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** Exercise**

1. A particle executes simple harmonic motion with an amplitude of 10 cm and time period 6 s. At

t = 0 it is at position x = 5 cm going towards positive x-direction. Write the equation for the displacement x at time t. Find the magnitude of the acceleration of the particle at t = 4 s.

2. The position, velocity and acceleration of a particle executing simple hamonic motion are found to have magnitudes 2 cm, 1 m/s and 10 m/s2 at a certain instant Find the amplitude and the time period of the motion.

2. 4.9 cm, 0.28 s

3. The particle executes simple harmonic motion with an amplitude of 10 cm. At what distance from the mean position are the kinetic and potential energies equal ?

4. The maximum speed and acceleration of a particle executing simple harmonic motion are

10 cm/s and 50 cm/s2. Find the position (s) of the particle when the speed is 8 cm/s.

4. ± 1.2 cm from the mean position

5. A particle having mass 10 g oscillates according to the equation x = (2.0 cm) sin [(100 s–1) t+ p/6]. Find (a) the amplitude, the time period and the spring constant (b) the position, the velocity and the acceleration at

t = 0.

5. (a) 2.0 cm, 0.063 s, 100 N/m (b) 1.0 cm, 1.73 m/s, 100 m/s2

6. The equation of motion of a particle started at t = 0 is given by x = 5 sin (20 t + p/3) where x is in centimetre and t in second. When does the particle

(a) first come to rest

(b) first have zero acceleration

(c) first have maximum speed ?

7. Consider a particle moving in simple harmonic motion according to the equation

where x is in centimetre and t in second. The motion is started at t = 0. (a) When does the particle come to rest for the first time ? (b) When does the acceleration have its maximum magnitude for the first time ? (c) When does the particle come to rest for the second time ?

8. Consider a simple harmonic motion of time period T. Calculate the time taken for the displacement to change value from half the amplitude to the amplitude.

8. T/6

9. The pendulum of a clock is replaced by a spring - mass system with the spring having spring constant 0.1 N/m. What mass should be attached to the spring ?

10. A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum i.e., a pendulum having frequency same as that of the block.

11. A block of mass 0.5 kg hanging from a vertical spring executes simple harmonic motion of amplitude 0.1 m and time period 0.314 s. Find the maximum force exerted by the spring on the block.

12. A body of mass 2 kg suspended through a vertical spring executes simple harmonic motion of period 4s. If the oscillations are stopped and the body hangs in equilibrium, find the potential energy stored in the spring.

12. 40 J

13. A spring stores 5 J of energy when stretched by 25 cm. It is kept vertical with the lower end fixed. A block fastened to its other end is made to undergo small oscillations. If the block makes 5 oscillations each second, what is the mass of the block ?

13. 0.16 kg

14. A small block of mass m is kept on a bigger block of mass M which is attached to a vertical spring of spring constant k as shown in the figure. The system oscillates vertically. (a) Find the resultant force on the smaller block when it is displaced through a distance x above its equilibrium position. (b) Find the normal force on the smaller block at this position. When is this force smallest in magnitude? (c) What can be the maximum amplitude with which the two blocks may oscillate together?

15. The block of mass m1 shown in figure is fastened to the spring and the block of mass m2 is placed against it. (a) Find the compression of the spring in the equilibrium position. (b) The blocks are pushed a further distance (2/k) (m1 + m2) g sin q against the spring and released. Find the position where the two blocks separate. (c) What is the common speed of blocks at the time of separation?

16. In figure k = 100 N/m, M = 1 kg and F = 10 N. (a) Find the compression of the spring in the equilibrium position. (b) A sharp blow by some external agent imparts a speed of 2 m/s to the block towards left. Find the sum of the potential energy of the spring and the kinetic energy of the block at this instant. (c) Find the time period of the resulting simple harmonic motion. (d) Find the amplitude. (e) Write the potential energy of the spring when the block is at the left extreme. (f) Write the potential energy of the spring when the block is at the right extreme.

The answers of (b), (e) and (f) are different. Explain why this does not violate the principle of conservation of energy.

16. (a) 10 cm (b) 2.5 J (c) p/5 s (b) 20 cm (e) 4.5 J (f) 0.5 J

17. Find the time period of the oscillation of mass m in figures a, b, c. What is the equivalent spring constant of the pair of springs in each case ?

(a) (b) (c)

18. The spring shown in figure is unstretched when a man starts pulling on the cord. The mass of the block is M. If the man exerts a constant force F, find (a) the amplitude and the time period of the motion of the block (b) the energy stored in the spring when the block passes through the equilibrium position and (c) the kinetic energy of the block at this position.

19. A particle of mass m is attatched to three springs A, B and C of equal force constant k as shown in figure. If the particle is pushed slightly against the spring C and released, find the time period of oscillations.

20. Repeat the previous exercise if the angle between each pair of springs is 120° initially.

21. The springs shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block.

,

22. Find the elastic potential energy stored in each spring shown in figure, when the block is in equilibrium. Also find the time period of vertical oscillation of the block.

23. The string, the spring and the pulley shown in figure are light. Find the time period of the mass m.

24. Solve the previous problem if the pulley has a moment of inertia I about its axis and the string does not slip over it.

25. Consider the situation shown in figure. Show that if the blocks are displaced slightly in opposite directions and released, they will execute simple harmonic motion. Calculate the time period.

26. A rectangular plate of sides a and b is suspended from a ceiling by two parallel strings of length L each (figure). The separation between the strings is d. The plate is displaced slightly in its plane keeping the strings tight. Show that it will excute simple harmonic motion. Find the time period.

27. A 1 kg block is executing simple harmonic motion of amplitude 0.1 m on a smooth horizontal surface under the restoring force of a spring of spring constant 100 N/m. A block of mass 3 kg is gently placed on it at the instant it passes through the mean position. Assuming that the two blocks move together, find the frequency and the amplitude of the motion.

28. The left block in figure moves at a speed v towards the right block placed in equilibrium. All collisions to take place are elastic and the surface are frictionless. Show that the motions of the two blocks are periodic. Find the time period of these periodic motions. Neglect the widths of the blocks.

29. Find the time period of the motion of the particle shown in figure. Neglect the small effect of the bend near the bottom.

30. All the surfaces shown in figure are frictionless. The mass of the car is M, that of the block is m and the spring has spring constant k. Initially, the car and the block are at rest and the spring is stretched through a length x0 when the system is released. (a) Find the amplitudes of the simple harmonic motion of the block and of the car as seen from the road. (b) Find the time period (s) of the two simple harmonic motions.

31. A uniform plate of mass M stays horizontally and symmetricaly on two wheels rotating in opposite directions (figure). The separation between the wheels is L. The friction coefficient between each wheel and the plate is µ. Find the time period of oscillation of the plate if it is slightly displaced along its length and released.

32. A pendulum having time period equal to two seconds is called a seconds pendulum. Those used in pendulum clocks are of this type. Find the length of a seconds pendulum at a place where

32. 1 m

33. The angle made by the string of a simple pendulum with the vertical depends on time as

33. 1m

34. The pendulum of a certain clock has time period 2.04 s. How fast or slow does the clock run during 24 hours ?

34. 28.8 minutes slow

35. A pendulum clock giving correct time at a place where g = 9.8 m/s2 is taken to another place where it loses 24 seconds during 24 hours. Find the value of g at this new place.

35. 9.795 m/s2

36. A simple pendulum is constructed by hanging a heavy ball by a 5.0 m long string. It undergoes small oscillations. (a) How many oscillations does it make per second ? (b) What will be the frequency if the system is taken on the moon where acceleration due to gravitation of the moon is 1.67 m/s2.

37. The maximum tension in the string of an oscillating pendulum is double of the minimum tension. Find the angular amplitude.

38. A small block oscillates back and forth on a smooth concave surface of radius R (figure). Find the time period of small oscillation.

39. A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period.

40. A simple pendulum of length 40 cm is taken inside a deep mine. Assume for the time being that the mine is 1600 km deep. Calculate the time period of the pendulum there. Radius of the earth = 6400 km.

40. 1.47 s

41. Assume that a tunnel is dug across the earth (radius = R) passing through its centre. Find the time a particle takes to cover the length of the tunnel if (a) it is projected into the tunnel with a speed of (b) it is released from a height R above the tunnel (c) it is thrown vertically upward along the length of tunnel with a speed of

42. Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance R/2 from the earth’s centre where R is the radius of the earth. The wall of the tunnel is frictionless. (a) Find the gravitational force exerted by the earth on a particle of mass m placed in the tunnel at a distance x from the centre of the tunnel. (b) Find the component of this force along the tunnel and perpendicular to the tunnel. (c) Find the normal force exerted by the wall on the particle. (d) find the resultant force on the particle. (e) Show that the motion of the particle in the tunnel is simple harmonic and find the time period.

43. A simple pendulum of length l is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator (a) is going up with an acceleration a0 (b) is going down with an acceleration a0 and (c) is moving with a uniform velocity.

44. A simple pendulum of length 1 feet suspended from the ceiling of an elevator takes p/3 seconds to complete one oscillation. Find the acceleration of the elevator.

44. 4 feet/s2 upwards

45. A simple pendulum fixed in a car has a time period of 4 seconds when the car is moving uniformly on a horizontal road. When the accelerator is pressed, the time period changes to 3.99 seconds. Making an approximate analysis, find the acceleration of the car.

45. g/10

46. A simple pendulum of length l is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation.

47. The ear-ring of a lady shown in figure. has a 3 cm long light suspension wire. (a) Find the time period of small oscillations if the lady is standing on the ground. (b) The lady now sits in a merry-go-round moving at 4 m/s in a circle of radius 2 m. Find the time period of small oscillations of the ear-ring.

47. (a) 0.34s (b) .30 s

48. Find the time period of small oscillations of the following systems. (a) A metre stick suspended through the 20 cm mark. (b) A ring of mass m and radius r suspended through a point on its perphery. (c) A uniform square plate of edge a suspended through a corner. (d) A uniform disc of mass m and radius r suspended through a point r/2 away from the centre.

49. A uniform rod of length l is suspended by an end and is made to undergo small oscillations. Find the length of the simple pendulum having the time period equal to that of the rod.

49. 2l/3

50. A uniform disc of radius r is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for it to have minimum time period ?

51. A hollow sphere of radius 2 cm is attached to an 18 cm long thread to make a pendulum. Find the time period of oscillation of this pendulum. How does it differ from the time period calculated using the formula for a simple pendulum ?

51. 0.89 s, it is about 0.3% larger than the calculated value

52. A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude 2° and time period 2s. Find (a) the radius of the circular wire, (b) the speed of the particle farthest away from the point of suspnsion as it goes through its mean position, (c) the acceleration of this particle as it goes through its mean position and (d) the acceleration of this particle when it is at an exetreme position. Take g = p2 m/s2.

52. (a) 50 cm (b) 11 cm/s

(c) 1.2 cm/s2 towards the point of suspension (d) 34 cm/s2 towards the mean position

53. a uniform disc of mass m and radius r is suspended through a wire attached to its centre. If the time period of the torsional oscillations be T, what is the torsional constant of the wire.

54. Two small balls, each of mass m are connected by a light rigid rod of length L. The system is suspended from its centre by a thin wire of torsional constant k. The rod is rotated about the wire through an angle q0 and released. Find the tension in the rod as the system passes through the mean position.

55. A particle is subjected to two simple harmonic motions of same time period in the same direction. The amplitude of the first motion is 3.0 cm and that of the second is 4.0 cm. Find the resultant amplitude if the phase difference between the motions is (a) 0°, (b) 60°, (c) 90°.

55. (a) 7.0 cm (b) 6.1 cm (c) 5.0 cm

56. Three simple harmonic motions of equal amplitudes A and equal time periods in the same direction combine. The phase of the second motion is 60° ahead of the first and the phase of the third motion is 60° ahead of the second. Find the amplitude of the resultant motion.

56. 2 A

57. A particle is subjected to two simple harmonic motions given by

where x is in centimeter and t in second. Find the displacement of the particle at (a) t = 0.0125, (b) t = 0.025.

57. (a) – 2.41 cm (b) 0.27 cm

58. A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of 45° with the X-axis. The two motions are given by