Problem 66 - College Physics

Question

A piece of steel string is under tension. (a) If the tension doubles, the transverse wave speed (1) doubles, (2) halves, (3) increases by \(\sqrt{2},\) (4) decreases by \(\sqrt{2}\). Why? (b) If the linear mass density of a 10.0 -m length of string is \(0.125 \mathrm{~kg} / \mathrm{m}\) and it is under a tension of \(9.00 \mathrm{~N}\), what is the transverse wave speed in the string? (c) What are its waves' natural frequencies?

Step-by-Step Solution

Answer

(a) Transverse wave speed increases by \( \sqrt{2} \). (b) The wave speed is 8.49 m/s. (c) Fundamental frequency is 0.4245 Hz.

Step 1: Understanding the Wave Speed and Tension Relationship

The speed of a transverse wave on a string is given by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( v \) is the wave speed, \( T \) is the tension in the string, and \( \mu \) is the linear mass density. If the tension doubles, \( T \) becomes \( 2T \). The new wave speed \( v' = \sqrt{\frac{2T}{\mu}} \) equates to \( \sqrt{2} \cdot \sqrt{\frac{T}{\mu}} = \sqrt{2} \cdot v \). Thus, the correct choice is (3), the speed increases by \( \sqrt{2} \).

Step 2: Calculating Transverse Wave Speed

We are given that \( T = 9.00 \, \text{N} \) and \( \mu = 0.125 \, \text{kg/m} \). The wave speed is \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{9.00}{0.125}} = \sqrt{72} \). Calculating further, \( v \approx 8.49 \, \text{m/s} \).

Step 3: Finding Natural Frequencies

Natural frequencies depend on the length of the string \( L = 10.0 \, \text{m} \). The fundamental frequency \( f_1 \) is given by \( f_1 = \frac{v}{2L} \). For the fundamental mode, substituting the calculated speed, \( f_1 = \frac{8.49}{2 \times 10.0} = 0.4245 \, \text{Hz} \). Higher harmonics can be found using \( f_n = n \cdot f_1 \), where \( n \) is a positive integer.

Key Concepts

Transverse Waves

Transverse waves are a type of wave where particles of the string move perpendicular to the direction of the wave's travel. Imagine flicking a skipping rope up and down; the wave travels horizontally along the rope, while the rope itself moves up and down. This movement creates a wave with characteristic oscillations, similar to water waves moving across the surface of a pool.
These waves require a medium, like a string, to travel through. The properties of this medium significantly affect how fast the wave will move. While in this article, we're focusing on strings, transverse waves also appear in other contexts, like light waves, though those don't need a medium.
Understanding how these waves work is crucial for various applications, including musical instrument design and various engineering fields.

Wave Speed Formula

The speed of a transverse wave on a string can be calculated using the formula: \[ v = \sqrt{\frac{T}{\mu}} \]where \(v\) is the wave speed, \(T\) is the tension in the string, and \(\mu\) is the linear mass density of the string.

This relationship shows how increasing the tension in the string increases the wave speed. If we double the tension, the formula becomes:\[ v' = \sqrt{\frac{2T}{\mu}} = \sqrt{2} \times v \]indicating that the wave speed increases by a factor of \(\sqrt{2}\).
On the other hand, the linear mass density refers to how much mass there is per unit length of the string. A heavier string (higher \(\mu\)) slows down waves, whereas a lighter string (lower \(\mu\)) allows them to travel faster.
By understanding this formula, one can predict how changes to the string's physical properties will affect wave movement, essential for tuning instruments and other practical uses.

Natural Frequencies

Natural frequencies, also known as resonant frequencies, are the frequencies at which an object or system naturally oscillates. For a string of a given length, the fundamental frequency is the simplest standing wave with the longest wavelength.
Using our previous example, the fundamental frequency is given by\[ f_1 = \frac{v}{2L} \]where \(f_1\) is the fundamental frequency, \(v\) is the wave speed, and \(L\) is the length of the string. In this case, the fundamental frequency for a string with a speed of \(8.49\, \text{m/s}\) and a length of \(10.0\, \text{m}\) is approximately \(0.4245\, \text{Hz}\).

Strings can also vibrate at higher harmonics, which are multiples of the fundamental frequency, such as \(2f_1, 3f_1\), and so on. These harmonics add richness to the sound and are significant in musical tuning.
Comprehending natural frequencies is vital for understanding resonance in musical instruments and architectural structures, among others.