Question

A piano string sounds a middle A by vibrating primarily at 220 Hz.a)Calculate its period.b)Calculate its angular frequency.c)Calculate the period for a soprano singing a "high A," two octaves up, which is four times the frequency of the piano string.d)Calculate the angular frequency for a soprano singing a "high A," two octaves up, which is four times the frequency of the piano string.

Answer 1

a) 4.5 ms

The period of a wave is given by:

$T=\frac{1}{f}$

where f is the frequency.

For the note in this problem, f = 220 Hz, so the period of the wave is

$T=\frac{1}{f}=\frac{1}{220 Hz}=4.5\cdot 10^{-3} s = 4.5 ms$

b) 1381.6 rad/s

The angular frequency is given by:

$\omega=2 \pi f$

where f is the frequency.

In this problem, f = 220 Hz, so the angular frequency is

$\omega=2 \pi (220 Hz)=1381.6 rad/s$

c) 1.1 ms

The frequency of the "high A" is four times the frequency of the piano string, so

$f=4 \cdot 220 Hz=880 Hz$

And so, its period is

$T=\frac{1}{f}=\frac{1}{880 Hz}=1.1\cdot 10^{-3} s=1.1 ms$

d) 5526.4 rad/s

The angular frequency is given by:

$\omega=2 \pi f$

where f is the frequency.

For this note, f = 880 Hz, so the angular frequency is

$\omega=2 \pi (880 Hz)=5526.4 rad/s$