Standing-wave vibrations are set up in a crystal goblet with four nodes and four antin- odes equally spaced around the 26.3 cm cir- cumference of its rim.
If transverse waves move around the glass at 900 m/s, an opera singer would have to produce a high harmonic with what frequency in order to shatter the glass with a resonant vibration? Answer in units of kHz.
To find the frequency required to shatter the glass with a resonant vibration, we need to determine the wavelength of the transverse waves that can form in the glass and then use the formula for calculating frequency.
The standing-wave pattern with four nodes and four antinodes suggests that we have a quarter-wavelength resonant mode. In this mode, the distance between two consecutive nodes (or antinodes) corresponds to a quarter-wavelength.
Given that there are four nodes and four antinodes equally spaced around the 26.3 cm circumference of the rim, we can infer that the quarter-wavelength corresponds to one-fourth of the circumference.
So, the quarter-wavelength is equal to:
Quarter-wavelength = Circumference / 4 = 26.3 cm / 4 = 6.575 cm = 0.06575 m
Now, we can use the wave equation:
v = λ * f
where:
v is the velocity of the wave (900 m/s),
λ is the wavelength (0.06575 m), and
f is the frequency that we need to calculate.
Rearranging the equation, we have:
f = v / λ = 900 m/s / 0.06575 m = 13692.63 Hz
To express the answer in kilohertz (kHz), we divide the frequency by 1000:
f = 13692.63 Hz / 1000 = 13.69263 kHz
Therefore, an opera singer would have to produce a high harmonic with a frequency of approximately 13.69263 kHz in order to shatter the glass with a resonant vibration.