Simple Harmonic Motion

• Describing Oscillations


  1. An oscillation is defined as the repetitive variation with time t of the displacement x of an object about the equilibrium position (x = 0)

  2. Equilibrium position (x = 0) is the position when there is no resultant force acting on an object

  3. Displacement (x) of a wave is the distance of a point on the wave from its equilibrium position

  4. Period (T) or time period is the time interval for one complete repetition

  5. Amplitude (x) is the maximum value of the displacement on either side of the equilibrium position

  6. Wavelength (λ) is the length of one complete oscillation measured from the same point on two consecutive waves

  7. Frequency (f) is the number of oscillations per second
• If the oscillations have a constant period, they are said to be isochronous
• Amplitude


  The maximum value of the displacement on either side of the equilibrium position
• Phase Difference

  1. Phase is a useful way to think about waves

  2. One complete oscillation = 1 wavelength = 360° = 2π radians

  3. The phase difference between two waves is a measure of how much a point or a wave is in front or behind another

  4. When the crests of each wave, or the troughs of each wave are aligned, the waves are in phase

  5. When the crest of one wave aligns with the trough of another, they are in antiphase

  6. Phase difference is measured in fractions of a wavelength, degrees or radians
• Frequency

  The number of oscillations per second
• Wavelength


  The length of one complete oscillation measured from the same point on two consecutive waves
• Phase difference can be calculated from two different points on the same wave or the same point on two different waves
• Phase difference


  The phase difference between two points can be described as: In phase is 360 or 2π radians, In anti-phase is 180 or π radians
• Angular frequency, ⍵, in oscillations is equivalent to angular velocity in circular motion
• A child on a swing performs 0.2 oscillations per second. Calculate the period of the oscillation. T = 1/f = 1/0.2 = 5.0 s
• Angular frequency


  The rate of change of angular displacement with respect to time
• Simple harmonic motion (SHM) is a specific type of oscillation where the acceleration of a body is proportional to its displacement but acts in the opposite direction
• Defining equation of SHM

  a ∝ −x
• Conditions for SHM

  1. The acceleration is proportional to the displacement

  2. The acceleration is in the opposite direction to the displacement
• Simple Harmonic Motion (SHM)


  • SHM is a specific type of oscillation
  • Acceleration of a body is proportional to its displacement but acts in the opposite direction
• An object in SHM will have a restoring force acting on it to return it to its equilibrium position
• All undamped SHM graphs are represented by periodic functions described by sine and cosine curves
• Restoring force and acceleration act in the same direction in SHM
• The displacement, velocity, and acceleration graphs in SHM are all 90° out of phase with each other
• The amplitude of oscillations A can be found from the maximum value of x in the displacement-time SHM graph
• The time period of oscillations T can be found from reading the time taken for
• Displacement-time SHM graph


  • The amplitude of oscillations A can be found from the maximum value of x
  • The time period of oscillations T can be found from reading the time taken for one full cycle
  • The graph might not always start at 0
  • If the oscillation starts at the positive or negative amplitude, the displacement will be at its maximum
• Velocity-time SHM graph

  • It is 90° out of phase with the displacement-time graph
  • Velocity is equal to the rate of change of displacement
  • The velocity of an oscillator at any time can be determined from the gradient of the displacement-time graph
  • An oscillator moves the fastest at its equilibrium position
  • The velocity is at its maximum when the displacement is zero
• Acceleration-time SHM graph

  • The acceleration graph is a reflection of the displacement graph on the x-axis
  • When a mass has positive displacement (to the right) the acceleration is in the opposite direction (to the left) and vice versa
  • It is 90° out of phase with the velocity-time graph
  • Acceleration is equal to the rate of change of velocity
  • The acceleration of an oscillator at any time can be determined from the gradient of the velocity-time graph
  • The maximum value of the acceleration is when the oscillator is at its maximum displacement
• The velocity is at its maximum when the displacement x = 0
• The displacement, velocity and acceleration graphs in SHM are all 90° out of phase with each other
• Equation for time period of a mass-spring system

  T = 2π sqrt(m/k)
• Simple Harmonic Motion (SHM)

  Acceleration is given by a = -ω^2x where ω is the angular frequency
• Equation for angular frequency


  ω = sqrt(k/m)
• Equilibrium position


  The position where the object is at rest
• Newton’s Second Law

  F = ma where F is the force, m is the mass of the object, and a is the acceleration
• The time period equation applies to both vertical and horizontal mass-spring systems
• Combining equations for force and acceleration


  ma = -kx, a = -k/m x
• The higher the spring constant k, the stiffer the spring and the shorter the time period of oscillation
• The time period of a mass-spring system is independent of the force of gravity, leading to the same time period on Earth and the Moon
• The spring constant k is in Hooke's Law, where F = kx
• Simple pendulum


  A type of simple harmonic oscillator consisting of a string and a bob at the end
• Simple pendulum


  • Can be modelled as simple harmonic motion when the angle of oscillation is small
  • Restoring force is -mg sin θ where m is the mass of the bob, g is acceleration due to gravity, and θ is the angle between the bob and the vertical
  • Acceleration can be expressed as a = -g sin θ
  • For small angles, sin θ ≅ θ can be used
  • Displacement x is equal to the length of the arc made by the bob as it moves through an angle θ
• For small values of x, the condition for Simple Harmonic Motion (SHM) is satisfied as the restoring force is proportional to -x
• For large values of x, the acceleration of a simple pendulum is not proportional to the displacement