Problem 47 - College Physics

Question

guitar string under a tension of \(50.000 \mathrm{~N}\) has a mass per unit length of \(0.10000 \mathrm{~g} / \mathrm{cm}\). What is the highest resonant frequency that can be heard by a person capable of hearing frequencies up to \(20000 \mathrm{~Hz}\) ?

Step-by-Step Solution

Answer

The highest resonant frequency that can be heard by a person is either the calculated fundamental frequency if it's less than or equal to 20000 Hz or just 20000 Hz if the calculated fundamental frequency is higher.

Step 1: Evaluate Knowns

First, we evaluate the known quantities. The tension on the string is \(T = 50,000 N\), and the string's mass per unit length, or linear density, is \(\mu = 0.10000 g/cm = 0.00100 kg/m\) (converted from g/cm to kg/m for consistency of units). We don't have the length of the string, but we don't need it for this problem as we're looking for the maximum frequency, not a specific frequency for a given length. We're asked for the highest frequency that can be heard, which will be the fundamental frequency.

Step 2: Apply the Formula for Fundamental Frequency

Next, we apply the formula for the fundamental frequency: \(f_1=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}\). As mentioned before, we don't need to know the length of the string \(L\), as we're searching for the maximum frequency, which does not depend on the length of the string. Hence, we modify the formula to \(f_1=\sqrt{\dfrac{T}{4\mu}}\), to find the maximum frequency.

Step 3: Substitute Known Values

Then, we substitute the known values into the equation: \(f_1=\sqrt{\dfrac{50000 N}{4 \cdot 0.00100 kg/m}}\).

Step 4: Calculate and Check the Result

When we solve the above equation, we have the result in Hz. We check if this calculated fundamental is less than or equal to the highest frequency a person can hear, 20000 Hz. If it's greater, then 20000 Hz becomes the highest resonant frequency that can be heard by a person.

Key Concepts

Resonant Frequency

In physics, the resonant frequency is a natural "sweet spot" at which an object or medium tends to vibrate. For a musical instrument like a guitar, the resonant frequency refers to the specific frequencies at which the strings successfully vibrate to emit sounds. When a guitar string is plucked, it vibrates at certain frequencies, and its loudest vibration occurs at its resonant frequency. This frequency results in standing waves that resonate or sustain for an extended time, producing clear and rich sound.

For any stringed instrument, the highest resonant frequency that can be audible is determined by the listener's hearing capability. In the exercise, it is capped at 20,000 Hz, which is generally the upper limit of human hearing. The aim was to calculate whether a resonant frequency on our guitar string can reach this limit.

String Tension

String tension refers to the force applied to stretch a string which directly affects its ability to produce sound. Higher tension usually means a higher pitch of the sound. This is because the molecules in the string are more tightly packed, allowing them to vibrate more rapidly when the string is plucked or struck.

In the exercise, the string tension is stated as 50,000 Newtons (N), a substantial amount ensuring that the string is quite taut. This immense tension is what facilitates high-frequency vibrations, because it forces the string to return to its original position more quickly, leading to shorter and faster vibrations, hence higher frequencies.

• A high tension implies stronger and crisper sound quality.
• Adjusting tension allows musicians to tune their instruments.

Linear Density

Linear density is an essential factor in determining the vibration frequency of a string. It is the mass of the string per unit length, often expressed in kilograms per meter (kg/m). A denser string will typically vibrate less frequently, resulting in a lower pitch. Simply put, a thin string vibrates faster than a thick one.

In our exercise, the linear density is given as 0.00100 kg/m which is relatively low. This low linear density means the string can vibrate more easily and quickly, which is crucial for achieving higher resonant frequencies.

• Lower linear density leads to higher pitch sounds.
• Higher linear density yields sound with more bass.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system naturally vibrates. For a vibrating string, the fundamental frequency forms the first harmonic, which is essentially the primary tone heard.

In the earlier solution of our problem, we derived the fundamental frequency using the formula: \(f_1 = \sqrt{\frac{T}{4\mu}}\). This formula accounts for the tension \(T\) in the string and its linear density \(\mu\), allowing us to determine how fast the string vibrates without needing its length.

Understanding fundamental frequency helps in musical acoustics and engineering, and forms the basis for calculating overtones. It's key to both forming the main tone and understanding the harmonic series that follows.

• The fundamental frequency is crucial in musical tuning.
• Defines the main note of a stringed instrument's sound.