Question

Length of a steel wire is \(11 \mathrm{~m}\) and its mass is \(2.2 \mathrm{~kg}\). What should be the tension in the wire so that the speed of a transverse wave in it is equal to the speed of sound in dry air at \(20^{\circ} \mathrm{C}\) temperature? (A) \(2.31 \times 10^{4} \mathrm{~N}\) (B) \(2.25 \times 10^{4} \mathrm{~N}\) (C) \(2.06 \times 10^{4} \mathrm{~N}\) (D) \(2.56 \times 10^{4} \mathrm{~N}\)

Short answer

The correct answer is (A) \(2.31 \times 10^{4} \mathrm{~N}\).

Step-by-step solution

Calculate the Linear Density of the Wire

Linear density is defined as mass per unit length. We are given the length (\(L\)) of the steel wire as \(11 \mathrm{~m}\) and its mass (\(M\)) as \(2.2 \mathrm{~kg}\). The linear density (\(\mu\)) can be found using the following equation: \[\mu = \frac{M}{L}\] Plugging in the values, we get: \[\mu = \frac{2.2}{11} = 0.2 \mathrm{~kg/m}\]

Find the Speed of Sound in Dry Air at \(20^{\circ} \mathrm{C}\)

The speed of sound in dry air can be calculated using the following equation: \[v = 331.4 + 0.6T\] Where \(v\) is the speed of sound, and \(T\) is the temperature in Celsius. Plugging in the given temperature, we get: \[v = 331.4 + 0.6 \times 20 = 343 \mathrm{~m/s}\]

Calculate the Tension in the Wire Using the Wave Speed Formula

The wave speed (\(v\)) in a string can be calculated using the following equation: \[v = \sqrt{\frac{T}{\mu}}\] Where \(T\) is the tension in the string, and \(\mu\) is the linear density. We aim to find the tension that results in a wave speed equal to the speed of sound in dry air (\(343 \mathrm{~m/s}\)). Rearranging the equation to find tension, we get: \[T = v^{2} \times \mu\] Plugging in the values, we get: \[T = 343^{2} \times 0.2 = 23589.4 \mathrm{~N}\]

Find the Closest Option

Now that we have calculated the tension in the wire, we can compare our result with the given options. The closest option is: (A) \(2.31 \times 10^{4} \mathrm{~N}\) So, the correct answer is (A).

Key concepts

Linear Density

Linear density is an important property of strings in wave mechanics. It helps to determine how waves travel across a string. Simply put, linear density is the mass of the string per unit length. This means it tells us how heavy a string is for each meter of its length.
For example, if you have a steel wire that is 11 meters long and has a total mass of 2.2 kilograms, you can find its linear density by dividing the mass by the length. This is given by the formula \[ \mu = \frac{M}{L} \]where:

• \( M \) is the mass of the string.
• \( L \) is the length of the string.
• \( \mu \) is the linear density.

Using this formula, you would find that the linear density of the steel wire is \( 0.2 \text{ kg/m} \). This means that for every meter of the wire, it has a mass of 0.2 kilograms.
Linear density helps us calculate the wave speed on the string, as it is a crucial factor in the wave speed formula.

Tension in Strings

Tension in a string is the force that is applied to it to keep it taut. This force is vital because it directly affects the speed at which waves travel through the string.
The wave speed in the string is determined not only by its tension but also by its linear density. The relationship is captured in the wave speed formula:\[ v = \sqrt{\frac{T}{\mu}} \]where:

• \( v \) is the wave speed.
• \( T \) is the tension in the string.
• \( \mu \) is the linear density.

To find the tension necessary for the wave speed to match a certain value (like the speed of sound), we can rearrange this formula to solve for tension:\[ T = v^{2} \times \mu \]By substituting in the values for wave speed and linear density, one can find the tension needed for specific conditions. For example, if a wave speed of 343 m/s is desired, and linear density is 0.2 kg/m, the tension required is 23589.4 N. It's critical to apply the correct tension to achieve the desired wave speed.

Speed of Sound

Understanding the speed of sound is crucial when studying wave speed in strings, especially when you want a wave to match the speed of sound, like in this exercise.
The speed of sound in air varies with temperature. At 20°C, you can calculate it using the formula:\[ v = 331.4 + 0.6T \]where:

• \( v \) is the speed of sound in meters per second (m/s).
• \( T \) is the temperature in degrees Celsius.

At 20°C, this equation tells us that the speed of sound is 343 m/s. This speed is crucial because it serves as a benchmark for wave speed in other mediums like strings.
The goal is to set the parameters of a string, such as its tension, so that the wave speed on the string matches this value. Achieving the correct wave speed in a string allows us to create waves that transfer sound efficiently, which is essential in many applications, from musical instruments to engineering solutions.