Page 301 - Physics Coursebook 2015 (A level)

Chapter 19: Oscillations
QUESTION
4a Figure 19.9b shows two oscillations which are out of phase. By what fraction of an oscillation are they out of phase?
b Why would it not make sense to ask the same question about Figure 19.9c?
Simple harmonic motion
There are many situations where we can observe the special kind of oscillations called simple harmonic motion (s.h.m.). Some are more obvious than others. For example, the vibrating strings of a musical instrument show s.h.m. When plucked or bowed, the strings move back and forth about the equilibrium position of their oscillation. The motion of the tethered trolley in Figure 19.3 and that
of the pendulum in Figure 19.4 are also s.h.m. (Simple harmonic motion is defined in terms of the acceleration and displacement of an oscillator – see pages 294–5.)
Here are some other, less obvious, situations where simple harmonic motion can be found:
■■ When a pure (single tone) sound wave travels through air, the molecules of the air vibrate with s.h.m.
■■ When an alternating current flows in a wire, the electrons in the wire vibrate with s.h.m.
■■ There is a small alternating electric current in a radio or television aerial when it is tuned to a signal, in the form of electrons moving with s.h.m.
■■ The atoms that make up a molecule vibrate with s.h.m. (see, for example, the hydrogen molecule in Figure 19.11a).
Step 1 Measure the time interval t between two corresponding points on the graphs.
t =17ms
Step2 DeterminetheperiodTforonecomplete
oscillation. T= 60ms
Hint: Remember that a complete oscillation is when the object goes from one side to the other and back again.
Step3 Nowyoucancalculatethephasedifferenceasa fraction of an oscillation.
phase difference = fraction of an oscillation Therefore: t 17
phase difference = T = 60 = 0.283 oscillation
Step4 Converttodegreesandradians.Thereare360°
and 2π rad in one oscillation.
phase difference = 0.283 × 360° = 102° ≈ 100° phase difference = 0.283 × 2π rad
a
0
b
c
Time
Time
Time
0
0
Figure 19.9 Illustrating the idea of phase difference.
WORKED EXAMPLE
1 Figure 19.10 shows displacement–time graphs for two identical oscillators. Calculate the phase difference between the two oscillations. Give your answer in degrees and in radians.
20 10 0 –10 –20
10 20 30 40 50 60 70 Time / ms
Figure 19.10 The displacement–time graphs of two oscillators with the same period.
=1.78rad≈1.8rad
289
Displacement / cm
Displacement Displacement Displacement