Circular motion and oscillations

• Frequency (f) = 1/Time period (T)
• 1 radian is equal to the arc length divided by the radius of the circle, in other words arc length = rΘ
• The period T is the time taken for one complete revolution.
• The angular velocity ω is measured in radians per second.
• The frequency f is the number of revolutions per second.
• Angle in radians = pi/180 x angle in degrees
• Angular velocity = ω = Θ/t
• Linear speed = v = ωr
• Angular velocity = ω = 2pi/T
• Angular velocity (ω) = 2pi x frequency (f)
• The definition of angular velocity is?

  The angle a object rotates through per second
• Centripetal acceleration (a) = linear velocity squared (v^2) / radius (r)

  OR

  Centripetal acceleration (a) = angular velocity squared (ω^2) x radius (r)
• Centripetal force (F) = mass (m) x linear velocity squared (v^2) / radius (r)

  OR

  Centripetal force (F) = mass (m) x angular velocity squared (ω^2) x radius (r)
• • Measure the mass of the bung and the washers.
  • Attach the bung to the end of some string, thread the string through a plastic tube and attach the washers to the other end.
  • Measure and make a reference mark on the string from the bung
  • Line up the reference mark with the tip and spin the bung in a horizontal circle.
  • Measure the time taken for the bung to make make ten rotations then divide time by 10
  • Once you have an accurate value for T. Do ω = 2pi/T to find the angular velocity. Then find centripetal force using F = mω^2 x r
  • Centripetal force should be equal to the weight of the washers.
• Centripetal acceleration is the acceleration towards the centre of a circle
• Centripetal force is the net force perpendicular to the velocity
• Simple harmoninc motion is defined as:

  • an oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint and is directed towards the midpoint
• acceleration (a) = - angular velocity squared (-ω^2) x displacement (x)
• Displacement, x, varies as a cosine or sine wave with a maximum value, A (amplitude)
• Velocity, v, is the gradient of the displacement-time graph. It has maximum value of ωA, where ω is the angular frequency of the oscillation. This can be calculated using ω = 2pi*f OR ω = 2pi/T
• Acceleration, a, is the gradient of the velocity-time graph. It has a maximum value of ω^2*A
• Phase difference is a measure of how much one wave lags behind another. This can be measured in radians or degrees.
• • An object in SMH exchanged Potential energy and Kinetic energy as it oscillates.
  • The type of potential can be gravitational or elastic
  • As the object moves towards the equilibrium, the restoring force does work on the object and so transfers some potential to kinetic.
  • When is object is moving away from equilibrium, all the kinetic is transferred to potential.
  • At equilibrium, the objects potential is said to be zero and its kinetic energy at its max – therefore velocity is at its max.
  • At max displacement the objects potential is at its max and its kinetic is zero, so its velocity is zero.
• The sum of the potential and kinetic energy is called the mechanical energy and is constant
• Maximum velocity – Equilibrium

  Minimum velocity – Maximum displacement

  Maximum acceleration – Maximum displacement

  Minimum acceleration – Equilibrium
• displacement (X) = amplitude (A) cos(angular velocity (ω) x time (t) )

  OR

  displacement (X) = amplitude (A) sin(angular velocity (ω) x time (t) )
• The equation is cos if the oscillation starts at maximum and is sin if it starts at equilibrium
• maximum acceleration (a_max) = angular velocity squared (ω^2) * amplitude (A)
• maximum velocity (V_max)= angular velocity (ω) x amplitude (A)
• Free vibrations involve no transfer of energy to or from the surroundings
• In free vibrations an object will oscillate at its resonant frequency
• Forced vibrations happen when there's an external driving force. The frequency of this force is called the driving frequency
• When the driving frequency is at a 90 degree phase difference from the natural frequency the system is is resonance
• In practice any oscillating system loses energy to the surroundings, this is called damping
• lightly damped systems take a long time to stop while heavily damped system take less time to stop
• Critical dampening reduces the amplitude in the shortest possible time. Example is car suspension
• systems with heavy dampening are overdamped. They take longer to return to equilibrium that a critically damped system. An example would be a heavy door so they don't slam shut to quickly
• Lightly damped systems have a very sharp resonance peak. Their amplitude only increases dramatically when the driving frequency is very close to the natural frequency
• Heavily damped systems have a flatter resonance peak. Their amplitude doesn't increase much near their natural frequency