Question

A particular steel guitar string has a mass per unit length of \(1.93 \mathrm{~g} / \mathrm{m}\). a) If the tension on this string is \(62.2 \mathrm{~N},\) what is the wave speed on the string? b) For the wave speed to be increased by \(1.00 \%\), how much should the tension be changed?

Short answer

Answer: The tension should be changed by approximately 1.20 N.

Step-by-step solution

Recall the wave speed formula

The wave speed \(v\) on a string can be determined using the formula: \(v = \sqrt{\frac{T}{\mu}}\) where \(T\) is the tension on the string and \(\mu\) is the mass per unit length of the string.

Calculate the wave speed of the string

We are given the mass per unit length \(\mu = 1.93\ \text{g/m}\) and tension \(T = 62.2\ \text{N}\). To find the wave speed, we should first convert \(\mu\) to consistent units (kg/m): \(\mu = 1.93\ \text{g/m} \cdot \frac{1\ \text{kg}}{1000\ \text{g}} = 0.00193\ \text{kg/m}\) Now we plug the given values into the wave speed formula: \(v = \sqrt{\frac{62.2\ \text{N}}{0.00193\ \text{kg/m}}} \approx 179.73\ \text{m/s}\) So, the wave speed on the string is approximately \(179.73\ \text{m/s}\).

Calculate the desired increase in wave speed

We want to increase the wave speed by \(1.00\%\). To find the target wave speed, we can calculate: \(v_{target} = v \cdot (1 + \frac{\Delta \%}{100}) = 179.73\ \text{m/s} \cdot 1.01 \approx 181.53\ \text{m/s}\) The target wave speed is approximately \(181.53\ \text{m/s}\).

Determine the required tension for the target wave speed

To find the tension required to achieve the target wave speed, we can rearrange the wave speed formula for tension: \(T_{target} = \mu \cdot v_{target}^2 = 0.00193\ \text{kg/m} \cdot (181.53\ \text{m/s})^2 \approx 63.40\ \text{N}\) The required tension for the target wave speed is approximately \(63.40\ \text{N}\).

Calculate the required change in tension

Now we can calculate the required change in tension to achieve the target wave speed: \(\Delta T = T_{target} - T = 63.40\ \text{N} - 62.2\ \text{N} \approx 1.20\ \text{N}\) To achieve a \(1.00 \%\) increase in wave speed, the tension should be changed by approximately \(1.20\ \text{N}\).

Key concepts

Wave Speed Formula

Understanding the wave speed on a string is crucial for a variety of applications, from musical instruments to engineering designs. The speed at which a wave travels along a string, or its wave speed, is calculated using the wave speed formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
In this formula, \( v \) represents the wave speed, \( T \) denotes the tension in the string, and \( \mu \) is the mass per unit length of the string. It's clear that wave speed increases with greater tension and decreases with increased mass per unit length. For instance, think of a guitar string; when you tighten the string (increase the tension), the pitch of the note (which corresponds to the wave speed) goes up.

• The tension \( T \) refers to the stretching force applied along the string.
• The mass per unit length \( \mu \) is how much mass is distributed along a unit of the string's length.

To apply this formula, ensure that the units for tension are in newtons (N) and mass per unit length in kilograms per meter (kg/m). Proper unit conversion is essential to obtain a correct result, as demonstrated in the step-by-step textbook solution.

Tension in Strings

Tension is a force and is measured in newtons (N). It's important in the context of wave speed on a string because it is directly proportional to the wave speed squared. As the force causing the string to stretch, tension determines the medium's resistance to deformation and the restitution speed, which are crucial for wave propagation.

• High tension increases wave speed, producing higher-pitched sounds in musical strings.
• Low tension slows down wave speed, resulting in lower-pitched sounds.

In physics problems, such as calculating how much tension is needed to achieve a certain wave speed, this concept is essential. If you're asked to change the wave speed by a percentage, like in our exercise, the change in tension is necessary. This relationship is rooted in the elasticity and mass distribution of the material, highlighting the significance of understanding both the physical properties of the string and the mathematical relationship described by the wave speed formula.

Mass Per Unit Length

The mass per unit length, symbolized as \( \mu \), is pivotal to determining the wave speed on a string, as seen in the formula \( v = \sqrt{\frac{T}{\mu}} \). It represents the linear density or the amount of mass distributed along a single unit of the string's length and is expressed in kilograms per meter (kg/m).

• Thicker or denser strings, which have more mass per unit length, will produce waves that travel more slowly.
• Thinner or lighter strings result in faster wave propagation.

In the context of the exercise, converting the given mass per unit length from grams per meter (g/m) to kilograms per meter (kg/m) is fundamental for consistency with the units of tension (N). Calculating wave speed or changes required in tension to alter the wave speed hinges on the accurate calculation of this property, stressing the importance of meticulous conversion and application of the mass per unit length in the wave speed formula.