Problem 23 - College Physics

Question

\(\bullet\) (a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern having the same frequency of the note that is sung. If someone sings the note \(B\) flat that has a frequency of 466 \(\mathrm{Hz}\) , how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords? (b) Hearing. When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that typical humans can hear has a period of 50.0\(\mu \mathrm{s} .\) What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) Vision. When light having vibrations with angular frequency ranging from \(2.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) to \(4.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light? (d) Ultrasound. High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as x rays do. To detect a small objects such as tumors, a frequency of around 5.0 \(\mathrm{MHz}\) is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

Step-by-Step Solution

Answer

(a) 2.15 ms, 2928 rad/s; (b) 20,000 Hz, 125,664 rad/s; (c) 2.3 to 1.3 fs, 4.3 to 7.5 \(\times 10^{14}\) Hz; (d) 0.2 \(\mu\)s, 31.4 Mrad/s.

Step 1: Calculating Time for One Cycle (Singing B Flat)

The frequency \(f\) of the note \(B\) flat is given as 466 Hz. The time for one complete cycle, called the period \(T\), is the reciprocal of the frequency.\[ T = \frac{1}{f} = \frac{1}{466} \text{ seconds} \approx 0.00215 \text{ seconds (or 2.15 ms)}. \]

Step 2: Calculating Angular Frequency (Singing B Flat)

The angular frequency \(\omega\) is given by the formula \(\omega = 2\pi f\). Applying the given frequency:\[ \omega = 2\pi \times 466 \approx 2928 \text{ rad/s}. \]

Step 3: Calculating Frequency (Hearing)

The period \(T\) of the highest pitch is given as 50.0 \( \mu s\). Convert \( \mu s \) to seconds: \(50 \mu s = 50 \times 10^{-6}\, \text{seconds}.\) Then, calculate the frequency:\[ f = \frac{1}{T} = \frac{1}{50 \times 10^{-6}} \approx 20,000 \text{ Hz}. \]

Step 4: Calculating Angular Frequency (Hearing)

Use the frequency from Step 3 to determine the angular frequency \(\omega = 2\pi f\).\[ \omega = 2\pi \times 20,000 \approx 125,664 \text{ rad/s}. \]

Step 5: Calculating Limits of Period and Frequency (Vision)

For vision, the range of angular frequencies \(\omega\) is from \(2.7 \times 10^{15}\) to \(4.7 \times 10^{15}\) rad/s. The frequency \(f\) is \(f = \frac{\omega}{2\pi}\).1. Lower limit: \[ f_{\text{min}} = \frac{2.7 \times 10^{15}}{2\pi} \approx 4.3 \times 10^{14}\, \text{Hz}. \]2. Upper limit:\[ f_{\text{max}} = \frac{4.7 \times 10^{15}}{2\pi} \approx 7.5 \times 10^{14}\, \text{Hz}. \]The period \(T\) is \(T = \frac{1}{f}\):1. Lower limit:\[ T_{\text{max}} = \frac{1}{4.3 \times 10^{14}} \approx 2.3 \times 10^{-15}\, \text{s}. \]2. Upper limit:\[ T_{\text{min}} = \frac{1}{7.5 \times 10^{14}} \approx 1.3 \times 10^{-15}\, \text{s}. \]

Step 6: Calculating Period and Angular Frequency (Ultrasound)

High-frequency sound waves at 5.0 MHz are used for ultrasound. Convert MHz to Hz: \(5.0\, \text{MHz} = 5.0 \times 10^{6}\, \text{Hz}.\)1. Period \(T\): \[ T = \frac{1}{5.0 \times 10^{6}} \approx 2.0 \times 10^{-7}\, \text{s}. \]2. Angular Frequency \(\omega\):\[ \omega = 2\pi \times 5.0 \times 10^{6} \approx 3.14 \times 10^{7} \text{ rad/s}. \]

Key Concepts

Vibrations in Singing

When you sing a note, your vocal cords vibrate at specific frequencies that match the musical note. For example, when a singer hits a B flat note, the frequency of their vocal cords' vibration is around 466 Hz. This means the cords complete 466 vibrating cycles every second. The time it takes to complete one cycle is known as the period, calculated as the reciprocal of the frequency. In this instance, it is approximately 0.00215 seconds (or 2.15 milliseconds). These vibrations create sound waves that our ears perceive as musical notes. The relationship between frequency and pitch means higher frequencies correspond to higher pitches.
Understanding this concept helps us appreciate the complexity of human voice and how singers control their pitch so effectively.

Angular Frequency

Angular frequency, denoted as ω (omega), provides another way to describe how often an object or wave completes a cycle, but it does so in terms of radians per second rather than cycles per second. To calculate angular frequency from regular frequency, the formula is \(\omega = 2\pi f\). This conversion effectively translates between the cyclical movement described by frequency and the rotational equivalent in radians, linking linear and angular motion. For instance, in the singing example with a B flat note at 466 Hz, the angular frequency is approximately 2928 rad/s.
This measure is particularly important in physics and engineering when dealing with systems involving circular or oscillatory motion, such as waves or rotating machines.

Audible Frequency Range

Humans have an audible frequency range typically between 20 Hz and 20,000 Hz. This range includes various sounds we encounter daily, from low bass in music to high-pitched notes. Our hearing is most sensitive in the mid-range frequencies, crucial for understanding human speech. However, the highest frequency audible to most people, particularly as we age, is around 20,000 Hz, represented by a period of about 50 microseconds.
The frequency and its reciprocal relationship with the period affect how we perceive pitch. For higher frequencies, the period is shorter, leading to higher pitches, as demonstrated by the vibrations of the eardrum responding to these frequencies.

Ultrasound Frequency

Ultrasound refers to sound waves with frequencies higher than the upper limit of human hearing, typically above 20,000 Hz. In medical settings, ultrasound frequencies are often in the megahertz range (1 MHz = 1 million Hz), such as 5.0 MHz. These high-frequency sound waves are crucial for medical imaging because they can penetrate tissues without the harmful effects of radiation.

• The period of ultrasound waves at 5.0 MHz is about 0.2 microseconds, as the waves complete millions of cycles per second.
• Angular frequency for these waves is around 31.4 million radians per second.

The small wavelength and high frequency of ultrasound allow for detailed imaging, vital for diagnosing and monitoring health conditions without invasive procedures.

Light Frequency Range

The light we see is only a small portion of the electromagnetic spectrum. Our eyes detect frequencies from around 4.3 x 10^14 Hz to 7.5 x 10^14 Hz, translating into the visible light range. The corresponding periods are exceptionally small, ranging from approximately 1.3 x 10^-15 to 2.3 x 10^-15 seconds. This high frequency corresponds to the many cycles per second required to stimulate our eyes' photoreceptor cells.
Within the visible spectrum, different frequencies correspond to different colors, from red (lower frequencies) to violet (higher frequencies). This is why changes in frequency affect what color we perceive, demonstrating the fascinating interplay between wave physics and human vision.