Calculate the fundamental frequency (f0) of the wire with a mass per unit length (μ) of 0.00025 kg/m and length of 0.4 m.
Understand the Problem
The question is asking us to calculate the fundamental frequency of a wire based on its mass per unit length and length. To solve this, we will use the formula for the fundamental frequency, which is given by f0 = (1/2L) * sqrt(T/μ), where T is the tension in the wire and μ is the mass per unit length. However, the tension is not provided, so we cannot directly calculate f0 without making assumptions.
Answer
$$ f_0 = \frac{1}{2} \sqrt{\frac{g}{L}} $$
Answer for screen readers
The fundamental frequency is given by the formula: $$ f_0 = \frac{1}{2} \sqrt{\frac{g}{L}} $$
Steps to Solve
- Identify known variables We will denote our known variables, where:
- $\mu$ is the mass per unit length of the wire.
- $L$ is the length of the wire.
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Express tension in terms of weight Since tension (T) is often due to the weight of the wire, we can express it as: $$ T = mg $$ where $m$ is the mass of the wire, and $g$ is the acceleration due to gravity (approximately $9.81 , m/s^2$).
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Calculate the mass of the wire The mass ($m$) of the wire can be found using: $$ m = \mu \cdot L $$ Now substitute $m$ in the tension formula: $$ T = (\mu \cdot L) \cdot g $$
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Substitute tension in the frequency formula Substituting $T$ into the fundamental frequency formula gives: $$ f_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} = \frac{1}{2L} \sqrt{\frac{\mu L g}{\mu}} $$ This simplifies to: $$ f_0 = \frac{1}{2L} \sqrt{L g} = \frac{1}{2} \sqrt{\frac{g}{L}} $$
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Final expression for frequency Thus, the final expression for the fundamental frequency of the wire can be simplified as: $$ f_0 = \frac{1}{2} \sqrt{\frac{g}{L}} $$
The fundamental frequency is given by the formula: $$ f_0 = \frac{1}{2} \sqrt{\frac{g}{L}} $$
More Information
This formula shows how the fundamental frequency of a wire is inversely related to the square root of its length. As the length of the wire increases, the frequency decreases. The acceleration due to gravity ($g$) is approximately $9.81 , m/s^2$, which is essential for calculating specific numerical values.
Tips
Common mistakes include:
- Forgetting to convert units correctly when dealing with mass or length.
- Misunderstanding the relationship between tension, mass per unit length, and weight of the wire.
- Incorrectly applying the formula by not simplifying properly.
To avoid these issues, ensure unit consistency and double-check each step of your calculations.
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