Problem 114 - College Algebra with Modeling and Visualization

Question

The frequency \(F\) of \(a\) vibrating string is directly proportional to the square root of the tension \(T\) on the string and inversely proportional to the length \(L\) of the string. Give two ways to double the frequency \(F\).

Step-by-Step Solution

Answer

Increase tension to 4 times or reduce length to half.

Step 1: Understand the Relationship

The problem states that the frequency \( F \) is directly proportional to the square root of the tension \( T \), and inversely proportional to the length \( L \) of a string. Mathematically, this relationship can be written as \( F = k \frac{\sqrt{T}}{L} \), where \( k \) is the constant of proportionality.

Step 2: Set Up the Initial Frequency Equation

Assuming the initial conditions are such that the initial frequency is \( F_1 = k \frac{\sqrt{T}}{L} \). Our goal is to double this frequency to achieve \( F_2 = 2F_1 \).

Step 3: First Method: Increase Tension

To double the frequency, we can increase the tension. We know \( F = k \frac{\sqrt{T}}{L} \), so to make \( F_2 = 2F_1 \), the new tension \( T_2 \) should be such that \( \sqrt{T_2} = 2\sqrt{T} \). Solving for \( T_2 \), we have \( T_2 = 4T \).

Step 4: Second Method: Decrease Length

Alternatively, to double the frequency, we can decrease the length. Knowing \( F = k \frac{\sqrt{T}}{L} \), we set the new length \( L_2 \) to enhance frequency to \( F_2 = 2F_1 \). Solving \( 2F = k \frac{\sqrt{T}}{L_2} \), we get \( L_2 = \frac{L}{2} \). The frequency doubles by halving the length.

Key Concepts

Proportionality Constant

In mathematics and physics, a proportionality constant is a value that relates two variables that are directly proportional. When we say that a variable, say frequency ( \( F \)), is directly proportional to another variable, for example, the square root of tension ( \( \sqrt{T} \)), it implies there is a constant ( \( k \)) effectively linking them. This can be written in the formula \[ F = k \frac{\sqrt{T}}{L} \] where \( k \) is the proportionality constant, and \( L \) represents another variable, the length, which is inversely proportional. **Importance of \( k \) in Direct Variation:** * It gives a precise mathematical relation between frequency, tension, and length. * Without \( k \), the equation wouldn't accurately represent real-world behavior of vibrating strings. By understanding this concept, we can manipulate the variables to achieve desired outcomes, like doubling the frequency, by adjusting either the tension or length accordingly.

Tension in Physics

Tension is a force that stretches a material, such as a string in this context, and is crucial in physics and engineering. When discussing a vibrating string, the tension directly influences the vibration frequency. As tension increases, the square root of tension becomes more significant, influencing frequency calculations through the equation: \[ F = k \frac{\sqrt{T}}{L} \] **Understanding Tension's Role in Frequency:** * Higher tension increases the frequency, making the string vibrate faster. * To double the frequency, the tension must be quadrupled, as shown mathematically by ensuring \( T_2 = 4T \). This relationship explains why musical instruments, such as guitars, sound higher in pitch when strings are tightened. It's the increase in frequency driven by greater tension.

Frequency and Wavelength

Frequency and wavelength are fundamental concepts in wave dynamics, closely associated with how waves travel through different media. In the case of a vibrating string: * **Frequency (\(F\))** is the number of vibrations per second, measured in Hertz (Hz). * **Wavelength** is the distance between two consecutive points in phase on the wave, such as from peak to peak. **Relation Between Frequency and Wavelength:** * When the length of the string decreases, the frequency increases if everything else remains constant. That's why halving the length doubles the frequency, as in \( L_2 = \frac{L}{2} \). * The speed of the wave on a string (\( v \)) is found using the equation \( v = F \times \lambda \), where \( \lambda \) is the wavelength. **Why This Matters:** Waves with higher frequencies have shorter wavelengths. In practical terms, shortening a guitar string by pressing on it raises its pitch, inviting sound waves with higher frequencies and lower wavelengths.