Oscillations

• Equilibrium position —> A system is in equilibrium when all the forces are balanced. No resultant force.
• Oscillating motion:

  • In each case the object starts in an equilibrium position. A force is then applied to the object, displacing it and it begins to oscillate
• Displacement —> the distance from the equilibrium position
• Amplitude —> the maximum displacement from the equilibrium position
• Period —>the time taken to complete one full oscillation
• Frequency —> the number of complete oscillations per unit time
• Angular frequency —> the motion of an oscillating object and is closely related to the angular velocity of an object in circular motion
• Simple harmonic motion —> oscillating motion for which the acceleration of the object is given by a = -w2x
• • the acceleration a of the object is directly proportional to its displacement x, that is a is directly proportional to x
  • the minus sign means that the acceleration of the object acts in the direction opposite to the displacement ( it returns the object to the equilibrium position)
• • v = +- w \/ a^2 - x^2
  • The velocity at any particular displacement has a positive or a negative value depending on the direction in which the oscillator is moving
  • The velocity can vary between zero ( at x=A) to its maximum values +- v max, at the equilibrium position
  • At the equilibrium position, x=0 so the equation becomes Vmax = wA
• Pendulum:

  • At the amplitude the pendulum is briefly stationary & has zero kinetic energy
  • All its energy is in the form of potential energy
  • As the pendulum falls it loses potential energy & gains kinetic energy
  • It has maximum velocity, and so maximum kinetic energy, as it moves though its equilibrium position
  • As the pendulum passes through the equilibrium position, it has no potential energy
• What is the formula for elastic potential energy (Ep) in a spring-mass system?

  Ep = 1/2kx^2
• What shape does a graph of elastic potential energy (Ep) against displacement (x) take?

  A parabola
• What is the relationship between elastic potential energy (Ep) and displacement (x)?

  Ep is directly proportional to \(x^2\)
• What is the value of elastic potential energy (Ep) when displacement (x) is zero?

  Ep = 0
• What is the maximum value of elastic potential energy (Ep) when displacement (x) is equal to amplitude (A)?


  Ep = \frac{1}{2}kA^2
• What happens to the glider when displacement (x) equals amplitude (A)?

  The glider will be stationary for an instant with no kinetic energy (KE)
• What is the total energy of the oscillator in a spring-mass system?


  The total energy is equal to \frac{1}{2}kA^2
• How is the kinetic energy (EK) of the glider at any instant calculated?

  EK = 1/2kA^2 - 1/2kx^2
• What is the simplified formula for kinetic energy (EK) in terms of amplitude (A) and displacement (x)?


  EK = 1/2k(A^2 - x^2)
• What shape does a graph of kinetic energy (EK) against displacement (x) take?

  An inverted parabola
• Damping --> An oscillation is damped when an external force that acts on the oscillator has the effect of reducing the amplitude of its oscillations
• • A pendulum moving through air experiences air resistance, which damps the oscillations until eventually the pendulum comes to rest.
• Light damping:

  • When the damping forces are small, the amplitude of the oscillator gradually decreases with time, but the period of the oscillations is almost unchanged.
  • The case when the pendulum oscillates in air
• Heavy damping:

  • For larger damping forces, the amplitude decreases significantly, and the period of the oscillations also increases slightly
  • Would occur for a pendulum oscillating in water
• • In all cases of damped motion, the kinetic energy of the oscillator is transferred to other forms (usually heat)
• Free oscillations:

  • When a mechanical system is displaced from its equilibrium position & then allowed to oscillate without any external forces, its motion is referred to as free oscillation
  • The frequency of the free oscillations is known as the natural frequency of the oscillator
• Forced oscillation:

  • A forced oscillation is one in which a periodic driver force is applied to an oscillator
  • The object will vibrate at the frequency of the driving force ( the driving frequency)
• • If the driving frequency is equal to the natural frequency of an oscillating object, then the object will resonate
  • This will cause the amplitude of the oscillations to increase dramatically, and if not damped, the system may break
• Resonance:

  • Occurs when the driving frequency of a forced oscillation is equal to the natural frequency of the oscillating object
• For a forced oscillator with negligible damping, at resonance:

  driving frequency = natural frequency of the forced oscillator
• • When an object resonates, the amplitude of the oscillations increases considerably
  • If the system is not damped, the amplitude will increase to the point at which the object fails
  • The greatest possible transfer of energy from the driver to the forced oscillator occurs at the resonant frequency --> the amplitude is maximum
• • Damping a forced oscillation has the effect of reducing the maximum amplitude at resonance
  • The degree of damping also has an effect on the frequency of the driver when maximum amplitude occurs
• • For light damping, the maximum amplitude occurs at the natural frequency f0 of the forced oscillator
• As the amount of damping increases:

  • the amplitude of vibration at any frequency decreases
  • the maximum amplitude occurs at a lower frequency than f0
  • the peak on the graph becomes flatter & broader