with their multiple options for their answers. Choose the only one options which ismost correct.

P14-16 : Paragraph for the Q. No 14 to 16

There is a solid sphere of uniform density and radius R. Assume the gravitational potential atinfinity to be zero. The center of the sphere is at C

14-Consider the masses in the portion in a region enclosed by a very thin spherical shell of meanradius R /2 and thickness x ( 0) , ratio of the angular momentum of the ring to that of the disc abouttheir centers will be

(a) 1 : 1 (b) 1 : 2 (c) 2 : 1 (d) 1 : 4

18- The ratio of the kinetic energyof the ring to that of the disc at any time t will be(a) 1:1 (b) 1 : 2 (c) 2 : 1 (d) 4 : 1

19- The number of revolutions of the ring to that of disc up to certain time t will be

(a) 4 : 1 (b) 1 : 2 (c) 2 : 1 (d) 1:4

SECTION-4 ( Matrix- Match Type )20-A particle of mass m ,is moving on a circular path of radius R with constant speed. The timeperiod of the motion is T. Let the center is at point C and there is a fixed point P located on thecircumference of the circle and the at time t =0, the particle at distance 2R from the P .Somephysical quantities related to the particle are mentioned in the column I and their description are

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given in the column II. Match the items in the column I with the correct descriptions given in thecolumn II and indicate your answer by darkening appropriate bubbles in the 4 x4 matrix given .

Column I Column II(A) Angular momentum about C (p) becomes zero twice in each cycle

(B) Torque about C (q) is always zero

(C) Angular momentum about P (r) remains constant and non zero

(D) Torque about P (s) is 2 m R2 / T at the time t =T/4

21-Four identical , uniformly dense rods , each of mass M and length L are lying on smoothhorizontal table. They are subjected to the forces as shown in the column I. The center of each rodis represented by the point c and in the second and third fig i.e. (B) and (C) , the forces P/2 are atequal distances L /4 from the center c . Based on the response of the rods , the possible features ofthe motion of the rod are given in the column II. Match the items in the column I with the correctdescriptions in the column II and indicate your answer by darkening appropriate bubbles in the 4x4 matrix given .

Column I Column II(A) (p) the rod moves without rotation

c P

(B) (q)the rod rotates without translation

cP/2 P/2

P(C) (r) the rod both rotates and translates

P c

(D) (s) acceleration of the point c is P/Mc

22- Satellites A and B of masses M and 2M respectively, are rotating in the circular orbit of radii Rei.e. very close to earth and 2Re respectively , under the effect of earths force. The orbit of thesatellite A is in equatorial place and the orbit of satellite B contains poles of the earth In the

following column I, some physical quantity related to the satellites is mentioned and in the columnII, some statement or value of these quantities in relation to A and B are given .Match the items inthe column I with the correct descriptions in the column II and indicate your answer by darkeningappropriate bubbles in the 4 x4 matrix given. Column I Column II(a) kinetic energy of A and B (p) are of same magnitude

(b) square of the speed of A & B (q ) are in ratio 2 : 1

(c) centripetal force on A and B (r) is 0.5 M g Re for each

(d) angular momentum of A and B (s) are always mutually perpendicular