Simple harmonic motion

• Vibration and oscillation are interchangeable terms for a body moving back and forth or to and fro about a point
• Simple Harmonic Motion (SHM) is a special kind of vibratory or oscillatory motion discussed in this chapter
• Examples of systems executing SHM include:

  • Motion of mass attached to a spring on a horizontal frictionless surface
  • Motion of a ball placed in a bowl
  • Motion of a bob attached to a string
• In a horizontal mass-spring system, if the spring is stretched or compressed through a small displacement x from its mean position, it exerts a force F on the mass
• According to Hooke’s law, the force exerted by the spring is directly proportional to the change in length x of the spring
• The spring constant k is a measure of the stiffness of the spring, with stiff springs having a large value of k and soft springs having a small value of k
• The acceleration of a mass attached to a spring is directly proportional to its displacement from the mean position, making the horizontal motion of a mass-spring system an example of simple harmonic motion
• A spider detects its prey due to vibrations produced in the web
• The negative sign in Eq. 10.1 means that the force exerted by the spring is always directed opposite to the displacement of the mass
• The spring force always acts towards the mean position, sometimes called a restoring force
• A restoring force always pushes or pulls the object performing oscillatory motion towards the mean position
• The magnitude of the restoring force decreases with the distance from the mean position and becomes zero at O
• The restoring force exerted by the spring on the mass pulls it towards the position O
• The mass gains speed as it moves towards the mean position and its speed becomes maximum at O
• The mass does not stop at the mean position O but continues its motion and reaches the extreme position B
• The restoring force acting on the mass towards the mean position steadily increases in strength as the mass moves from O to B
• The mass finally comes briefly to rest at the extreme position B
• The speed of the mass decreases as it moves towards the extreme position B
• The mass continues to oscillate back and forth about the mean position O, known as Simple Harmonic Motion (SHM)
• Simple Harmonic Motion (SHM) occurs when the net force is directly proportional to the displacement from the mean position and is always directed towards the mean position
• An object oscillates about a fixed position (mean position) such that its acceleration is directly proportional to its displacement from the mean position and is always directed towards the mean position, its motion is called SHM
• Important features of SHM:

  • A body executing SHM always vibrates about a fixed position
  • Acceleration is always directed towards the mean position
  • Magnitude of acceleration is directly proportional to its displacement from the mean position
  • Velocity is maximum at the mean position and zero at the extreme positions
• Vibration: One complete round trip of a vibrating body about its mean position is called one vibration
• Time Period (T): The time taken by a vibrating body to complete one vibration is called time period
• Frequency (f): The number of vibrations or cycles of a vibrating body in one second is called its frequency. It is reciprocal of time period i.e., f = 1/T
• Amplitude (A): The maximum displacement of a vibrating body on either side from its mean position is called its amplitude
• Damped oscillations occur when the oscillations of a system are reduced due to the presence of friction or resistance
• Shock absorbers in automobiles are an example of damped motion
• Waves are disturbances in the medium causing particles to undergo vibratory motion about their mean position
• Two categories of waves are mechanical waves and electromagnetic waves
• Mechanical waves require a medium for propagation, examples include water waves, sound waves, and waves on strings and springs
• Electromagnetic waves do not require a medium for propagation, examples include radiowaves, television waves, X-rays, heat, and light waves
• Longitudinal waves can be produced on a spring (slinky) where disturbances move along the length of the slinky with regions called compressions
• Longitudinal waves consist of compressions and rarefactions

  • Compressions: loops of the spring are close together
  • Rarefactions: loops of the spring are spaced apart
  • Particles of the medium are closer together in compressions and spaced apart in rarefactions
  • Wavelength: distance between two consecutive compressions
  • Particles of the medium move back and forth along the direction of propagation of wave
• Transverse waves can be produced with a slinky

  • Vibratory motion of particles of the medium is perpendicular to the direction of propagation of waves
  • Crests are the highest points, troughs are the lowest points from the mean position
  • Wavelength: distance between two consecutive crests or troughs
  • Crests and troughs move perpendicular to the direction of the wave
• Energy can be transferred through waves

  • Vibrating force disturbs particles of the medium and sets them in motion
  • Energy transferred from one place to another in the form of a wave
  • Amount of energy carried by the wave depends on the amplitude of the wave
• Velocity of a wave = distance/time

  • Time period T is reciprocal of frequency f
  • Velocity of wave = frequency x wavelength
• Ripple tank is used to produce water waves and study their characteristics

  • Waves can be produced on the surface of water by a vibrator
• Simple harmonic motion (SHM) is to and fro oscillatory motion

  • Acceleration of the body is directly proportional to the displacement from the mean position
  • Examples of SHM: mass attached to a spring, simple pendulum, ball inside a bowl
  • Time period of a simple pendulum depends on length and is independent of mass and amplitude
  • Frequency is the number of cycles completed in one second, reciprocal of time period
  • Amplitude is the maximum displacement from mean position
• Wave transfers energy without transferring matter