Course: Simple Harmonic Motion

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Physics By KAILASH SHARMA 1 | P a g e

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Course: Simple Harmonic Motion

Presented by Kailash Sharma

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EXERCISE-I Part-I

(Single Correct type Questions)

1. A particle performing SHM takes time equal to T (time period of SHM) in consecutive appearances at a particular point. This point is :

(A) An extreme position (B) The mean position

(C) Between positive extreme and mean position (D) Between negative extreme and mean position

2. A particle executing linear SHM. Its time period is equal to the smallest time interval in which particle acquires a particular velocity v, the magnitude of v may be :

(A) zero (B) Vmax (C)V^max

2 (D) V^max

2

3. If F is a force, v is velocity, a vector is acceleration vector and r vector is displacement vector w.r.t mean position than which of the following quantities are always non-negative in a simple harmonic motion along a straight line ?

(A) F.a (B) v.r (C) a.r (D) F.r

4. A simple harmonic motion having an amplitude A and time period T is represented by the equation:

y = 5 sin π (t + 4) m

Then the value of A (in m) and T (in sec) are:

(A) A = 5; T = 2 (B) A = 10; T = 1 (C) A = 5; T = 1 (D) A = 10; T = 2 5. The maximum acceleration of particle in SHM is made two times keeping the maximum speed to

be constant. It is possible when

(A) amplitude of oscillation is doubled while frequency remains constant (B) amplitude of doubled while frequency is halved

(C) frequency is doubled while amplitude is halved

(D) frequency is doubled while amplitude remains constant

6. A small mass executes linear SHM about O with amplitude a and period T. Its displacement from O at time T/8 after passing through O is :

(A) a

8 (B) a

2 2 ^(C)

a

2 (D) a

2

7. The displacement of a body executing SHM is given by x = A sin (2πt + π/3). The first time from t = 0 when the velocity is maximum is

(A) 0.33 sec (B) 0.16 sec (C) 0.25 sec (D) 0.5 sec

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8. A particle performing SHM is found at its equilibrium at t = 1 sec. and it is found to have a speed of 0.25 m/s at t = 2 sec. If the period of oscillation is 6 sec. Calculate amplitude of oscillation

(A) 3

2m (B) 3

4m (C)6

m (D) 3

8m

9. Two SHM’s are represented by y = a sin (ωt – kx) and y = b cos (ωt – kx). The phase difference between the two is :

(A) 2

 (B)

4

 (C)

6

 (D) 3

4



10. How long after the beginning of motion is the displacement of a harmonically oscillating particle equal to one half its amplitude if the period is 24s and particle starts from rest.

(A) 12s (B) 2s (C) 4s (D) 6s

11. The magnitude of average acceleration in half time period from equilibrium position in a simple harmonic motion is

(A) 2A2

 ^(B)

A 2

2



 ^(C)

A 2

2



 (D) Zero

12. The time taken by a particle performing SHM to pass from point A to B where its velocities are same is 2 seconds. After another 2 seconds it returns to B. The time period of oscillation is (in seconds)

(A) 2 (B) 8 (C) 6 (D) 4

13. The angular frequency of motion whose equation is d y²

4 9y 0

dt + = is (y = displacement and t = time)

(A) 9

4 (B) 4

9 (C) 3

2 (D) 2

3

14. Time period of a particle executing SHM is 8 sec. At t = 0 it is at the mean position. The ratio of the distance covered by the particle in the 1^st second to the 2^nd second is :

(A) 1

2 1+ ^{(B) 2} ^(C)

1

2 ^{(D) 2}⁺¹

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15. A particle of mass 1 kg is undergoing S.H.M for which graph between force and displacement (from mean position) as shown. Its time period, in seconds is:

(A) π/3 (B) 2π/3 (C) π/6 (D) 3/π

16. A particle executes SHM on a straight line path. The amplitude of oscillation is 2cm. when the displacement of the particle from the mean position is 1 cm, the numerical value of magnitude of acceleration is equal to the numerical value of magnitude of velocity. The frequency of SHM (in second^–1) is:

(A) 2 3 (B) 2

3

 (C) 3

2 ^(D)

1 2 3

17. A particle executes SHM with time period T and amplitude A. The maximum possible average velocity in time T

4 is (A) 2A

T (B) 4A

T (C) 8A

T (D) 4 2A

T

18. A particle executes SHM of period 1.2 sec. and amplitude 8 cm. Find the time it takes to travel 3 cm from positive extremely of its oscillation.

(A) 0.28 sec (B) 0.32 sec. (C) 0.17 sec. (D) 0.42 sec.

19. A particle performs SHM with a period T and amplitude a. The mean velocity of the particle over the time interval during which it travels a distance a/2 from the extreme position is

(A) a/T (B) 2a/T (C) 3a/T (D) a/2T

20. A particle moves along the x-axis according to: x = A.[1 + sinωt]. What distance does it travel between t = 0 and t = 2.5 π/ω ?

(A) 4A (B) 6A (C) 5A (D) None

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21. Two particles undergo SHM along parallel lines with the same time period (T) and equal amplitude.

At a particular instant, one particle is at its extreme position while the other is at its mean position.

They move in the same direction. They will cross each other after a further time

(A) T/8 (B) 3T/8 (C) T/6 (D) 4T/3

22. Two particles are in SHM in a straight line about same equilibrium position. Amplitude A and time period T of both the particle are equal. At time t = 0, one particle is at displacement y1 = +A and the other at y2 = – A/2, and they are approaching towards each other. After what time they cross each other?

(A) T/3 (B) T/4 (C) 5T/6 (D) T/6

23. Two particles execute SHM of same amplitude of 20 cm with same period along the same line about the same equilibrium position. The maximum distance between the two is 20 cm. Their phase difference in radians is

(A) 2 3

 (B)

2

 (C)

3

 (D)

4



24. Two particles A and B perform SHM along the same straight line with the same amplitude ‘a’ same frequency ‘f’ and same equilibrium position ‘O’. The greatest distance between them is found to be 3a/2. At some instant of time they have the same displacement from mean position. What is the displacement ?

(A) a

2 (B) a 7

4 ^(C)

3 a

2 ^(D)

3a 4

25. Two pendulums have time period T and 5T/4. They start SHM at the same time from the mean position. After how many oscillations of the smaller pendulum they will be again in the same phase:

(A) 5 (B) 4 (C) 11 (D) 9

26. Two particles are in SHM on same straight line with amplitude A and 2A and with same angular frequency ω. It is observed that when first particle is at a distance A

2 from origin and going toward mean position, other particle is at extreme position on other side of mean position. Find phase difference between the two particles.

(A) 45° (B) 90° (C) 135° (D) 180°

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27. A body performs simple harmonic oscillations along the straight line ABCDE with C as the midpoint of AE. Its kinetic energies at B and D are each one fourth of its maximum value. If AE = 2R, the distance between B and D is

(A) 3 R

2 ^(B)

R

2 ^(C) 3 R (D) 2 R

28. Two particles P and Q describe simple harmonic motions of same period, same amplitude, along the same line about the same equilibrium position O. When P and Q are on opposite sides of O at the same distance from O they have the same speed of 1.2 m/s in the same direction, when their displacements are the same they have the same speed of 1.6 m/s in opposite directions. The maximum velocity in m/s of either particle is

(A) 2.8 (B) 2.5 (C) 2.4 (D) 2

29. A mass at the end of spring executes harmonic motion about an equilibrium position with an amplitude A. Its speed as it passes through the equilibrium position is V. If extended 2A and released, the speed of the mass passing through the equilibrium position will be

(A) 2V (B) 4V (C) V

2 (D) V

4

30. A particle starts oscillating simple harmonically from its equilibrium position then, the ratio of kinetic energy and potential energy of the particle at the time T/12 is : (T = time period)

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

31. If the potential energy of a harmonic oscillator of mass 2 kg on its equilibrium position is 5 joules and the total energy is 9 joules when the amplitude is one meter the period of the oscillator (in sec) is:

(A) 1.5 (B) 3.14 (C) 6.28 (D) 4.67

32. The K.E. and P.E of a particle executing SHM with amplitude A will be equal when its displacement is

(A) 2 A (B) A

2 (C) A

2 ^(D)

2A 3

33. A point particle of mass 0.1 kg is executing S.H.M. of amplitude of 0.1 m. When the particle passes through the mean position, its kinetic energy is 8 × 10^–3 J. The equation of motion of this particle when the initial phase of oscillation is 45° can be given by

(A) 0.1 cos 4t 4

 +

 

  (B) 0.1 sin 4t 4

 + 

 

  (C) 0.4 sin t

4

 +

 

  (D) 0.2 sin 2t

2

+ 

 

 

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Use referral code KAILASH10 to get 10% discount on subscriptions on Unacademy Plus 34. For a particle performing SHM :

(A) The kinetic energy is never equal to the potential energy (B) The kinetic energy is always equal to the potential energy

(C) The average kinetic energy in one time period is equal to the average potential in this period (D) The average kinetic energy in any time interval is equal to average potential energy in that interval

35. Acceleration a versus time t graph of a body in SHM is given by a curve shown below. T is the time period. Then corresponding graph between kinetic energy KE and time t is correctly represented by

(A) (B)

(C) (D)

36. A particle performs S.H.M. of amplitude A along a straight line. When it is at a distance 3 2 ^A from mean position, its kinetic energy gets increased by an amount 1 ² ²

m A

2  due to an impulsive force. Then its new amplitude becomes:

(A) 5

2 A ^(B)

3A

2 ^(C) 2 A (D) 5 A

37. Two spring mass systems have equal mass and spring constant k1 and k2. If the maximum velocities in two systems are equal then ratio of amplitude of 1st to that of 2nd is :

(A) k / k₁ ₂ (B) k1/k2 (C) k2/k1 (D) k / k₂ ₁

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38. A toy car of mass m is having two similar rubber ribbons attached to it as shown in the figure. The force constant of each rubber ribbon is k and surface is frictionless. The car is displaced from mean position by x cm and released. At the mean position the ribbons are undeformed. Vibration period is

(A)

2

m(2k)

2 k (B)

2

1 m(2k)

2 k (C) m

2 k (D) m

2 k k +

39. A mass of 1 kg attached to the bottom of a spring has a certain frequency of vibration. The following mass has to be added to it in order to reduce the frequency by half :

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

40. A ball of mass m kg hangs from a spring of spring constant k. The ball oscillates with a period of T seconds. If the ball is removed, the spring is shortened (w.r.t. length in mean position) by

(A)

2 2

gT metre

(2 ) (B)

2 2

3T g metre

(2 ) (C) Tm

metre

k (D) Tk

metre m

41. A smooth inclined plane having angle of inclination 30° with horizontal has a mass 2.5 kg held by a spring which is fixed at the upper end as shown in figure. If the mass is taken 2.5 cm up along the surface of the inclined plane, the tension in the spring reduces to zero. If the mass is then released, the angular frequency of oscillation in radian per second is

(A) 0.707 (B) 7.07 (C) 1.414 (D) 14.14

42. A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is T. If the spring is divided in two equal parts and one part is used to continue the simple harmonic motion, the time period will

(A) remain T (B) become 2T (C) become T/2 (D) become T/ 2

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43. Four massless springs whose force constants are 2k, 2k, k and 2k respectively are attached to a mass M kept on a frictionless plane (as shown in figure). If the mass M is displaced in the horizontal direction, then the frequency of the system.

(A) 1 k

2 4M (B) 1 4k

2 M (C) 1 k

2 7M (D) 1 7k

2 M

44. The total mechanical energy of a particle of mass m executing SHM with the help of a spring is E =1

2 mω²A². If the particle is replaced by another particle of mass m/2 while the amplitude A remains same. New mechanical energy will be :

(A) 2 E (B) 2E (C) E/2 (D) E

45. A block of mass m is resting on a piston as shown in figure which is moving vertically with a SHM of period 1 s. The minimum amplitude of motion at which the block and piston separate is :

(A) 0.25 m (B) 0.52 m (C) 2.5 m (D) 0.15 mA

46. The potential energy of a particle of mass 'm' situated in a unidimensional potential field varies as U(x) = U0 [1 – cosax], where U0 and a are constants. The time period of small oscillations of the particle about the mean position

(A)

0

2 m

 aU (B)

0

2 am

 U (C)

2 0

2 m

 a U (D)

2

0

2 a m

 U

47. A spring mass system performs S.H.M. If the mass is doubled keeping amplitude same, then the total energy of S.H.M. will become:

(A) double (B) half (C) unchanged (D) 4 times

48. A plank with a small block on top of it is under going vertical SHM. Its period is 2 sec. The minimum amplitude at which the block will separate from plank is :

(A) 2

10

 (B)

2

10

 (C)

2

20

 (D)

10



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49. Find the ratio time periods of two identical springs if they are first joined in series and then in parallel and a mass m is suspended from them:

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

50. In an elevator, a spring clock of time period Ts (mass attached to a spring) and a pendulum clock of time period Tp are kept. If the elevator accelerates upwards

(A) TS well as TP increases (B) TS remain same, TP increases (C) TS remain same, TP decreases (D) TS well as TP decreases

51. Two bodies P and Q of equal mass are suspended from two separated massless spring of force constants k1 & k2 respectively. If the maximum velocity of them are equal during their motion, the ratio of amplitude of P to Q is :

(A) ¹

2

k

k (B) ²

1

k

k (C) ²

1

k

k (D) ¹

2

k k

52. The spring in figure A and B are identical but length in A is three times each of that in B. the ratio of period TA/TB is:

(A) 3 (B) 1

3 (C) 3 (D) 1

3

53. A 2 K g block moving with 10 m/s strikes a spring of constant π² N/m attached to 2 Kg block at rest kept on a smooth floor. The time for which rear moving block remain in contact with spring will be

(A) 2 sec (B) 1

2sec ^{(C) 1 sec} ^(D)

1sec 2

54. In the above question 53, the velocity of the rear 2 kg block after it separates from the spring will be:

(A) 0 m/s (B) 5 m/s (C) 10 m/s (D) 7.5 m/s

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55. A simple pendulum is oscillating in a lift. If the lift is going down with constant velocity, the time period of the simple pendulum is T1. If the lift is going down with some retardation its time period is T2, then

(A) T1 > T2

(B) T1 < T2

(C) T1 = T2

(D) depends upon the mass of the pendulum bob

56. Two pendulums begin to swing simultaneously. The first pendulum makes 9 full oscillations when the other makes 7. Find the ratio of length of the two pendulums.

(A) 49

81 (B) 7

9 (C) 50

81 (D) 1

2

57. Two pendulums at rest start swinging together. Their lengths are respectively 1.44 m and 1 m. They will again start swinging in same phase together after (of longer pendulum) :

(A) 1 vibration (B) 3 vibrations (C) 4 vibrations (D) 5 vibrations 58. A scientist measures the time period of a simple pendulum as T in a lift at rest. If the lift moves up

with acceleration as one fourth of the acceleration of gravity, the new time period is

(A) T

4 (B) 4T (C) 2

5T ^(D)

5T 2

59. A simple pendulum has some time period T. What will be the percentage change in its time period if its amplitude is decreased by 5%?

(A) 6 % (B) 3 % (C) 1.5 % (D) 0 %

60. A simple pendulum with length and bob of mass m executes SHM of small amplitude A. The maximum tension in the string will be

(A) mg(1 + A/ ) (B) mg (1 + A/ )² (C) mg[1 + (A/ )²] (D) 2 mg

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61. A solid ball of mass m is made to fall from a height H on a pan suspended through a spring of spring constant K as shown in figure. If the ball does not rebound and the pan is massless, then amplitude of oscillation is

(A) mg

K (B)

mg 2HK 1/2

K 1 mg

 + 

 

 

(C)

mg 2HK 1/2

K mg

 

+  

  (D)

mg 2HK 1/2

1 1

K mg

   

 + +  

   

 

62. Two plates of same mass are attached rigidly to the two ends of a spring as shown in figure. One of the plates rests on a horizontal surface and the other results a compression y of the spring when it is in equilibrium state. The further minimum compression required, so that after the force causing compression is removed the lower plate is lifted off the surface, will be :

(A) 0.5 y (B) 3y (C) 2y (D) y

63. The right block in figure moves at a speed V towards the left block placed in equilibrium. All the surfaces are smooth and all the collisions are elastic. Find the time period of periodic motion.

Neglect the width of the blocks.

(A) m 2L

2k v

 + (B) m L

2k v

 + (C) m L

2k v

 − (D) m L

k v

 +

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64. A small bob attached to a light inextensible thread of length l has a periodic time T when allowed to vibrate as a simple pendulum. The thread is now suspended from a fixed and O of a vertical rigid rod of length 3

4

l (as in figure). If now the pendulum performs periodic oscillations in this arrangement, the periodic time will be

(A) 3T

2 (B) T

2 (C) T (D) 2T

65. A system of two identical rods (L-shaped) of mass m and length l are resting on a peg P as shown in the figure. If the system is displaced in its plane by small angle θ, find the period of oscillations:

(A) 2

2 3gl

(B) 2 2

2 3gl

(C) 2

2 3gl

(D) 3

 3gl

66. A ring of diameter 2m oscillates as a compound pendulum about a horizontal axis passing through a point at its rim. It oscillates such that its centre move in a plane which is perpendicular to the plane of the ring. The equivalent length of the simple pendulum is

(A) 2m (B) 4m (C) 1.5 m (D) 3m

67. A man is swinging on a swing made of 2 ropes of equal length L and in direction perpendicular to the plane of paper. The time period of the small oscillations about the mean position is

(A) L

2 2g (B) 3 L

2 2g (C) L

2 2 3 g (D) L

 g

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68. A rod whose ends are A and B of length 25 cm is hanged in vertical plane. When hanged from point A and point B the time periods calculated are 3 sec and 4 sec respectively. Given the moment of inertia of rod about axis perpendicular to the rod is the ratio 9 : 4 points A and B. find the distance of the centre of mass from point A.

(A) 9 cm (B) 5 cm (C) 25 cm (D) 20 cm

69. A wire frame in the shape of an equilateral triangle is hinged at one vertex so that it can swing freely in a vertical plane, with the plane of the Δ always reaming vertical. The side of the frame is

1 m

3 . The time period in seconds of small oscillations of the frame will be (A)

2

 (B)  2 (C)

6

 (D)

5



70. A circular disc has a tine hole in it, at a distance z from its center. Its mass is M and radius R (R >z).

A horizontal shaft is passed through the hole and held fixed so that the disc can freely swing in the vertical plane. For small disturbance, the disc performs SHM whose time period is minimum for z =

(A) R/2 (B) R/3 (C) R / 2 (D) R / 3

71. A particle is subjected to two mutually perpendicular simple harmonic motion such that its x and y coordinates are given by

x = 2 sin ωt; y = 2 sin t 4

 + 

 

 

The path of the particle will be:

(A) an ellipse (B) a straight line (C) a parabola (D) a circle 72. The amplitude of the vibrating particle due to superposition of two SHMs,

y₁ sin t and y₂ sin t is : 3

 

=  +  = 

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

73. Two simple harmonic motions y1 = Asin ωt and y2 = A cosωt are superimposed on a particle of mass. The total mechanical energy of the particle is :

(A) 1 ² ²

m A

2  (B) m²A² (C) 1 ² ²

m A

4  (D) zero

74. Vertical displacement of a plank with a body of mass ‘m’ on it is varying according to law y=sin t  + 3 cos t . The minimum value of ω for which the mass just breaks off the plank and the moment it occurs first after t = 0 are given by: (y is positive vertically upwards)

(A) g 2 2, 6 g

 (B) g 2

,3 g 2

 (C) g 2

2 3, g

 (D) 2

2g, 3g



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Use referral code KAILASH10 to get 10% discount on subscriptions on Unacademy Plus 75. Equations y = 2A cos² ωt and y = A(sin t + 3 cos t) represent the motion of two particles.

(A) Only one of these is S.H.M (B) Ratio of maximum speeds is 2 : 1

(C) Ratio of maximum speeds is 1 : 1 (D) Ratio of maximum accelerations is 1 : 4

76. A block of mass ‘m’ is attached to a spring in natural length of spring constant ‘k’. The other end A of the spring is moved with a constant velocity v away from the block. Find the maximum extension in the spring.

(A)

1 mv2

4 k (B)

mv2

k (C)

1 mv2

2 k (D)

mv2

2 k

77. In the above question 76, find the amplitude of oscillations of the block in the reference frame of point A of the spring

(A)

1 mv2

4 k (B)

1 mv2

2 k (C)

mv2

k (D)

mv2

2 k

78. A mass m, which is attached to a spring with spring constant k, oscillates on a horizontal table, with

amplitude A. At an instant when the spring is stretched by 3 A

2 , a second mass m is dropped vertically onto the original mass and immediately sticks to it. What is the amplitude of the resulting motion ?

(A) 3

2 A ^(B)

7 A

8 (C) 13

16A (D) 2

3A

79. For a particle acceleration is defined as 5xiˆ a | x |

= − for x ≠ 0 and a = for x = 0. If the particle is 0 initially at rest (a, 0) what is period of motion of the particle.

(A) 4 2a / 5 sec. (B) 8 2a / 5 sec.

(C) 2 2a / 5 sec. (D) can’t be determined 80. A particle free to move along the x-axis has potential energy given by

U(x) = k[1 – exp (–x²)] for −   + where k is a positive constant of appropriate dimensions. x , Then

(A) at points away from the origin, the particle is in unstable equilibrium.

(B) for any finite non-zero value of x, there is a force directed away from the origin (C) If its total mechanical energy is k/2, it has its minimum kinetic energy at the origin.

(D) for small displacements from x=0, the motion is simple harmonic.

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81. In the figure shown, the spring are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end. Rotational SHM of the rod (Mass m, length L) will occur only if

(A) k > mg/3L (B) k > 2mg/3L (C) k > 2mg/5L (D) k > 0 82. What is the angular frequency of oscillations of the rod in the above problem if k = mg/L ?

(A) (3/2).[k/m]^1/2 (B) (3/4).[k/m]^1/2 (C) [2k/5m]^1/2 (D) None of these 83. The bob in a simple pendulum of length is released at t = 0 from the position of small angular

displacement θ0. Linear displacement of the bob at any time t from the mean position is given by

(A) ₀ g

cos t

 (B) g ₀

t cos  (C) g sin  ₀ (D) ₀ g

sin t



84. A rod of mass M and length L is hinged at its one end and carries a particle of mass m at its lower end. A spring of force constant k1 is installed at distance a from the hinge and another of force constant k2 at a distance b as shown in the figure. If the whole arrangement rests on a smooth horizontal table top, the frequency of vibration is

(A)

2 2

1 2

2

1 k a k b

2 L (m M)

3 +

 +

(B) 1 k² k¹

2 M m

+

 + (C)

2

2 1 2

k k a

1 b

2 4M m

3 +

 +

(D)

2 2

1 2

k k b

1 a

2 4m M

3 +

 +

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85. A particle moves along the X-axis according to the equation x = 10 sin³(πt). The amplitudes and frequencies of component SHMs are

(A) amplitude 30/4, 10/4 ; frequencies 3/2, 1/2 (B) amplitude 30/4, 10/4 ; frequencies 1/2, 3/2 (C) amplitude 10, 10 ; frequencies 1/2, 1/2 (D) amplitude 30/4, 10 ; frequencies 3/2, 2 86. The amplitude of a particle due to superposition of following S.H.Ms. Along the same line is

X1 = 2 sin 50 πt ; X2 = 10 sin (50 πt + 37º) X3 = – 4 sin 50 π t ; X4 = – 12 cos 50 πt

(A) 4 2 (B) 4 (C) 6 2 (D) none of these

87. When an oscillator completes 100 oscillations its amplitude reduced to 1

3 of initial value. What will be its amplitude, when it completes 200 oscillations :

(A) 1

8 (B) 2

3 (C) 1

6 (D) 1

9

88. The damping force on an oscillator is directly proportional to the velocity. The units of the constant of proportionality are :

(A) kgms^–1 (B) kgms^–2 (C) kgs^–1 (D) kgs

89. In forced oscillation of a particle, the amplitude is maximum for a frequency ω1 of the force, while the energy is maximum for a frequency ω2 of the force. What is the relation between ω1 and ω2 ? 90. For the damped oscillator shown in Fig , the mass of the block is 200 g, k = 80 N m^–1and the

damping constant b is 40 g s^–1 Calculate

(a) The period of oscillation,

(b) Time taken for its amplitude of vibrations to drop to half of its initial value (c) The time for the mechanical energy to drop to half initial value.

(18)

Part-II

Previous Years JEE Main Questions (2006-2020)

1. The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7mm, is 4.4 m/s. The period of oscillation is

(A) 0.1 s (B) 100 s (C) 0.01 s (D) 10 s

[AIEEE 2006]

2. Starting from the origin a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy by 75% of the total energy -

(A) 1

3s (B) 1

12s (C) 1

6s (D) 1

4s

[AIEEE 2006]

3. A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency . The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time-

(A) for an amplitude of g²/² (B) at the highest position of the platform (C) at the mean position of the platform (D) for an amplitude of g/^

[AIEEE 2006]

4. The displacement of an object attached to a spring and executing simple harmonic motion is given by x = 2 × 10^–2 cos t meters. The time at which the maximum speed first occurs is-

(A) 0.5 s (B) 0.75 s (C) 0.125 s (D) 0.25 s

[AIEEE 2007]

5. A point mass oscillates along the x-axis according to the law x = x0 cos (t – /4). If the acceleration of the particle is written as a = A cos (t + ), then

(A) ₀,

4

=  = −

A x (B) ₀, ²

4

=   = 

A x

(C) ₀ ²,

4

=   = −

A x (D) ₀, ²

4

=   = 

A x

[AIEEE 2007]

6. Two springs, of force constant k1 and k2, are connected to a mass m as shown. The frequency of oscillation of the mass if f. If both k1 and k2 are made four times their original values, the frequency of oscillation becomes

(A) f

2 (B) f

4 (C) 4f (D) 2f

[AIEEE 2007]

(19)

7. A particle of mass m executes simple harmonic motion with amplitude ‘a’ and frequency ‘v’. The average kinetic energy during its motion from the position of equilibrium to the end is

(A) ²m a² v² (B) 1 ² ² m a v

4 (C) 4²m a² v² (D) 4²m a² v²

[AIEEE 2007]

8. If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?

(A) aT/x (B) aT + 2v (C) at/V (D) a²T² + 4^v²

[AIEEE 2009]

9. A mass M, attached to a horizontal spring, executes S.H.M. with amplitude A1. When the mass M passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude A2. The ratio of ¹

2

A A

 

 

  is (A) M

M + m (B) M + m

M (C)

M 1/ 2

M + m

 

 

  (D)

M + m 1/ 2

M

 

 

 

[AIEEE 2011]

10. Two particles are executing simple harmonic motion of the same amplitude A and frequency  along the x-axis. Their mean position is separated by distance X0 (X0 > A). If the maximum separation between then is (X0 + A), the phase difference between their motion is:

(A) π

2 (B) π

3 (C) π

4 (D) π

6

[AIEEE 2011]

11. An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M.

The piston and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is V0 and its pressure is P0. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executed a simple harmonic with frequency.

(A) 1 V MP⁰ ₂ ⁰

2π Α γ (B)

2 0

0

1 Α γP

2π MV (C) ⁰

0

MV 1

2π ΑγP (D) ⁰

0

AγP 1 2π V M

[JEE Main 2013]

12. A particle moves with simple harmonic motion in a straight line. In first  s, after starting from rest it travels a distance a, and in next  s it travels 2a, in same direction, then

(A) time period of oscillations is 8 (B) amplitude of motion is 4a (C) time period of oscillations is 6 (D) amplitude of motion is 3a

[JEE Main 2014]

(20)

13. A pendulum of a uniform wire of cross sectional area A has time period T. When an additional mass M is added to its bob, the time period changes to TM. If the Young’s modulus of the material of the wire is Y then 1

Y is equal to : (g = gravitational acceleration) (A)

2

TM A

1 T Mg

 −  

   

 

  (B)

T 2 A

1 T_M Mg

   

 −   

   

 

(C)

2

TM A

T 1 Mg

  − 

  

 

  (D)

2

TM Mg

T 1 A

  − 

  

 

 

[JEE Main 2015]

14. For a simple pendulum, a graph is plotted between its kinetic energy (KE) and potential energy (PE) against its displacement d. Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

(A) (B)

(C) (D)

[JEE Main 2015]

15. A particle performs simple harmonic motion with amplitude A. Its speed is trebled at the instant that it is at a distance 2A

3 from equilibrium position. The new amplitude of the motion is:

(A) 7A

3 (B) A

3 41 (C) 3A (D) A 3

[JEE Main 2016]

16. In amplitude modulation, sinusoidal carrier frequency used is denoted by c and the signal frequency is denoted by m. The bandwidth (m) of the signal is such that m<<c. Which of the following frequencies is not contained in the modulated wave?

(A) c – m (B) m (C) c (D) m + c

[JEE Main 2017]

(21)

17. A particle is executing simple harmonic motion with a time period T. At time t = 0, it is at its position of equilibrium. The kinetic energy – time graph of the particle will look like

(A) (B)

(C) (D)

[JEE Main 2017]

18. A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of 10¹² s^–1. What is the force constant of the bonds connecting one atom with the other ? (Mole wt. of silver = 108 and Avogardo number = 6.02 × 10²³ gm mole^–1 )

(A) 6.4 N m^–1 (B) 7.1 N m^–1 (C) 2.2 N m^–1 (D) 5.5 N m^–1

[JEE Main 2018]

19. Two simple harmonic motion, as shown here, are at right angles. They are combined to form Lissajous figures.

x(t) = A sin (at + δ) y(t) = B sin (bt)

Identify the correct match below.

Parameters Curve Parameters Curve

(A) A = B, a = b; δ = π/2 Line (B) A ≠ B, a = b; δ = 0 Parabola (C) A ≠ B, a = 2b; δ = π/2 Circle (D) A ≠ B, a = b; δ = π/2 Ellipse

[JEE Main 2018]

20. A particle executes simple harmonic motion and is located at x = a, b and c at times t0, 2t0 and 3t0

respectively. The frequency of the oscillation is

(A) ¹

0

1 a b

2 t cos 2c

−  + 

 

   (B) ¹

0

1 2a 3c

2 t cos b

−  + 

 

  

(C) ¹

0

1 a 2b

2 t cos 3c

−  + 

 

   (D) ¹

0

1 a c

2 t cos 2b

−  + 

 

  

[JEE Main 2018]

(22)

21. A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits torsional oscillations; If two masses each of 'm' are attached at distance 'L/2' from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to:

(A) 0.17 (B) 0.37 (C) 0.57 (D) 0.77

[JEE Main-2019]

22. A particle is executing simple harmonic motion (SHM) of amplitude A, along the x-axis, about x = 0. When its potential Energy (PE) equals kinetic energy (KE), the position of the particle will be:

(A) A/2 (B) A/22 (C) A/2 (D) A

[JEE Main-2019]

23. A particle executes simple harmonic motion with an amplitude of 5 cm. When the particle is at 4 cm from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is:

(A) 7/3  (B) 3/8  (C) 4/3 (D) 8/3

[JEE Main-2019]

24. A particle undergoing simple harmonic motion has time dependent displacement given by x (t) = A sin t/90. The ratio of kinetic to potential energy o the particle at t = 210s will be

(A) 1

9 (B) 1 (C) 1

3 (D) 3

[JEE Main-2019]

25. A body of mass 1 kg falls freely from a height of 100 m, on a platform of mass 3 kg which is mounted on a spring having spring constant k = 1.25 × 10⁶ N/m. The body sticks to the platform and the spring’s maximum compression is found to be x. Given that g = 10 ms^-2, the value of x will be close to:

(A) 40 cm (B) 2 cm (C) 80 cm (D) 8 cm

[JEE Main-2019]

26. A simple pendulum of length 1 m is oscillating with an angular frequency 10 rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency of 1 rad/s and an amplitude of 10^–2 m. The relative change in the angular frequency of the pendulum is best given by:

(A) 10^–3 rad/s (B) 1 rad/s (C) 10^–1 rad/s (D) 10^–5 rad/s

[JEE Main-2019]

27. A pendulum is executing simple harmonic motion and its maximum kinetic energy is K1. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is K2 then:

(A) K2 = 2K1 (B) K2 = K1/2 (C) K2 = K1/4 (D) K2 = K1

[JEE Main-2019]

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28. The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of a simple pendulum on the Earth is 2s. The period of oscillation of the same pendulum on the planet would be:

(A) 3 / 2s (B) 2 / 3 s (C) 3/2 s (D) 23 s

[JEE Main-2019]

29. Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length l and mass m. the rod is pivoted at its centre ‘O’ and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

(A) 1 3k

2 m (B) 1 2k

2 m (C) 1 6k

2 m (D) 1 k

2 m

[JEE Main-2019]

30. A simple pendulum, made of a string of length l and a bob of mass m, is released from a small angle

0. It strikes a block of mass M, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle 1. Then M is given by:

(A) ⁰ ¹

0 1

m 2

 +  

 −  

  (B) ⁰ ¹

0 1

m −  

 +  

  (C) ⁰ ¹

0 1

m +  

 −  

  (D) ⁰ ¹

0 1

m 2

 −  

 +  

 

[JEE Main-2019]

31. A simple harmonic motion is represented by: y = 5 (sin3t + 3 cos3t) cm, The amplitude and time period of the motion are:

(A) 10cm, 2/3 s (B) 10 cm, 3/2 s (C) 5 cm, 3/2 s (D) 5 cm, 2/3 s

[JEE Main-2019]

32. A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to 1/1000 of the original amplitude is close to:

(A) 50 s (B) 100 s (C) 20 s (D) 10 s

[JEE Main-2019]

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33. A simple pendulum oscillating in air has period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is 1/16th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:

(A) 1

2T 10 (B) 1

2T 14 (C) 1

4T 14 (D) 1

4T 15

[JEE Main-2019]

34. The displacement of a damped harmonic oscillator is given by x(t) = e^–0.1tcos (10t + ) Here t it is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to:

(A) 27s (B) 13s (C) 7s (D) 4s

[JEE Main-2019]

35. A simple pendulum of length L is placed between the plates of a parallel plate capacitor having electric field E, as shown in figure. Its bob has mass m and change q, the time period of the pendulum is given by:

(A) 2

2

2 L

g qE m



 

+  

 

(B) L

2

g qE m

  + 

(C) L

2

g qE m

  − 

(D) 2 2

2

2 L

g q E m



−

[JEE Main-2019]

36. A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude 0. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance l (l << L), is close to.

(A) Mgl (B) Mgl

2

1 0

2

 + 

 

  (C) Mgl(1 + 02) (D) Mgl(1 – 02) [JEE Main-2019]

37. A particle executes simple harmonic motion (amplitude = A) between x = –A and x = +A. The time taken for it to go from A to A

2 is T1 and to go from A

2 to A is T2. Then -

(A) T1 < T2 (B) T1 > T2 (C) T1 = T2 (D) T1 = 2T2

[JEE Main-2020]

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EXERCISE-II Part-I

Section-A (Multiple Correct type Questions)

1. A particle moves on the X-axis according the equation x = x0 sin²t. The motion is simple harmonic

(A) with amplitude x0/2 (B) with amplitude 2x0 (C) with time period 2π

ω (D) with time period π

ω

2. The speed v of a particle moving along a straight line, when it is at a distance (x) from a fixed point of the line is given by

v² = 108–9x²

(assuming mean position to have zero phase constant) (all quantities are in cgs units):

(A) the motions is uniformly accelerated along the straight line

(B) the magnitude of the acceleration at a distance 3cm from the fixed point is 27 cm/s² (C) the motion is simple harmonic about the given fixed point.

(D) the maximum displacement from the fixed points 4 cm.

3. The potential energy of a particle of mass 0.1 kg moving along the x-axis, is given by U = 5x(x–4)J, where x is in meters. It can be concluded that

(A) the particle is acted upon by a constant force (B) the speed of the particle is maximum at x = 2m (C) the particle executed SHM

(D) the period of oscillation of the particle is (/5) sec

4. A particle free to move along the x-axis has potential energy given by U(x)=k[1 e− ⁻^x²] for –∞ ≤ x ≤ x + ∞, where k is a positive constant appropriate dimensions. Then select the incorrect options:

(A) at point away from the origin, the particle is in unstable equilibrium.

(B) for any finite non-zero value of x, there is a force directed away from the origin (C) if its total mechanical energy is k/2, it has its minimum kinetic energy at the origin.

(D) for small displacement from x = 0, the motion is simple harmonic

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5. A ball is hung vertically by a thread of length ‘’ from a point ‘P’ of an inclined wall that makes an angle ‘’ with the vertical. The thread with the ball is then deviated through a small angle ‘’

( > ) and set free. Assuming that wall to be perfectly elastic, the period of such pendulum is/are

(A) ¹ α

2 sin

g β

 −  

  

   (B) π ¹ α

2 sin

g 2 β

 −  

 +  

  

(C) ¹ α

2 cos

g β

 −  

  

   (D) ¹ α

2 cos

g β

 − − 

  

 

 

6. The position of a particle at time t moving in x-y plane is given by r = +(iˆ 2 j) A cos ω t .ˆ Then, the motion of the particle is:

(A) on a straight line (B) on an ellipse (C) periodic (D) SHM

7. Part of a simple harmonic motion is graphed in the figure, where y is the displacement from the mean position. The correct equation describing the S.H.M. is

(A) y = 4 cos (0.6t) (B) 10 π

y 2sin t

3 2

 

=  − 

(C) 10 π

y 4sin t

3 2

 

=  +  (D) 10 π

y 2cos t

3 2

 

=  + 

8. A particle is executing SHM with amplitude A, time period T, maximum acceleration a0 and maximum velocity v0. Its starts from mean position at t = 0 and at time t, it has the displacement A/2, acceleration a and velocity v then

(A) t = T/12 (B) a = a0/2 (C) v = v0/2 (D) t = T/8

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Use referral code KAILASH10 to get 10% discount on subscriptions on Unacademy Plus 9. The amplitude of a particle executing SHM about O is 10 cm. Then:

(A) When the K.E. is 0.64 of its max. K.E. its displacement is 6cm from O.

(B) When the displacement is 5 cm from O its K.E. is 0.75 of its max.P.E.

(C) Its total energy at any point is equal to its maximum K.E.

(D) Its velocity is half the maximum velocity when its displacement is half the maximum displacement.

10. A particle starts from a point P at a distance of A/2 from the mean position O & travels towards left as shown in the figure. If the time period of SHM, executed about O is T and amplitude A then the equation of motion of particle is:

(A) 2π π

x A sin t

T 6

 

=  +  (B) 2π 5π

x A sin t

T 6

 

=  + 

(C) 2π π

x A cos t

T 6

 

=  +  (D) 2π π

x A cos t

T 3

 

=  + 

11. A particle of mass m performs SHM along a straight line with frequency f and amplitude A.

(A) The average kinetic energy of the particle is zero.

(B) The average potential energy is m ²t²A².

(C) The frequency of oscillation of kinetic energy is 2f.

(D) Velocity function leads acceleration by /2.

12. Two particles are in SHM with same angular frequency and amplitude A and 2A respectively along same straight line with same mean position. They cross each other at position A/2 distance from mean position in opposite direction. The phase between them is:

(A) 5π ¹ 1

6 sin 4

−  

−    (B) π ¹ 1

6 sin 4

−  

−   

(C) 5π ¹ 1 6 cos 4

−  

   (D) π ¹ 1

6 cos 4

−  

−  

 

13. A particle is executing SHM of amplitude A, about the mean position x = 0. Which of the following cannot be a possible phase difference between the positions of the particle at x = +A/2 and

x= −A / 2.

(A) 75ô (B) 165ô (C) 135ô (D) 195ô

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14. A spring has natural length 40 cm and spring constant 500 N/m. A block of mass 1 kg is attached at one end of the spring and other end of the spring is attached to ceiling. The block released from the position, where the spring has length 45 cm.

(A) the block will perform SHM of amplitude 5 cm.

(B) the block will have maximum velocity 30 5 cm/sec.

(C) the block will have maximum acceleration 15 m/s² (D) the minimum potential energy of the spring will be zero.

15. Two spring with negligible masses and force constant of K1 = 200 Nm^–1 and K2 = 160 Nm^–1 are attached to the block of mass m = 10 kg as shown in the figure. Initially the block is at rest, at the equilibrium position in which both springs are neither stretched nor compressed. At time t = 0, a sharp impulse of 50 Ns is given to the block with a hammer.

(A) Period of oscillations for the mass m is π 3s.

(B) Maximum velocity of the mass m during its oscillation is 5 ms^–1 (C) Data are insufficient to determine maximum velocity.

(D) Amplitude of oscillation is 0.42m

16. The two blocks shown here rest on a frictionless surface. If they are pulled apart by a small distance and released at t = 0, the time when 1 kg block comes to rest can be

(A) 2π

3 sec. (B)  sec. (C) π

2sec. (D) π

9sec.

17. In case (i) a ring is supported at a point on its periphery by a small peg. The ring is in vertical plane.

In case (ii), the ring is hinged about a point on its periphery:

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Use referral code KAILASH10 to get 10% discount on subscriptions on Unacademy Plus (A) If the ring undergoes small oscillations in its plane, the period of oscillation in case (i) is more

than in case (ii)

(B) If the ring undergoes small oscillations in its own plane, the period of oscillation in both cases is equal.

(C) The angular velocity required to be imparted to the ring so that it completes vertical circle in case (i) is more than in case (ii)

(D) The angular velocity required to be imparted to the ring so that it completes vertical circle is same in both case.

18. A disc of mass 3m and a disc of mass m are connected by a massless spring of stiffness k. The heavier disc is placed on the ground with the spring vertical and lighter disc on top. From its equilibrium position, the upper disc is pushed down by a distance  and released. Then

(A) if  > 3mg/k, the lower disc will bounce up

(B) if  = 2mg/k, maximum normal reaction from ground on lower disc = 6 mg (C) if  = 2mg/k, maximum normal reaction from ground on lower disc = 4 mg (D) if  > 4mg/k, the lower disc will bounce up

19. At a smooth horizontal table between two identical fixed stretched springs is a small ball (see Figure). The length of the springs in the free state is . The ball is shifted slightly from the equilibrium position and it begins to perform vibrations – once along the axis OX, second – along the y-axis. The time period for these motions is Tx and Ty respectively. Neglect gravity.

(A) Motion along x axis is simple harmonic (B) Motion along y axis is simple harmonic

(C) _x m

T 2π

= 2k (D) _y m

T 2π

= 2k

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Section-B

(Comprehension type Questions) Paragraph for Qus 1 to 3

A particle of mass ‘m’ moves on a horizontal smooth line AB of length ‘a’ such that when particle is at any general point P on the line two forces act on it. A force mg(AP)

a towards A and another force 2mg(BP)

a towards B.

1. Find its time period when released from rest from mid-point of line AB.

(A) 3a

T 2π

= g (B) a

T 2π

= 2g (C) a

T 2π

= g (D) a

T 2π

= 3g 2. Find the minimum distance of the particle from B during the motion.

(A) a

6 (B) a

4 (C) a

3 (D) a

8

3. If the force acting towards A stops acting when the particle is nearest to B then find the velocity with which it crosses point B.

(A) 2ga

3 (B) 2ga

6 (C) 2ga

5 (D) ga

3

Paragraph for Qus 4 to 5

Spring of spring constant k is attached with a block of mass m1 as shown in figure. Another block of mass m2 is placed against m1 and both masses lie on smooth incline plane.

4. Find the compression in the spring when the system is in equilibrium.

(A) (m¹ m ) gsin θ² 2k

+ (B)(m¹ m ) g sin θ²

k +

(C) (m¹ m )g² k

+ (D) 2(m¹ m ) g sin θ²

k +

(31)

Use referral code KAILASH10 to get 10% discount on subscriptions on Unacademy Plus 5. From the equilibrium position the blocks are pushed a further distance 2 ₁ ₂

(m m )g sin θ

k + against

the spring and released. Find the common speed of blocks when they separate.

(A) 1 ₁ ₂

(m m g sin θ 3k

 

 + 

  (B) 2 ₁ ₂

(m m ) g sin θ k

 

 + 

 

(C) 3 ₁ ₂

(m m ) g sin θ k

 

 + 

  (D) 1 ₁ ₂

(m m ) g sin θ k

 

 + 

 

Section-C

[MATRIX TYPE]

1. Two blocks A and B of mass m and 2m connected by a light spring of spring constant k lie at rest on a fixed smooth horizontal surface. Initially the spring is unstressed. Now at time t = 0 both the blocks are imparted horizontal velocities towards each other of magnitude 2u and u as shown in figure. In the subsequent motion, the only horizontal force acting on blocks is due to spring. Match the conditions in column-I with the instants of time they occurs as given in column-II.

Column-I Column-II

(A) The length of spring is least at time (P) π 2m

t = 2 3k (B) The lengths of spring is maximum at time (Q) 2m

t π

= 3k (C) The acceleration of both blocks is zero simultaneously at

time (R) 3m

t π

= 2k

(S) π 3m

t= 2 2k

(32)

PART-II

(Subjective type Questions)

1. The figure shows the displacement – time graph of a particle executing SHM. If the time period of oscillation is 2s, then the equation of motion is given by x = ________________.

2. Two particles A and B execute SHM along the same line with the same amplitude a, same frequency and same equilibrium position O. if the phase difference between them is  = 2 sin^–1 (0.9), then find the maximum distance between the two.

3. A block of mass 0.9 kg attached to a spring of force constant k is lying on a frictionless floor. The spring is compressed to 2 cm and the block is at a distance 1 / 2 cm from the wall as shown in the figure. When the block is released, it makes elastic collision with the wall and its period of motion is 0.2 sec. Find the approximate value of k.

4. Ram has been caught red handed doing notorious activities and is punished to run back and forth in a 20 m long corridor from room number 109 to the other end. Ram starts running from room number 109, touches the other end, returns & touches the door of room number 109 & so on for 65 minutes continuously after which he drops down exhausted. His speed is seen to be v= 100−x m/s² where x is the distance (in m) from the centre of the corridor. How many times did he touch the other end during this time interval. (Take :  = 3.14)

5. A force F = –10 x + 2 acts on a particle of mass 0.1 kg, where ‘x’ is in m and F in Newton, If it is released from rest at x = –2 m, find:

(a) amplitude; (b) time period; (c) equation of motion.

6. A block is kept on a horizontal table. The table is undergoing simple harmonic motion of frequency 3 Hz in a horizontal plane. The coefficient of static friction between block and the table surface is 0.72. Find the maximum amplitude of the table at which the block does not slip on the surface.