Page # 1This 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 xy plane centerd at origin and two semi circular rings of radius r lying in xy and yz plane 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) 2 5 mr 2 (C) 2 3 mr 2 (D) 4 mr 2 2. A uniform thin hemispherical shell is kept at rest and in equilibrium on an inclined 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 free to rotate in vertical plane about a fixed horizontal axis passing through frictionless hinge. The stick is held in a horizontal position. At what distance x from the hinge should a second mass M = m be attached to the stick, so that stick falls as fast as possible when released from rest (A) L 2 (B) L 3 (C) L ) 1 2 ( & (D) L ) 1 3 ( & 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 Page # 2 4. A uniform rod of mass m and length L is hinged at one end and free to rotate in horizontal plane. All the surfaces are smooth. A particle of same mass m collides with the rod perpendicular to the length of rod with a speed V 0 . The coefficient of restitution for the collision is e = 2 1 . If hinge reaction during the collision is zero then the value of x is : (A) No such value of x is possible (B) x = 2 L (C) x = 3 L 2 (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 that the block does not slide, the minimum speed V so that the block topples is : (A) t . h g 2 / 1 a w ( ( ) * + + , - ' ' (B) t . h g 2 / 1 w a ( ( ) * + + , - ' ' (C) t . h g 2 / 1 ( ) * + , - (D) None of these 6. An oscillation is superposition of three harmonic oscillations and decribed by the equation x = A sin2" . 1 t where A changes with time according to A = A 0 (1 + cos2" . 2 t) with A 0 to be constant. The frequencies of pure harmonic oscillations forming this oscillation are : (A) . 1 ,. 2 , | | 2 1 . & . (B) . 1 , | | 2 1 . & . , . 1 + . 2 (C) . 1 ,. 2 , | | 1 2 . & . (D) . 1 , . 2 , . 1 + . 2 7. The natural frequency of the system shown in figure is: {The pulleys are smooth and massless.} (A) M k 2 1 " (B) M k 2 2 " (C) M k 1 " (D) M k 4 1 " 8. A uniform disc of mass m is attached to a spring of spring constant k as shown in figure and there is sufficient friction to prevent slipping of disc. Time period of small oscillations of disc is: (A) k m 2" (B) k 3 m 2 2" (C) k 2 m 3 " (D) k 3 m 2 " Page # 3 9. A particle is executing simple harmonic motion in a conservative force field. The total energy of simple harmonic motion is given by E = ax 2 + bv 2 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 be zero} (A) amplitude of S.H.M is a E (B) Maximum velocity of the particle during S.H.M is b E (C) Time peried of motion is a b ! 2 (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 is hinged at one end about a smooth hinge and it performs oscillations of small angle in vertical plane. The length of the equivalent simple pendulum is: (A) 2 3! (B) 3 10! (C) 3 5! (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 of elasticity of rod) (A) AY 2 ! !/ (B) AY 3 3 2 ! !/ (C) AY 3 2 ! !/ (D) 3 2 AY 3 ! !/ 12. A thin uniform elastic rod of natural length L, density ' and Young's modulus Y is immersed completely in vertical position in a fluid of density 2' by applying a vertically downward force at its top end such that the top end 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, neglect atmospheric pressure) (A) Net compression in rod is Y 4 g L 2 ' (B) Net compression in rod is Y 2 g L 3 2 ' (C) Net compression in rod is Y 2 g L 2 ' (D) Net compression in rod is Y g L 2 ' 13. A uniform metal rod (fixed at both ends) of 2 mm 2 cross-section is cooled from 40 ºC to 20 ºC. The co- efficient of the linear expansion of the rod is 12 0 10 &6 per degree & it’s young modulus of elasticity is 10 11 N/m 2 . The energy stored per unit volume of the rod is: (A) 2880 J/m 3 (B) 1500 J/m 3 (C) 5760 J/m 3 (D) 1440 J/m 3 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) v L 2 + 2 " Y A L m (B) v L 2 + 2 " Y A L m 2 (C) v L 2 + " Y A L m (D) v L 2 15. Two forces F 1 and F 2 act on a thin uniform elastic rod placed in space. Force F 1 acts at right end of rod and F 2 acts exactly at centre of rod as shown (both forces act parallel to length of the rod). F 2 F 1 C (i) F 1 causes extension of rod while F 2 causes compression of rod. (ii) F 1 causes extension of rod and F 2 also causes extension of rod. (iii) F 1 causes extension of rod while F 2 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 Page # 4 16. A uniform disc of mass m and radius R is free to rotate about its fixed horizontal axis without friction. There is sufficient friction between the inextensible light string and disc to prevent slipping of string over disc. At the shown instant extension in light spring is K mg 3 , 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 3 g 4 (B) Tension in the string continuously increases till extension in the spring reaches maximum value. (C) Acceleration of the block just after release g 4 3 (D) Angular acceleration of disc just after release is R 3 g 4 17. A uniform solid cylinder of mass m, radius R is at rest on an extremely rough horizontal surface. Now a horizontal force F = kt (where k = positive constant and t = time), is applied at the highest point on the cylinder. Assume that the cylinder performs pure rolling . 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 m 3 kt 2 2 (C) speed of the particle which is at highest point of the cyllinder after time t will be m 3 kt 4 2 (D) If the coefficient of friction between the ground and the cylinder is 1, the cylinder will start sliding at k mg 3 t 1 2 . 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 = / & 2 A (B) Amplitude of SHM is 2 A / (C) At any time t displacement of particle from mean position is,x = ( ( ) * + + , - & / / 2 3 t sin A 2 (D) The equilibrium position is x = 2 2 A 3 / 20. Figure shows roughly how the force F between two adjacent atoms in a solid varies with inter atomic separation r. Which of the following statements are correct ? (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 the axis of r. Page # 5 21. A 20 gm particle is subjected to two simple harmonic motions x 1 = 2 sin 10 t, x 2 = 4 sin (10 t + 3 " ). where x 1 & x 2 are in metre & t is in sec. (A) The displacement of the particle at t = 0 will be 3 2 m. (B) Maximum speed of the particle will be 7 20 m/s. (C) Magnitude of maximum acceleration of the particle will be 200 7 m/s 2 . (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 by X = A cos /t. There is sufficient friction present in the surface of contact that can prevent the slipping between the cylinder and platform. maximum torque acting on the cylinder during its motion in N-m is 34 Find the value of 33. [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 end A 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 AP AB ( ) * + , - 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 from horizontal 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 line PQ from the instant of breakage. (g = 10 m/s 2 ) (hinge is smooth and there is 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 kg R ) 3 k ( 2 2 5 " . 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 sin/t. The time period of SHM is T and the coefficient of friction between the block and the platform is 1. The condition that the block does not slip on the platform at any instant is 2 2 gT A x" 6 1 then write the value of ‘x’. 27. Two particles P 1 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 amplitudes is 12 cm then find the minimum time after which the separation between the particles become 6 cm. 28. Two opposite forces F 1 = 120N and F 2 = 80N act on an heavy elastic plank of modulus of elasticity y = 2×10 11 N/m 2 and length L = 1m placed over a smooth horizontal surface. The cross-sectional area of plank is A = 0.5m 2 . If the change in the length of plank is x × 10 –9 m, then find x ? F 1 F 2 29. A ring of radius r made of wire of density ' is rotated about a stationary vertical axis passing 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 7. Ignore gravity. (Take 7 ' = 4 m 2 /s 2 and r = 1m) Page # 6 30. The length of an elastic string is 5 metre when the longitudinal tension is 4 N and 6 metre when the tension is 5 N. If the length of the string (in metre) is "2X" when the longitudinal tension is 9 N (assume Hooke’s law 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 as shown. maximum length of AB so that all the planks remain in equilibrium is 4 L 7 . STATEMENT-2 : For equilibrium, combined centre of mass of plank-2 and plank-3 must have horizontal distance 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 mean position. 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 m g k 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 2 u . Due to this the block starts oscillatory motion. 34. Velocity of block just after bullet comes out of block is, (A) 4 u (B) 2 u (C) 2u (D) 3u 35. Amplitude of oscillations of block is : (A) 2 mg k (B) mg k (C) 3 2 mg k (D) 2mg k 36. Maximum speed of block during oscillations is : (A) 2 g k m (B) k g m (C) 3 2 g k m (D) 2 k g m COMPREHENSION - 2 A rod of mass 'm and length L is attached to a L shaped plank at 'A'. rod can move freely about A. A string is tied between rod and plank as shown in figure. Whole system is moving with a constant acceleration g in x-direction Page # 7 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) 2 5 mg (D) 4 mg 5 39. If string is cut at any instant then the angular acceleration of rod (with respect to the plank) at that instant is (A) ! 3 g 2 (B) ! g 6 (C) ! 3 g 2 (D) ! 2 g 3 COMPREHENSION - 3 A uniform rod of mass M and length L, area of cross section A is placed on a smooth horizontal surface. Forces acting on the rod are shown 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) AY 6 L F 7 2 (B) AY 6 L F 11 2 (C) AY 6 L F 5 2 (D) AY 2 L F 3 2 43. In column-I some situations are shown and in column-II information about their resulting motion is given. Select the 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 acceleration 1 kg and radius 1 m, 1 s = 0.05 of centre of mass of body. (Q) A uniform body of mass 1 kg, r = m 2 1 , (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 perpendicular to plane of paper) = 2 kg m 2 , 1 s = 0.3 (R) A uniform solid cylinder released on a (3) Body rotates clockwise. fixed incline plane m = 2 kg, R = 1 m, 1 s = 5 2 . (S) A uniform body of mass 1 kg (4) Body rotates anticlockwise. r = m 2 1 , R = 1 m, I(about axis passting through centre and perpendicular to plain of paper) = 2kgm 2 1 s = 0.5 (String tightly wound on inner radius is pulled). Page # 8 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 total energy 2J. x is positive if measured towards right from O. At t = 0, particle is at O and moves towards right. Then select 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 the block. In column I some changes are given and column II respective effect is written. Then select the correct answer 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