Unit 9

• Simple harmonic motion (SHM)


  A fundamental oscillation which appears naturally in various phenomena
• Conditions for an object to oscillate in SHM

  • Periodic oscillations
  • Acceleration proportional to its displacement
  • Acceleration in the opposite direction to its displacement

  
• Isochronous


  The time period stays constant, so frequency must also be constant
• Calculating the period of simple harmonic oscillations

  T = 2π/ω

  
• Acceleration of an oscillator in SHM

  a = -ω^2x

  
• Acceleration and displacement in SHM

  Acceleration reaches maximum when displacement is at maximum

  
• Displacement and acceleration in SHM

  Acceleration is in the opposite direction to displacement
• Restoring force in SHM

  Directly proportional, but in the opposite direction, to the displacement of the object from the equilibrium position

  
• The acceleration of an object in SHM is directly proportional to the negative displacement</b>

  
• Displacement equation for SHM starting from equilibrium

  x = x0sin(ωt)

  
• Displacement equation for SHM starting from amplitude

  x = x0cos(ωt)

  
• The two displacement equations represent the same SHM, the difference is the starting position
• Summary of equations and graphs for SHM

  • Displacement, velocity, acceleration

  
• Calculating displacement of an oscillator in SHM

  Use displacement equation with known values

  
• Velocity of an oscillator in SHM starting from equilibrium


  v = ωx0cos(ωt)

  
• Velocity of an oscillator in SHM starting from maximum displacement

  v = -ωx0sin(ωt)

  
• Maximum speed of oscillation

  vmax = ωx0

  
• Acceleration of an oscillator in SHM starting from equilibrium

  a = -ω^2x0sin(ωt)

  
• Acceleration of an oscillator in SHM starting from maximum displacement

  a = -ω^2x0cos(ωt)

  
• Maximum acceleration of oscillation

  amax = ω^2x0
• Displacement-velocity relation for SHM: v = ±ω√(x0^2 - x^2)

  
• Exam tip: Keep positive and negative numbers distinct, use radians mode for trigonometric functions
• Energy changes in SHM

  Constantly exchanged between kinetic energy and potential energy

  
• Kinetic energy is maximum when displacement is zero, potential energy is maximum at maximum displacement

  
• Total energy of a simple harmonic system always remains constant
• Energy-time graph for SHM

  • Kinetic and potential energies vary periodically in opposite directions
  • Total energy is constant

  
• Sum of the kinetic and potential energies


  Total energy of an oscillator in simple harmonic motion

  
• The kinetic and potential energy of an oscillator in SHM vary periodically
• Energy-time graph for SHM

  • Both the kinetic and potential energies are represented by periodic functions (sine or cosine) which are varying in opposite directions to one another
  • When the potential energy is 0, the kinetic energy is at its maximum point and vice versa
  • The total energy is represented by a horizontal straight line directly above the curves at the maximum value of both the kinetic or potential energy
  • Energy is always positive so there are no negative values on the y axis
  • Kinetic and potential energy go through two complete cycles during one period of oscillation

  
• Energy-displacement graph for SHM

  • Displacement is a vector, so, the graph has both positive and negative x values
  • The potential energy is always at a maximum at the amplitude positions x and zero at the equilibrium position (x = 0)
  • The kinetic energy is the opposite: it is zero at the amplitude positions x and maximum at the equilibrium position, x = 0
  • The total energy is represented by a horizontal straight line above the curves

  
• Calculating kinetic energy in SHM

  1. Use the expression for velocity, v = -ωx0sin(ωt)

  2. EK = 1/2 mv^2

  3. EK = 1/2 m(-ωx0sin(ωt))^2

  4. EK = 1/2 mω^2x0^2 sin^2(ωt)

  5. Maximum kinetic energy is EK(max) = 1/2 mω^2x0^2

  
• Calculating potential energy in SHM

  1. Use the expressions for velocity and displacement

  2. x = x0sin(ωt)

  3. v = ωx0cos(ωt)

  4. Use the trigonometric identity sin^2(ωt) + cos^2(ωt) = 1

  5. EP = ET - EK

  6. EP = 1/2 mω^2x0^2 - 1/2 mω^2x0^2 sin^2(ωt)

  7. EP = 1/2 mω^2x0^2 cos^2(ωt)

  8. Since x = x0sin(ωt), EP = 1/2 mω^2x^2

  9. Maximum potential energy is EP(max) = 1/2 mω^2x0^2

  
• The total energy of a system undergoing simple harmonic motion is ET = 1/2 mω^2x0^2

  
• Deriving the kinetic energy-displacement relation for SHM

  1. Use the displacement-velocity relation v = ±ω(x0^2 - x^2)^1/2

  2. Substitute into the kinetic energy equation EK = 1/2 mv^2

  3. EK = 1/2 mω^2(x0^2 - x^2)

  
• You may be expected to draw as well as interpret energy graphs against time or displacement in exam questions
• Key equations related to simple harmonic motion

  • Kinetic energy, EK = 1/2 mω^2x0^2 sin^2(ωt)
  • Potential energy, EP = 1/2 mω^2x0^2 cos^2(ωt)
  • Total energy, ET = 1/2 mω^2x0^2
  • Kinetic energy-displacement relation, EK = 1/2 mω^2(x0^2 - x^2)
  • Period of a simple pendulum, T = 2π(L/g)^1/2
  • Period of a mass-spring system, T = 2π(m/k)^1/2

  
• Calculating the period and amplitude of a simple pendulum

  1. Use the graph to determine the maximum potential energy, EP(max)

  2. The amplitude, x0, is the maximum displacement, which occurs at the maximum potential energy

  3. Use the equation EP(max) = 1/2 mω^2x0^2 to find ω

  4. Then use the relation T = 2π/ω to find the period, T

  
• Calculating the maximum speed and kinetic energy of an oscillating mass on a spring


  1. Use the equation vmax = ωx0 to find the maximum speed

  2. Then use the equation EKmax = 1/2 mvmax^2 to find the maximum kinetic energy

  
• Calculating the total energy of an oscillating mass-spring system


  Use the equation ET = 1/2 mω^2x0^2, where ω = 2πf and f is the given frequency

  
• Most of the equations related to SHM are given in the data booklet for exams