Page 1 For JEE Advanced Assertion and Reason Directions : Choose the correct option. (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. (c) If Assertion is true, but the Reason is false. (d) If Assertion is false but the Reason is true. Q 1. Assertion : In x = A cos ?t, x is the displacement measured from extreme position. Reason : In the above equation x = A at time t = 0. Q 2. Assertion : A particle is under SHM along the x-axis. Its mean position is x = 2, amplitude is A = 2 and angular frequency ?. At t = 0, particle is at origin, then x-co-ordinate versus time equation of the particle will be x = - 2 cos ?t + 2. Reason : At t = 0, particle is at rest. Q 3. Assertion : A spring block system is kept over a smooth surface as shown in figure. If a constant horizontal force F is applied on the block it will start oscillating simple harmonically. Reason : Time period of oscillation is less than m 2 k ? . Q 4. Assertion : Time taken by a particle in SHM to move from x = A to x = 3A 2 is same as the time taken by the particle to move from 3A A x to x 22 ?? . Reason : Corresponding angles rotated in the reference circle are same in the given time intervals. Q 5. Assertion : Path of a particle in SHM is always a straight line. Reason : All straight line motions are not simple harmonic. Q 6. Assertion : In spring block system if length of spring and mass of block both are halved, then angular frequency of oscillations will remain unchanged. Reason : Angular frequency is given by k m ?? Q 7. Assertion : All small oscillations are simple harmonic in nature. Reason : Oscillations of spring block system are always simple harmonic whether amplitude is small or large. Q 8. Assertion : In x = A cos ?t, the dot product of acceleration and velocity is positive for time interval 0< t < 2 ? ? . Reason : Angle between them is 0°. Page 2 For JEE Advanced Assertion and Reason Directions : Choose the correct option. (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. (c) If Assertion is true, but the Reason is false. (d) If Assertion is false but the Reason is true. Q 1. Assertion : In x = A cos ?t, x is the displacement measured from extreme position. Reason : In the above equation x = A at time t = 0. Q 2. Assertion : A particle is under SHM along the x-axis. Its mean position is x = 2, amplitude is A = 2 and angular frequency ?. At t = 0, particle is at origin, then x-co-ordinate versus time equation of the particle will be x = - 2 cos ?t + 2. Reason : At t = 0, particle is at rest. Q 3. Assertion : A spring block system is kept over a smooth surface as shown in figure. If a constant horizontal force F is applied on the block it will start oscillating simple harmonically. Reason : Time period of oscillation is less than m 2 k ? . Q 4. Assertion : Time taken by a particle in SHM to move from x = A to x = 3A 2 is same as the time taken by the particle to move from 3A A x to x 22 ?? . Reason : Corresponding angles rotated in the reference circle are same in the given time intervals. Q 5. Assertion : Path of a particle in SHM is always a straight line. Reason : All straight line motions are not simple harmonic. Q 6. Assertion : In spring block system if length of spring and mass of block both are halved, then angular frequency of oscillations will remain unchanged. Reason : Angular frequency is given by k m ?? Q 7. Assertion : All small oscillations are simple harmonic in nature. Reason : Oscillations of spring block system are always simple harmonic whether amplitude is small or large. Q 8. Assertion : In x = A cos ?t, the dot product of acceleration and velocity is positive for time interval 0< t < 2 ? ? . Reason : Angle between them is 0°. Q 9. Assertion : In simple harmonic motion displacement and acceleration always have a constant ratio. Reason : displacement T2 acceleration ?? Q 10. Assertion : We can call circular motion also as simple harmonic motion. Reason : Angular velocity in uniform circular motion and angular frequency in simple harmonic motion have the same meanings. Answers 1. (d) 2. (b) 3. (c) 4. (a) 5. (d) 6. (d) 7. (d) 8. (a) 9. (a) 10. (d) Solutions 1. For, x = A cos at, particle is at x = A at t = 0, but the displacement is measured from mean position. So, assertion is false. 2. Assertion and reason are both true but reason is not correct explanation. 3. Assertion is true while time period of oscillation is so, the reason is false. 4. Reason correctly explains assertion. 5. In uniform circular motion also particle performs SHM, so assertion is false. 6. If spring is halved, then spring constant becomes twice and as mass is halved, then i.e., angular frequency gets double. Thus, assertion is false. 7. Assertion is false as if then motion will not be SHM. 8. For x = A cos ?t between t = 0 to 2 ? ? , particle moves from one extreme to mean position such that velocity and acceleration are in same direction, i.e., angle between them is zero. Such that Page 3 For JEE Advanced Assertion and Reason Directions : Choose the correct option. (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. (c) If Assertion is true, but the Reason is false. (d) If Assertion is false but the Reason is true. Q 1. Assertion : In x = A cos ?t, x is the displacement measured from extreme position. Reason : In the above equation x = A at time t = 0. Q 2. Assertion : A particle is under SHM along the x-axis. Its mean position is x = 2, amplitude is A = 2 and angular frequency ?. At t = 0, particle is at origin, then x-co-ordinate versus time equation of the particle will be x = - 2 cos ?t + 2. Reason : At t = 0, particle is at rest. Q 3. Assertion : A spring block system is kept over a smooth surface as shown in figure. If a constant horizontal force F is applied on the block it will start oscillating simple harmonically. Reason : Time period of oscillation is less than m 2 k ? . Q 4. Assertion : Time taken by a particle in SHM to move from x = A to x = 3A 2 is same as the time taken by the particle to move from 3A A x to x 22 ?? . Reason : Corresponding angles rotated in the reference circle are same in the given time intervals. Q 5. Assertion : Path of a particle in SHM is always a straight line. Reason : All straight line motions are not simple harmonic. Q 6. Assertion : In spring block system if length of spring and mass of block both are halved, then angular frequency of oscillations will remain unchanged. Reason : Angular frequency is given by k m ?? Q 7. Assertion : All small oscillations are simple harmonic in nature. Reason : Oscillations of spring block system are always simple harmonic whether amplitude is small or large. Q 8. Assertion : In x = A cos ?t, the dot product of acceleration and velocity is positive for time interval 0< t < 2 ? ? . Reason : Angle between them is 0°. Q 9. Assertion : In simple harmonic motion displacement and acceleration always have a constant ratio. Reason : displacement T2 acceleration ?? Q 10. Assertion : We can call circular motion also as simple harmonic motion. Reason : Angular velocity in uniform circular motion and angular frequency in simple harmonic motion have the same meanings. Answers 1. (d) 2. (b) 3. (c) 4. (a) 5. (d) 6. (d) 7. (d) 8. (a) 9. (a) 10. (d) Solutions 1. For, x = A cos at, particle is at x = A at t = 0, but the displacement is measured from mean position. So, assertion is false. 2. Assertion and reason are both true but reason is not correct explanation. 3. Assertion is true while time period of oscillation is so, the reason is false. 4. Reason correctly explains assertion. 5. In uniform circular motion also particle performs SHM, so assertion is false. 6. If spring is halved, then spring constant becomes twice and as mass is halved, then i.e., angular frequency gets double. Thus, assertion is false. 7. Assertion is false as if then motion will not be SHM. 8. For x = A cos ?t between t = 0 to 2 ? ? , particle moves from one extreme to mean position such that velocity and acceleration are in same direction, i.e., angle between them is zero. Such that as then = positive. Thus, reason is true explanation of assertion. 9. In SHM, a = - ? 2 x, So, reason is correct explanation of assertion. 10. Only uniform circular motion can be called SHM, so, assertion is false. Objective Questions Single Correct Option Q 1. A particle of mass 2 kg moves in simple harmonic motion and its potential energy U varies with position x as shown. The period of oscillation of the particle is (a) 2 s 5 ? (b) 22 s 5 ? (c) 2 s 5 ? (d) 4 s 5 ? Q 2. In the figure shown, a spring mass system is placed on a horizontal smooth surface in between two vertical rigid walls W 1 and W 2. One end of spring is fixed with wall W 1 and other end is attached with mass m which is free to move. Initially, spring is tension free and having natural length l 0. Mass m is compressed through a distance a and released. Taking the collision between wall W 2 and mass m as elastic and K as spring constant, the average force exerted by mass m on wall W 2 in one oscillation of block is (a) 2aK ? (b) 2ma ? (c) aK ? (d) 2aK m Q 3. Two simple harmonic motions are represented by the following equations y = 40 sin ?t and y2 = 10(sin ?t + c cos ?t). If their displacement amplitudes are equal, then the value of c (in appropriate units) is (a) 13 (b) 15 (c) 17 (d) 4 Q 4. A particle executes simple harmonic motion with frequency 2.5 Hz and amplitude 2 m. The speed of the particle 0.3 s after crossing, the equilibrium position is (a) zero (b) 2 ? m/s (c) 4 ? m/s (d) ? m/s Page 4 For JEE Advanced Assertion and Reason Directions : Choose the correct option. (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. (c) If Assertion is true, but the Reason is false. (d) If Assertion is false but the Reason is true. Q 1. Assertion : In x = A cos ?t, x is the displacement measured from extreme position. Reason : In the above equation x = A at time t = 0. Q 2. Assertion : A particle is under SHM along the x-axis. Its mean position is x = 2, amplitude is A = 2 and angular frequency ?. At t = 0, particle is at origin, then x-co-ordinate versus time equation of the particle will be x = - 2 cos ?t + 2. Reason : At t = 0, particle is at rest. Q 3. Assertion : A spring block system is kept over a smooth surface as shown in figure. If a constant horizontal force F is applied on the block it will start oscillating simple harmonically. Reason : Time period of oscillation is less than m 2 k ? . Q 4. Assertion : Time taken by a particle in SHM to move from x = A to x = 3A 2 is same as the time taken by the particle to move from 3A A x to x 22 ?? . Reason : Corresponding angles rotated in the reference circle are same in the given time intervals. Q 5. Assertion : Path of a particle in SHM is always a straight line. Reason : All straight line motions are not simple harmonic. Q 6. Assertion : In spring block system if length of spring and mass of block both are halved, then angular frequency of oscillations will remain unchanged. Reason : Angular frequency is given by k m ?? Q 7. Assertion : All small oscillations are simple harmonic in nature. Reason : Oscillations of spring block system are always simple harmonic whether amplitude is small or large. Q 8. Assertion : In x = A cos ?t, the dot product of acceleration and velocity is positive for time interval 0< t < 2 ? ? . Reason : Angle between them is 0°. Q 9. Assertion : In simple harmonic motion displacement and acceleration always have a constant ratio. Reason : displacement T2 acceleration ?? Q 10. Assertion : We can call circular motion also as simple harmonic motion. Reason : Angular velocity in uniform circular motion and angular frequency in simple harmonic motion have the same meanings. Answers 1. (d) 2. (b) 3. (c) 4. (a) 5. (d) 6. (d) 7. (d) 8. (a) 9. (a) 10. (d) Solutions 1. For, x = A cos at, particle is at x = A at t = 0, but the displacement is measured from mean position. So, assertion is false. 2. Assertion and reason are both true but reason is not correct explanation. 3. Assertion is true while time period of oscillation is so, the reason is false. 4. Reason correctly explains assertion. 5. In uniform circular motion also particle performs SHM, so assertion is false. 6. If spring is halved, then spring constant becomes twice and as mass is halved, then i.e., angular frequency gets double. Thus, assertion is false. 7. Assertion is false as if then motion will not be SHM. 8. For x = A cos ?t between t = 0 to 2 ? ? , particle moves from one extreme to mean position such that velocity and acceleration are in same direction, i.e., angle between them is zero. Such that as then = positive. Thus, reason is true explanation of assertion. 9. In SHM, a = - ? 2 x, So, reason is correct explanation of assertion. 10. Only uniform circular motion can be called SHM, so, assertion is false. Objective Questions Single Correct Option Q 1. A particle of mass 2 kg moves in simple harmonic motion and its potential energy U varies with position x as shown. The period of oscillation of the particle is (a) 2 s 5 ? (b) 22 s 5 ? (c) 2 s 5 ? (d) 4 s 5 ? Q 2. In the figure shown, a spring mass system is placed on a horizontal smooth surface in between two vertical rigid walls W 1 and W 2. One end of spring is fixed with wall W 1 and other end is attached with mass m which is free to move. Initially, spring is tension free and having natural length l 0. Mass m is compressed through a distance a and released. Taking the collision between wall W 2 and mass m as elastic and K as spring constant, the average force exerted by mass m on wall W 2 in one oscillation of block is (a) 2aK ? (b) 2ma ? (c) aK ? (d) 2aK m Q 3. Two simple harmonic motions are represented by the following equations y = 40 sin ?t and y2 = 10(sin ?t + c cos ?t). If their displacement amplitudes are equal, then the value of c (in appropriate units) is (a) 13 (b) 15 (c) 17 (d) 4 Q 4. A particle executes simple harmonic motion with frequency 2.5 Hz and amplitude 2 m. The speed of the particle 0.3 s after crossing, the equilibrium position is (a) zero (b) 2 ? m/s (c) 4 ? m/s (d) ? m/s Q 5. A particle oscillates simple harmonically with a period of 16 s. Two second after crossing the equilibrium position its velocity becomes 1 m/s. The amplitude is (a) m 4 ? (b) 82 m ? (c) 8 m ? (d) 42 m ? Q 6. A seconds pendulum is suspended from the ceiling of a trolley moving horizontally with an acceleration of 4 m/s 2 . Its period of oscillation is (a) 1.90 s (b) 1.70 s (c) 2.30 s (d) 1.40 s Q 7. A particle is performing a linear simple harmonic motion. If the instantaneous acceleration and velocity of the particle are a and v respectively, identify the graph which correctly represents the relation between a and v (a) (b) (c) (d) Q 8. In a vertical U-tube a column of mercury oscillates simple harmonically. If the tube contains 1 kg of mercury and 1 cm of mercury column weighs 20 g, then the period of oscillation is (a) 1s (b) 2 s (c) 2s (d) Insufficient data Q 9. A solid cube of side a and density ?0 floats on the surface of a liquid of density ?. If the cube is slightly pushed downward, then it oscillates simple harmonically with a period of (a) 0 a 2 g ? ? ? (b) 0 a 2 g ? ? ? (c) 0 a 2 1g ? ?? ? ? ?? ? ?? (d) 0 a 2 1g ? ?? ? ? ?? ? ?? Q 10. A particle of mass m is attached with three springs A, B and C of equal force constants k as shown in figure. The particle is pushed slightly against the spring C and released, the time period of oscillation will be (a) m 2 k ? (b) m 2 2k ? (c) m 2 3k ? (d) m 2 5k ? Q 11. A uniform stick of length l is mounted so as to rotate about a horizontal axis perpendicular to the stick and at a distance d from the centre of mass. The time period of small oscillations has a minimum value when d/l is (a) 1 2 (b) 1 12 (c) 1 3 (d) 1 6 Q 12. Three arrangements of spring-mass system are shown in figures (A), (B) and (C). If T1, T2 and T3 represent the respective periods of oscillation, then correct relation is Page 5 For JEE Advanced Assertion and Reason Directions : Choose the correct option. (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. (c) If Assertion is true, but the Reason is false. (d) If Assertion is false but the Reason is true. Q 1. Assertion : In x = A cos ?t, x is the displacement measured from extreme position. Reason : In the above equation x = A at time t = 0. Q 2. Assertion : A particle is under SHM along the x-axis. Its mean position is x = 2, amplitude is A = 2 and angular frequency ?. At t = 0, particle is at origin, then x-co-ordinate versus time equation of the particle will be x = - 2 cos ?t + 2. Reason : At t = 0, particle is at rest. Q 3. Assertion : A spring block system is kept over a smooth surface as shown in figure. If a constant horizontal force F is applied on the block it will start oscillating simple harmonically. Reason : Time period of oscillation is less than m 2 k ? . Q 4. Assertion : Time taken by a particle in SHM to move from x = A to x = 3A 2 is same as the time taken by the particle to move from 3A A x to x 22 ?? . Reason : Corresponding angles rotated in the reference circle are same in the given time intervals. Q 5. Assertion : Path of a particle in SHM is always a straight line. Reason : All straight line motions are not simple harmonic. Q 6. Assertion : In spring block system if length of spring and mass of block both are halved, then angular frequency of oscillations will remain unchanged. Reason : Angular frequency is given by k m ?? Q 7. Assertion : All small oscillations are simple harmonic in nature. Reason : Oscillations of spring block system are always simple harmonic whether amplitude is small or large. Q 8. Assertion : In x = A cos ?t, the dot product of acceleration and velocity is positive for time interval 0< t < 2 ? ? . Reason : Angle between them is 0°. Q 9. Assertion : In simple harmonic motion displacement and acceleration always have a constant ratio. Reason : displacement T2 acceleration ?? Q 10. Assertion : We can call circular motion also as simple harmonic motion. Reason : Angular velocity in uniform circular motion and angular frequency in simple harmonic motion have the same meanings. Answers 1. (d) 2. (b) 3. (c) 4. (a) 5. (d) 6. (d) 7. (d) 8. (a) 9. (a) 10. (d) Solutions 1. For, x = A cos at, particle is at x = A at t = 0, but the displacement is measured from mean position. So, assertion is false. 2. Assertion and reason are both true but reason is not correct explanation. 3. Assertion is true while time period of oscillation is so, the reason is false. 4. Reason correctly explains assertion. 5. In uniform circular motion also particle performs SHM, so assertion is false. 6. If spring is halved, then spring constant becomes twice and as mass is halved, then i.e., angular frequency gets double. Thus, assertion is false. 7. Assertion is false as if then motion will not be SHM. 8. For x = A cos ?t between t = 0 to 2 ? ? , particle moves from one extreme to mean position such that velocity and acceleration are in same direction, i.e., angle between them is zero. Such that as then = positive. Thus, reason is true explanation of assertion. 9. In SHM, a = - ? 2 x, So, reason is correct explanation of assertion. 10. Only uniform circular motion can be called SHM, so, assertion is false. Objective Questions Single Correct Option Q 1. A particle of mass 2 kg moves in simple harmonic motion and its potential energy U varies with position x as shown. The period of oscillation of the particle is (a) 2 s 5 ? (b) 22 s 5 ? (c) 2 s 5 ? (d) 4 s 5 ? Q 2. In the figure shown, a spring mass system is placed on a horizontal smooth surface in between two vertical rigid walls W 1 and W 2. One end of spring is fixed with wall W 1 and other end is attached with mass m which is free to move. Initially, spring is tension free and having natural length l 0. Mass m is compressed through a distance a and released. Taking the collision between wall W 2 and mass m as elastic and K as spring constant, the average force exerted by mass m on wall W 2 in one oscillation of block is (a) 2aK ? (b) 2ma ? (c) aK ? (d) 2aK m Q 3. Two simple harmonic motions are represented by the following equations y = 40 sin ?t and y2 = 10(sin ?t + c cos ?t). If their displacement amplitudes are equal, then the value of c (in appropriate units) is (a) 13 (b) 15 (c) 17 (d) 4 Q 4. A particle executes simple harmonic motion with frequency 2.5 Hz and amplitude 2 m. The speed of the particle 0.3 s after crossing, the equilibrium position is (a) zero (b) 2 ? m/s (c) 4 ? m/s (d) ? m/s Q 5. A particle oscillates simple harmonically with a period of 16 s. Two second after crossing the equilibrium position its velocity becomes 1 m/s. The amplitude is (a) m 4 ? (b) 82 m ? (c) 8 m ? (d) 42 m ? Q 6. A seconds pendulum is suspended from the ceiling of a trolley moving horizontally with an acceleration of 4 m/s 2 . Its period of oscillation is (a) 1.90 s (b) 1.70 s (c) 2.30 s (d) 1.40 s Q 7. A particle is performing a linear simple harmonic motion. If the instantaneous acceleration and velocity of the particle are a and v respectively, identify the graph which correctly represents the relation between a and v (a) (b) (c) (d) Q 8. In a vertical U-tube a column of mercury oscillates simple harmonically. If the tube contains 1 kg of mercury and 1 cm of mercury column weighs 20 g, then the period of oscillation is (a) 1s (b) 2 s (c) 2s (d) Insufficient data Q 9. A solid cube of side a and density ?0 floats on the surface of a liquid of density ?. If the cube is slightly pushed downward, then it oscillates simple harmonically with a period of (a) 0 a 2 g ? ? ? (b) 0 a 2 g ? ? ? (c) 0 a 2 1g ? ?? ? ? ?? ? ?? (d) 0 a 2 1g ? ?? ? ? ?? ? ?? Q 10. A particle of mass m is attached with three springs A, B and C of equal force constants k as shown in figure. The particle is pushed slightly against the spring C and released, the time period of oscillation will be (a) m 2 k ? (b) m 2 2k ? (c) m 2 3k ? (d) m 2 5k ? Q 11. A uniform stick of length l is mounted so as to rotate about a horizontal axis perpendicular to the stick and at a distance d from the centre of mass. The time period of small oscillations has a minimum value when d/l is (a) 1 2 (b) 1 12 (c) 1 3 (d) 1 6 Q 12. Three arrangements of spring-mass system are shown in figures (A), (B) and (C). If T1, T2 and T3 represent the respective periods of oscillation, then correct relation is (a) T1 > T2 > T3 (b) T3 > T2 > T, (c) T2 > T1 > T3 (d) T2 > T3 > T1 Q 13. Three arrangements are shown in figure. (A) A spring of mass m and stiffness k (B) A block of mass m attached to massless spring of stiffness k (C) A block of mass m 2 attached to a spring of mass m 2 and stiffness k If T1 T2 and T3 represent the period of oscillation in the three cases respectively, then identify the correct relation (a) T1<T2<T3 (b) T1<T3<T2 (c) T1>T3>T2 (d) T3 <T1<T2 Q 14. A block of mass M is kept on a smooth surface and touches the two springs as shown in the figure but not attached to the springs. Initially springs are in their natural length. Now, the block is shifted (l 0/2) from the given position in such a way that it compresses a spring and released. The time-period of oscillation of mass will be (a) M 2k ? (b) M 2 5k ? (c) 3M 2k ? (d) M 2k ? Q 15. A particle moving on x-axis has potential energy U = 2 - 20x + 5x 2 Joule along x-axis. The particle is released at x = - 3. The maximum value of x will be (x is in metre) (a) 5 m (b) 3 m (c) 7 m (d) 8 m Q 16. A block of mass m, when attached to a uniform ideal spring with force constant k and free length L executes SHM. The spring is then cut in two pieces, one with free length n L and other with free length (1 - n) L. The block is also divided in the same fraction. The smaller part of the block attached to longer part of the spring executes SHM with frequency f1. The bigger part of the block attached to smaller part of the spring executes SHM with frequency f 2. The ratio f 1/ f 2 isRead More

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