1. Mechanical Oscillations
© D Hoult 2010

2. If a body is to oscillate it must be acted on by a force which is at all times directed

3. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position

4. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position
The force is called the restoring force

5. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position
The force is called the restoring force
The simplest type of oscillation is called

6. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position
The force is called the restoring force
The simplest type of oscillation is called simple harmonic motion (s.h.m.)

7. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position
The force is called the restoring force
The simplest type of oscillation is called simple harmonic motion (s.h.m.)
If a body moves such that its

8. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position
The force is called the restoring force
The simplest type of oscillation is called simple harmonic motion (s.h.m.)
If a body moves such that its acceleration

9. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position
The force is called the restoring force
The simplest type of oscillation is called simple harmonic motion (s.h.m.)
If a body moves such that its acceleration is directly proportional to its displacement from a fixed point and is always directed

10. If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position
The force is called the restoring force
The simplest type of oscillation is called simple harmonic motion (s.h.m.)
If a body moves such that its acceleration is directly proportional to its displacement from a fixed point and is always directed towards that point, then the motion is s.h.m.

11. a a displacement

12. a a displacement
a = - (a constant) x

13. a a displacement
a = - (a constant) x
therefore the magnitude of the constant is given by

14. a a displacement
a = - (a constant) x
therefore the magnitude of the constant is given by a x

15. a a displacement
a = - (a constant) x
therefore the magnitude of the constant is given by a = x

16. a a displacement
a = - (a constant) x
therefore the magnitude of the constant is given by a F = x mx

17. a a displacement
a = - (a constant) x
therefore the magnitude of the constant is given by a F = x mx
i) the mass of the oscillating body

18. a a displacement
a = - (a constant) x
therefore the magnitude of the constant is given by a F = x mx
i) the mass of the oscillating body
ii) the force per unit displacement acting on the oscillating body

19. The amplitude is the maximum displacement from the equilibrium position

20. The amplitude is the maximum displacement from the equilibrium position
The frequency is the number of oscillations per unit time

22. Relation between Acceleration and Displacement

24. We will assume that at t = 0, the body has displacement, x = 0 (that is, the body was at its equilibrium position at t = 0)

29. The point p’ has acceleration, ac =

30. The point p’ has acceleration, ac = r w2

31. The acceleration of the oscillating body, p is equal to the component of the acceleration of p’ acting in a direction parallel to the line along which the body is oscillating

32. Acceleration of p is a =

33. Acceleration of p is a = ac sin q =

34. Acceleration of p is a = ac sin q = rw2 sin q

35. but, r sin q =

36. but, r sin q = x

37. So, the magnitude of the acceleration is a =

38. So, the magnitude of the acceleration is a = w2 x

39. acceleration is a =

40. acceleration is a = w2 x = 0

42. acceleration is a =

43. acceleration is a = w2 r

44. maximum acceleration at maximum displacement

46. Note that, when the displacement, x, is positive, the acceleration is negative and vice versa

47. Note that, when the displacement, x, is positive, the acceleration is negative and vice versa
The “s.h.m. equation” is usually written as

48. Note that, when the displacement, x, is positive, the acceleration is negative and vice versa
The “s.h.m. equation” is usually written as
a = - w2 x

49. Note that, when the displacement, x, is positive, the acceleration is negative and vice versa
The “s.h.m. equation” is usually written as
a = - w2 x
It is clear that the magnitude of the constant for a given oscillation can be found by simply measuring the

50. Note that, when the displacement, x, is positive, the acceleration is negative and vice versa
The “s.h.m. equation” is usually written as
a = - w2 x
It is clear that the magnitude of the constant for a given oscillation can be found by simply measuring the time period of the oscillation (and then using w = 2p/T)

51. Relation between Displacement and Time

53. x = r sin q

54. x = r sin q
q = w t

55. x = r sin q
q = w t
x = r sin wt

56. Relation between Velocity and Time

60. At any instant, the magnitude of v is equal to the magnitude of the

61. At any instant, the magnitude of v is equal to the magnitude of the component of v’ in a direction parallel to the line A - B

62. The angle between v’ and the line A - B is

63. The angle between v’ and the line A - B is

64. The angle between v’ and the line A - B is q

65. v =

66. v = v’ cos q

67. v = v’ cos q
v =

68. v = v’ cos q
v = r w cos wt

69. Relation between Acceleration and Time

70. From the definition of angular velocity q =

71. From the definition of angular velocity q = wt

72. From the definition of angular velocity q = wt so, acceleration of p is a = rw2 sin wt

73. and remembering that, when x is positive, a is negative and v.v.

74. the relation between acceleration and time is given by a = -rw2 sin wt