The document discusses oscillatory and vibratory motion, describing it as motion where an object moves back and forth about a mean position in a periodic fashion. It defines simple harmonic motion as oscillatory motion produced by a restoring force proportional to displacement. Key concepts discussed include restoring force, amplitude, frequency, period, displacement, velocity, acceleration, phase, energy conservation, and free vs forced oscillations. It also covers waves, types of waves, progressive waves, superposition, interference, beats, and stationary waves.

1. Oscillations
MUHAMMAD IMTIAZ
CS, YEAR 2
NAMAL COLLEGE, MIANWALI

2. Oscillatory / Vibratory motion
 The motion of an object in which object moves to and fro about its mean position,
is called oscillatory motion.
 If this motion repeats itself after equal intervals of time, Called Periodic motion.
 Examples:
 Simple pendulum motion
 Motion of steel ruler
 Steel ball rolling in a curved plate
 Mobile in vibration

3. Simple harmonic motion
 Oscillatory motion due to restoring force, Called simple harmonic motion.
 This restoring force is equal and opposite to the applied force.
 𝑭 𝒓 = -kx
 -ve sign shows that this force is directed towards mean position.
 Hook’s law:
 The restoring force is directly proportional to the displacement and always directed
towards the mean position.
 Acc. Produced due to restoring force can be determined by 2nd law of motion
 a 𝜶 – x
 Acc. Is directly proportional to displacement and directed towards mean position.

4. Terminologies related SHM
 Restoring force:
 The force which is equal and opposite to applied force, Restoring force.
 Instantaneous displacement:
 The displacement at any instant from the mean position, Instantaneous displacement.
 Amplitude:
 The maximum displacement of the body from the mean position, Amplitude.
 It is denoted by 𝑋0.

5. Continued…
 Vibration / cycle:
 A complete round trip of body performing SHM, Vibration / cycle.
 Time period:
 Time to complete one vibration, Time period.
 T = 2 / ω
 Frequency:
 Number of vibrations in unit time, Frequency.
 f = 1 / T
 Unit is vibrations per seconds, cycles per seconds or hertz(HZ, SI unit)

6. Displacement
 The distance of a body from its mean position is called displacement.
 Formula:
 x = 𝑥0sin𝛳 here 𝛳 = ωt
 x = 𝑥0sinωt
 Here 𝜭 gives the state of system in its vibrational motion. i.e. if
 𝛳 = 0, ‘p’ is at mean position
 𝛳 = 90 or / 2, ‘p’ competes one fourth of its cycle
 𝛳 = 180 or , ‘p’ competes half of its cycle
 𝛳 = 270 or 3 / 2, ‘p’ competes third fourth of its cycle
 𝛳 = 360 or 2 , ‘p’ competes one cycle

7. Velocity
 The velocity of an object executing SHM is directed along the motion of body.
 v = 𝑥0ω cos ωt 𝛳 = ωt
 The direction of velocity depends upon the value of ′𝜭’
 Here 𝛳 = 𝑥0
2 − 𝑥2 / 𝑥0
 v = ω 𝑥0
2 − 𝑥2
 From the above equation if x = 0,
 V = maximum if x = 0
 V = minimum if 𝑥0 = 0

8. Acceleration
 Acceleration of body executing SHM is directly proportional to displacement and
is directed towards the mean position.
 Formula:
 a = - ω 𝟐
x

9. Phase / phase angle
 The angle 𝛳 = ωt is the angle which specifies the displacement as well as the
direction of motion of the point executing SHM.
 i.e. if
 𝛳 = 0, ‘p’ is at mean position
 𝛳 = 90 or / 2, ‘p’ competes one fourth of its cycle
 𝛳 = 180 or , ‘p’ competes half of its cycle
 𝛳 = 270 or 3 / 2, ‘p’ competes third fourth of its cycle
 𝛳 = 360 or 2 , ‘p’ competes one cycle

10. Displacement, Velocity and acceleration
 As we know a = - kx / m and a = - ω 𝟐
x
 So ω = 𝑘/𝑚 thus
 Displacement:
 x = 𝑥0sin 𝑘/𝑚 t
 Velocity:
 v = 𝑘/𝑚 𝑥0
2 − 𝑥2
 Acceleration:
 a = -
𝑘x
𝑚
 Time period:
 T = 2 𝑚/𝑘

11. Energy conservation in SHM
 Statement:
 In SHM energy is energy of the vibrating system remains constant. i.e. energy is
converted into P.E and K.E but the total energy remains conserve.
 Formula:
 Total energy = ½ k𝒙 𝟎
𝟐

12. Free and forced oscillations
 Free oscillations:
 A body is said to be oscillate freely when there is no continuous force applied on a body
to oscillate. Only initial push is required.
 The frequency of the body is called its natural frequency. E.g. simple pendulum
 Forced oscillations:
 When there is continuous force required to oscillate the body, its oscillations are called
forced oscillations.
 The physical system under going forced vibrations is known as driven harmonic
oscillator.

13. GKK / HKK
 Resonance
 Damped oscillations
 Sharpness of resonance

14. Waves
 It is the disturbance in the medium through which energy is transported without
transporting the matter.
 Examples:
 sound waves,
 waves in spring,
 waves in rope,
 water waves,
 air waves etc.

15. Types of waves
 Mechanical waves:
 Waves which propagate by the oscillations of material particle, called mechanical
waves.
 E.g.
 Sound waves, air waves, spring waves, water waves
 Electromagnetic waves:
 Waves which propagate by the oscillations of electric and magnetic fields, called
Electromagnetic waves.

16. Progressive waves
 Waves which propagates energy by moving away from the source of disturbance,
travelling or progressive waves.
 These are of two types.
 Transverse waves:
 Waves in which particles of the medium have displacement perpendicular to the
direction propagation of waves.
 Longitudinal or compressional waves:
 Waves in which particles of the medium have displacement parellel to the direction
propagation of waves.

17. Superposition of waves
 The composition of two (or more) waves travelling through the same medium at
the same time, principle of superposition.
 Principle of superposition leads us to three different cases:
 Interference
 Beats
 Stationary waves

18. Interference
 Superposition of two waves having same frequency and travelling in the same
direction, interference.
 Constructive interference:
 Whenever path difference is an integral multiple of wavelength, interference of two
waves is called constructive interference.
 Δs = nλ
 Destructive interference:
 Whenever path difference is an odd integral multiple of half of wavelength, interference
of two waves is called destructive interference.
 Δs = (2n+1)λ/2

19. GKK / HKK
 Beats
 Reflection of waves
 Stationary waves
 Stationary waves in stretched string
 Stationary waves in air column
 Doppler effect

20. Thank you