Problem 76 - College Physics

Question

A student uses a 2.00 -m-long steel string with a diameter of \(0.90 \mathrm{~mm}\) for a standing wave experiment. The tension on the string is tweaked so that the second harmonic of this string vibrates at \(25.0 \mathrm{~Hz}\). (a) Calculate the tension the string is under. (b) Calculate the first harmonic frequency for this string. (c) If you wanted to increase the first harmonic frequency by \(50 \%,\) what would be the tension in the string? [Hint: See Table 9.2\(]\)

Step-by-Step Solution

Answer

(a) Use formula for tension: \( T = 4 f_2^2 L^2 \mu \); (b) Solve for \( f_1 \) using: \( f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \); (c) Increase \( f_1 \) by 50%, recalculating tension.

Step 1: Understand the problem

The exercise involves calculating the tension in a steel string, determining the frequency of its first harmonic, and adjusting tension to increase that frequency. Key given data include the string's length (2.00 m), diameter (0.90 mm), and the frequency of the second harmonic (25.0 Hz). The goal is to find tension values for different conditions.

Step 2: Calculate the mass per unit length

The mass per unit length \( \mu \) of the string is required for the calculations. The volume of a cylindrical string is \( V = \pi r^2 L \). The radius \( r \) is half the diameter, \( 0.90 \mathrm{~mm} = 0.0009 \mathrm{~m} \). Convert diameter to radius and calculate \( \mu \): \[ \mu = \frac{\text{mass}}{\text{length}} = \frac{\rho \pi r^2 L}{L} = \rho \pi r^2, \] substituting \( \rho \) (density of steel, usually around \( 7800 \mathrm{~kg/m}^3 \)).

Step 3: Determine second harmonic condition

For the second harmonic, the string forms one complete wave loop, and the frequency relation is given by \( f_2 = \frac{n}{2L} \sqrt{\frac{T}{\mu}} \), where \( n = 2 \) for the second harmonic. Solving for \( T \), the tension, rewrite the formula: \[ T = 4f_2^2 L^2 \mu. \] Substitute the known values (\( f_2 = 25.0 \mathrm{~Hz}, L = 2.00 \mathrm{~m}\), and \( \mu \) from Step 2).

Step 4: Calculate first harmonic frequency

The first harmonic frequency \( f_1 \) is determined by \( f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \). Use the tension \( T \) calculated in Step 3 and substitute \( \mu \) from Step 2 to calculate \( f_1 \).

Step 5: Calculate new tension for increased frequency

To increase the first harmonic frequency by 50%, multiply \( f_1 \) by 1.5 (i.e., \( f_1' = 1.5 f_1 \)). Use the relationship \( T' = 4(f_1')^2 L^2 \mu \) to calculate the new tension \( T' \).

Key Concepts

Harmonics

To comprehend standing waves in a string, it's crucial to understand harmonics. Harmonics refer to the fixed-frequency patterns of vibration that occur under specific conditions. These patterns depend on both the length of the string and the boundary conditions—how the string is attached at its ends. In this exercise, the steel string is clamped at both ends, allowing the creation of standing waves.
The fundamental frequency, or first harmonic, corresponds to a single segment vibrating between the endpoints. Higher harmonics, like the second harmonic, consist of multiple segments or loops. For example, the second harmonic has two segments, or loops, along the string.
Each harmonic has a distinct frequency, which increases with the harmonic number. The frequency of each harmonic can be calculated by understanding the system's physical conditions, including string length, tension, and mass per unit length.

String Tension

String tension is a critical variable in determining both the frequency of the harmonics and the formation of standing waves in a string. The tension of the string affects the speed at which waves travel along it and, consequently, the frequencies at which the string vibrates.
In our specific problem, the tension in the string must be calculated to know the frequency of harmonics. Using the formula for the second harmonic frequency\[ f_2 = \frac{n}{2L} \sqrt{\frac{T}{\mu}} \]where \( n = 2 \), one can manipulate the equation to solve for \( T \), the tension. The tension impacts how the harmonics, especially the second harmonic at 25 Hz, are manifested in physical terms.
Understanding how to calculate and adjust tension aids in manipulating the string's vibration to achieve desired frequencies. A change in tension inevitably alters the harmonic frequencies, allowing for precise control of wave phenomena.

Frequency Calculation

Calculating frequencies for different harmonics requires knowledge of specific physical parameters like string length (\(L\)), string tension (\(T\)), and mass per unit length (\(\mu\)). The exercise presents a situation where different frequencies need to be calculated based on these parameters.
The frequency of the first harmonic, often called the fundamental frequency, can be calculated with\[ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \],where \(T\) is the tension found in prior calculations. Frequency calculations allow for predicting how a string will vibrate under various conditions, such as altered tension aiming to increase frequency by a specific percentage.
By calculating new tension indicated by the formula for adjusted frequency\[ T' = 4(f_1')^2 L^2 \mu \],it's possible to modify the system to achieve a target frequency, providing insight into the interplay between physical properties and wave behavior.