Page 305 - Physics Coursebook 2015 (A level)

Chapter 19: Oscillations
In Figure 19.17, a single cycle of s.h.m. is shown, but with the x-axis marked with the phase of the motion in radians.
Equations of s.h.m.
The graph of Figure 19.15a, shown earlier, represents how the displacement of an oscillator varies during s.h.m. We have already mentioned that this is a sine curve. We can present the same information in the form of an equation. The relationship between the displacement x and the time t is as follows:
x = x0 sin ωt
where x0 is the amplitude of the motion and ω is its frequency. Sometimes the same motion is represented using a cosine function, rather than a sine function:
x = x0 cos ωt
The difference between these two equations is illustrated in Figure 19.19. The sine version starts at x = 0, i.e. the oscillating mass is at its equilibrium position when t = 0.
+xx 0
during one cycle.
xx
++ 0t0t
0 –x0
Phase/rad
2π
Figure 19.17 The phase of an oscillation varies from 0 to 2π
3π 22
π π
–
–
Figure 19.19 These two graphs represent the same simple harmonic motion. The difference in starting positions is related to the sine and cosine forms of the equation for x as a function of t.
x = x0sinωt QUESTIONS
x = x0cosωt
13
11 An object moving with s.h.m. goes through two complete cycles in 1.0 s. Calculate:
a the period T
b the frequency f
c the angular frequency ω.
12 Figure 19.18 shows the displacement–time graph for an oscillating mass. Use the graph to determine the following:
An atom in a crystal vibrates with s.h.m. with a frequency of 1014 Hz. The amplitude of its motion is 2.0 × 10−12 m.
a amplitude
b period
c frequency
d angular frequency
e f g
A
displacement at A velocity at B velocity at C.
C
0.3 0.4 0.5
a b
Sketch a graph to show how the displacement of the atom varies during one cycle.
Use your graph to estimate the maximum velocity of the atom.
0.20 0.10 0 –0.10 –0.20
0.1
0.2
0.6 0.7 0.8
0.9 Time / s
B
Figure 19.18 A displacement–time graph. For Question 12.
293
Displacement / m