All of the examples discussed so far show the same pattern of movement. The trolley accelerates as it moves towards the centre of the oscillation. It is moving fastest at the centre. It decelerates as it moves towards the end of the oscillation. At the extreme position, it stops momentarily, reverses its direction and accelerates back towards the centre again.

Amplitude, period and frequency

Many oscillating systems can be represented by a displacement–time graph like that shown in Figure 18.7. The displacement x varies in a smooth way on either side of the midpoint. The shape of this graph is a sine curve, and the motion is described as sinusoidal.
Notice that the displacement changes between positive and negative values, as the object moves through the equilibrium position. The maximum displacement from the equilibrium position is called the amplitude ${x_0}$ of the oscillation.

The displacement–time graph can also be used to show the period and frequency of the oscillation. The period T is the time for one complete oscillation. Note that the oscillating object must go from one side to the other and back again (or the equivalent). The frequency f is the number of oscillations per unit time, and so f is the reciprocal of T:

$frequency = \frac{1}{{period}} \equiv f = \frac{1}{T}$

The equation can also be written as:

$period = \frac{1}{{frequency}} \equiv T = \frac{1}{f}$

Question

 

3) From the displacement–time graph shown in Figure 18.8, determine the amplitude, period and frequency of the oscillations represented.

Phase

The term phase describes the point that an oscillating mass has reached within the complete cycle of an oscillation. It is often important to describe the phase difference between two oscillations. The graph of Figure 18.9a shows two oscillations that are identical except for their phase difference. They are out of step with one another. In this example, they have a phase difference of one-quarter of an oscillation.
Phase difference can be measured as a fraction of an oscillation, in degrees or in radians (see Worked example 1).

Figure 18.9: Illustrating the idea of phase difference.

WORKED EXAMPLE

1) Figure 18.10 shows displacement–time graphs for two identical oscillators. Calculate the phase difference between the two oscillations. Give your answer in degrees and in radians.

Step 1: Measure the time interval t between two corresponding points on the graphs.
$t = 17 ms$
Step 2: Determine the period T for one complete oscillation.
$T = 60 ms$
Hint: Remember that a complete oscillation is when the object goes from one side to the other and back again.
Step 3: Now you can calculate the phase difference as a fraction of an oscillation.
phase difference = fraction of an oscillation
Therefore:
$\begin{array}{l}
phase\,difference = \frac{t}{T}\\
 = \frac{{17}}{{60}}\\
 = 0.283\,oscillations
\end{array}$
Step 4: Convert to degrees and radians. There are 360° and 2π rad in one oscillation.
$\begin{array}{l}
phase\,difference = 0.283 \times {360^ \circ }\\
 = {102^ \circ } \approx {100^ \circ }
\end{array}$
$\begin{array}{l}
phase\,difference = 0.283 \times 2\pi \,rad\\
 = 1078\,rad \approx 108\,rad
\end{array}$