Describing Oscillations (DP IB Physics): Revision Note

Properties of Oscillations

• 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)

• The equilibrium position (x = 0) is the position when there is no resultant force acting on the object

  • This is the fixed central point that the object oscillates around

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 by the following properties:

  • displacement

  • amplitude

  • time period

  • frequency

  • angular frequency

Displacement

• 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)

Amplitude

• 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

Time period

• 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

Frequency

• 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 

Angular frequency

• 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)

Calculating Time Period of an Oscillation

• 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 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

The motion of an oscillating object can be analysed in terms of a fraction of an object in circular motion