Problem 44 - Fundamentals of Physics

Question

A \(125 \mathrm{~cm}\) length of string has mass \(2.00 \mathrm{~g}\) and tension \(7.00 \mathrm{~N}\). (a) What is the wave speed for this string? (b) What is the lowest resonant frequency of this string?

Step-by-Step Solution

Answer

(a) Wave speed is approximately 66.14 m/s. (b) Lowest resonant frequency is approximately 26.46 Hz.

Step 1: Calculate Linear Density

The linear density, \( \mu \), is the mass per unit length of the string. Calculate it using the formula \( \mu = \frac{m}{L} \), where \( m = 2.00 \mathrm{~g} = 0.00200 \mathrm{~kg} \) and \( L = 125 \mathrm{~cm} = 1.25 \mathrm{~m} \). Thus, \( \mu = \frac{0.00200 \mathrm{~kg}}{1.25 \mathrm{~m}} = 0.0016 \mathrm{~kg/m} \).

Step 2: Determine Wave Speed

The speed of a wave on a string is calculated using \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the string. With \( T = 7.00 \mathrm{~N} \) and \( \mu = 0.0016 \mathrm{~kg/m} \), we have \( v = \sqrt{\frac{7.00}{0.0016}} = \sqrt{4375} \approx 66.14 \mathrm{~m/s} \).

Step 3: Find Fundamental Frequency

The fundamental frequency or lowest resonant frequency of a string is given by \( f_1 = \frac{v}{2L} \). Using the wave speed \( v = 66.14 \mathrm{~m/s} \) and string length \( L = 1.25 \mathrm{~m} \), we find \( f_1 = \frac{66.14}{2 \times 1.25} \approx 26.46 \mathrm{~Hz} \).

Key Concepts

Linear Density

The concept of linear density is important when dealing with wave phenomena on a string. It represents the mass of the string per its unit length. To calculate it, you take the total mass of the string and divide it by its length. This can be expressed as \( \mu = \frac{m}{L} \). For the exercise, with a mass, \( m = 2.00 \mathrm{~g} \), and a length, \( L = 1.25 \mathrm{~m} \), the linear density becomes \( 0.0016 \mathrm{~kg/m} \). Linear density provides insight into how mass is distributed along the string, which is essential for understanding how waves propagate through it.
Understanding linear density helps in anticipating how a change in either mass or length influences wave speed and behavior on the string. A higher linear density would mean more mass per unit length, potentially slowing wave motion.

Wave Speed Calculation

Wave speed is a critical parameter for understanding wave dynamics. It indicates how fast a wave travels through a medium—in this case, a string. The wave speed on a string is determined by a straightforward formula: \( v = \sqrt{\frac{T}{\mu}} \). Here, \( T \) is the tension in the string and \( \mu \) is the previously calculated linear density.
In the problem, with a tension of \( 7.00 \mathrm{~N} \) and a linear density of \( 0.0016 \mathrm{~kg/m} \), substituting these values into the formula gives a wave speed of approximately \( 66.14 \mathrm{~m/s} \).
This calculation shows the dynamic relationship between tension and wave speed—higher tension leads to higher wave speed, assuming constant linear density. Thus, controlling tension is a practical way to manage wave speed on a string, which is especially useful in musical instruments.

Resonant Frequency

Resonant frequency refers to the frequencies at which a system naturally oscillates. For a string, the resonant frequencies are the harmonics that the string naturally supports. These are determined by the wave speed and the length of the string.
When a string vibrates at its resonant frequencies, these correspond to standing wave patterns. Such frequencies become outputs that don't seem to "lose energy," i.e., conserve energy effectively. The relationship is due to constructive interference where waves constructively build upon each other.
Identification of these frequencies is crucial because it governs how the system, like a musical instrument string, might naturally sound when plucked or driven by another frequency.

Fundamental Frequency

The fundamental frequency or the lowest resonant frequency is the simplest standing wave pattern that can form on a string. It is calculated using the formula \( f_1 = \frac{v}{2L} \), where \( v \) is the wave speed and \( L \) is the length of the string.
In our example, with a wave speed of \( 66.14 \mathrm{~m/s} \) and a string length of \( 1.25 \mathrm{~m} \), the fundamental frequency is approximately \( 26.46 \mathrm{~Hz} \).
This frequency defines the vibration mode where the string vibrates as a single segment, moving back and forth. It's crucial because it sets the base pitch of the string and all higher harmonics are multiples of this frequency.
The fundamental frequency is particularly important in music. It influences the tonal properties of instruments by setting the baseline pitch.