Describing Oscillations (HL IB Physics)

• An oscillation is defined as follows:

The repetitive variation with time t of the displacement x of an object about the equilibrium position (x = 0)  

A pendulum oscillates between A and B. On a displacement-time graph, the oscillating motion of the pendulum is represented by a wave, with an amplitude equal to x₀

• A particle undergoing an oscillation can be described using the following properties:
  • Equilibrium position (x = 0) is the position when there is no resultant force acting on an object
    • This is the fixed central point that the object oscillates around
  • Displacement (x) is the horizontal or vertical distance of a point on the wave from its equilibrium position
    • It is a vector quantity
    • It can be positive or negative depending on which side of the oscillation it is
    • It is measured in metres (m)
  • Period (T) or time period, is the time interval for one complete oscillation measured in seconds (s)
    • If the oscillations have a constant period, they are said to be isochronous

Diagram showing the time period of a wave

• • Amplitude (x₀) is the maximum value of the displacement on either side of the equilibrium position
    • Amplitude is measured in metres (m)

When the pendulum is in its extreme position this is its amplitude

• • Frequency (f) is the number of oscillations per second and it is measured in hertz (Hz)
    • Hz has the SI units per second s⁻¹ because  see below
  • Angular frequency (ω) is the rate of change of angular displacement with respect to time
    • It is measured in radians per second (rad s⁻¹)

• This equation relates the frequency and the time period of an oscillation: 

The equation linking time period and frequency

• Angular frequency (⍵) can be calculated using the equation:

• Where:
  • ⍵ = angular frequency (rad s^-1)
  • 2π = circumference of a circle
  • T = time period (s)
  • f = frequency of oscillation (Hz)
• The angular displacement of objects in oscillation can be determined by matching the displacement to an object in circular motion:
  • After moving from one amplitude position x = −A to the equilibrium position x = 0 the mass on the spring has moved an angular displacement of  of a circle =  =  radians
  • Continuing the oscillation from the equilibrium position to the other amplitude position the angular displacement is also radians
  • Continuing the oscillation back to the starting point means the mass travels a further angular displacement of  radians
  • Hence, the total angular displacement in one oscillation is  radians