Physics A Level
Chapter 18: Oscillations 18.6 Frequency and angular frequency
Physics A Level
Chapter 18: Oscillations 18.6 Frequency and angular frequency
The frequency f of s.h.m. is equal to the number of oscillations per unit time. As we saw earlier, f is related to the period T by:
$f = \frac{1}{T}$
We can think of a complete oscillation of an oscillator or a cycle of s.h.m. as being represented by $2\pi $ radians. (This is similar to a complete cycle of circular motion, where an object moves round through $2\pi $ radians.) The phase of the oscillation changes by $2\pi $ rad during one oscillation. Hence, if there are f oscillations in unit time, there must be $2\pi f$ radians in unit time. This quantity is the angular frequency of the s.h.m. and it is represented by the Greek letter $\omega $ (omega).
The angular frequency $\omega $ is related to frequency f by the equation:
$\omega = 2\pi f$
Since $f = \frac{1}{T}$, the angular frequency ω is related to the period T of the oscillator by the equation:
$\omega = \frac{{2\pi }}{T}$ or $T = \frac{{2\pi }}{\omega }$
Questions
7) Use the graphs shown in Figure 18.15 to determine the values of the following quantities:
a: amplitude
b: time period
c: maximum velocity
d: maximum acceleration.
8) State at what point in an oscillation the oscillator has zero velocity but acceleration towards the right.
9) Look at the $x–t$ graph of Figure 18.15a. When $t = 0.1 s$, what is the gradient of the graph? State the velocity at this instant.
10) Figure 18.16 shows the displacement–time ($x–t$) graph for an oscillating mass. Use the graph to determine the following quantities:
a: the velocity in $cm\,{s^{ - 1}}$ when $t = 0 s$
b: the maximum velocity in $cm\,{s^{ - 1}}$
c: the acceleration in $cm\,{s^{ - 2}}$ when $t = 1.0 s$.
In Figure 18.17, a single cycle of s.h.m. is shown, but with the x-axis marked with the phase of the motion in radians.
Questions
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 $\omega $.
12) Figure 18.18 shows the displacement–time graph for an oscillating mass. Use the graph to determine the following:
a: amplitude
b: period
c: frequency
d: angular frequency
e: displacement at A
f: velocity at B
g: velocity at C.
13) An atom in a crystal vibrates with s.h.m. with a frequency of ${10^{14}}Hz$. The amplitude of its motion is $2.0 \times {10^{ - 12}}m$.
a: Sketch a graph to show how the displacement of the atom varies during one cycle.
b: Use your graph to estimate the maximum velocity of the atom.