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Understanding how to calculate the wave speed is crucial when dealing with waves on a string or wire. The wave speed dictates how quickly the wave travels through the medium. We calculate it using the formula: \[ v = \sqrt{\frac{T}{\mu}} \] where \( v \) is the wave speed, \( T \) is the tension in the wire, and \( \mu \) is the linear density.- **Tension (\( T \))**: This is the force stretching the wire. It is usually measured in newtons (N).- **Linear Density (\( \mu \))**: This is a measure of mass per unit length, usually in kilograms per meter (kg/m). For example, in the given problem, the tension is 250 N, and the linear density is calculated as 0.01 kg/m. Plugging these values into the formula, we obtain the wave speed \( v \approx 158.11 \text{ m/s} \). This value gives us how fast each pulse of the wave travels along the wire.

Linear density, often represented by the Greek letter \( \mu \), is an essential parameter in wave-related calculations. It measures how much mass is distributed along a unit length of the string or wire. To compute linear density, use the formula:\[ \mu = \frac{\text{mass}}{\text{length}} \]Consider the wire in the exercise, which has:- **Mass**: 100 g = 0.1 kg (since converting grams to kilograms involves multiplying by 0.001)- **Length**: 10.0 mThus, the linear density becomes \( \mu = \frac{0.1 \text{ kg}}{10.0 \text{ m}} = 0.01 \text{ kg/m} \).This value is critical because it affects the wave speed. A higher linear density indicates that the wire has more mass per unit length, typically resulting in slower wave speed if the tension remains constant.

The fundamental frequency is the lowest frequency at which a system can naturally vibrate. It is crucial for solving problems involving standing waves. The formula to determine the fundamental frequency \( f_1 \) on a wire fixed at both ends is:\[ f_1 = \frac{v}{2L} \]Where:- **\( v \)** is the wave speed- **\( L \)** is the length of the wireFor the given problem, substituting the values:- **Wave Speed, \( v \)**: 158.11 m/s- **Length, \( L \)**: 10.0 mThe fundamental frequency \( f_1 \) is calculated as \( f_1 = \frac{158.11 \text{ m/s}}{2 \times 10.0 \text{ m}} \approx 7.91 \text{ Hz} \).This is the lowest or first harmonic frequency at which standing waves can form on the wire. Understanding this concept is essential to determine other harmonics, which are integer multiples of the fundamental frequency.