Knowledge-12 SHM

• SHM is an oscillation motion in which:

  1. the magnitude of the acceleration is proportional to the displacement from equilibrium position.
  2. the direction of the acceleration is always in the opposite direction to the displacement , directed back towards the the equilibrium position.
• The condition for SHM is:

  equilibrium is the midpoint of its motion and:

  • a is proportional to the -displacement.
• Graph of displacement against time (SHM):

  • is a cos graph
  • y-axis is the displacement from equilibrium
  • x-axis is the time
• Graph of velocity against time (SHM):

  • is a inverse sin graph
  • y-axis is the velocity from equilibrium
  • x-axis is the time
• Graph of acceleration against time (SHM):

  • is a inverse cos graph
  • y-axis is the acceleration from equilibrium
  • x-axis is the time
• the defining equation for SHM is :

  • acceleration a = -angular speed ^2 x displacement
• If an object has initial (t=0) velocity of zero and an initial displacement of +A :

  • displacement at time is x = Acos(angular speedxt)
  • velocity v = +/- angular speed ^2 x root (+A^2 -x^2)
  • max speed is = angular speed x A
  • max acceleration is = angular speed^2 x A
• Small angle approximation:

  • A simple pendulum will be in SHM if the angle it makes with the vertical is less than 10 degrees.
  • if angle is less than ten degrees then the small angle approximation can be used to show that the acceleration of the bob is proportional to the displacement from the equilibrium and always acts towards the equilibrium.
• The period of a simple pendulum in SHM is:

  • T = 2pie root (l/g)
• Mass-spring systems:

  The resultant force F due to the stretched string is known because it always acts towards the equilibrium position.

  For a spring with spring constant K stretched by a displacement x , the restoring force is :

  F = -kx
• the period of a mass-spring system is :

  • T = 2pie x root (m/k)
• There is a continuous and repeated transfer of energy from the potential energy to the kinetic energy and back again for an object in SHM.
• SHM:

  • E kinetic is max at zero displacement from equilibrium
  • E potential is max at max displacement from equilibrium
• Damping is the process by which an oscillating object loses energy to its surroundings.
• Damping is caused by damping forces such as air resistance or friction, which decrease the amplitude of the oscillations over time.
• Light damping:

  • involves a small damping force leading to a gradual change in amplitude.
• Critical damping:

  • critical damping causes the oscillations to stop in the shortest possible time , returning an object to its equilibrium without overshooting it.
• Heavy damping:

  • Heavy damping , sometimes called over damping , causes the object to slowly return to its equilibrium position without oscillating.
• Uses of damping:

  Damping can be used in suspension systems in vehicles and for cutting down unwanted sound in recording studios.
• Free vibration is one which occurs with no transfer of energy to or from the surroundings.

  • the only forces acting on freely oscillating objects are the ones providing the resultant restoring force.
  • has constant amplitude.
• A forced vibration is one where a periodic external driving force is applied.
• Natural frequency of an object is the frequency at which it would vibrate if left to vibrate freely.
• Resonance is the process by which a maximum in the amplitude of a force oscillation is produced.

  In the absence of of damping , resonance occurs when driving frequency is equal to the natural frequency.
• the time taken to complete one revolution can be found:

  • finding circumference and dividing by speed
  • diving the angle by angular speed
• To convert angular speed to speed you:

  1. times angular speed by radius of circle