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18. Waves & Sound

Standing Waves

Physics

18. Waves & Sound

Standing Waves

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5:49 minutes

Problem 38

Textbook Question

Tendons are, essentially, elastic cords stretched between two fixed ends. As such, they can support standing waves. A woman has a 20-cm-long Achilles tendon—connecting the heel to a muscle in the calf—with a cross-section area of 90 mm² . The density of tendon tissue is 1100 kg/m³. For a reasonable tension of 500 N, what will be the fundamental frequency of her Achilles tendon?

Verified step by step guidance

Step 1: Understand the problem. The Achilles tendon is modeled as a stretched string that can support standing waves. The fundamental frequency corresponds to the lowest frequency of vibration, where the tendon has a single antinode in the middle. The formula for the fundamental frequency is \( f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \( L \) is the length of the tendon, \( T \) is the tension, and \( \mu \) is the linear mass density.

Step 2: Calculate the linear mass density \( \mu \). The linear mass density is the mass per unit length of the tendon, given by \( \mu = \frac{m}{L} \). The mass \( m \) can be found using the formula \( m = \rho \cdot V \), where \( \rho \) is the density of the tendon tissue and \( V \) is its volume. The volume \( V \) is the product of the cross-sectional area \( A \) and the length \( L \): \( V = A \cdot L \).

Step 3: Substitute the given values to find \( \mu \). The density \( \rho \) is 1100 kg/m^3, the cross-sectional area \( A \) is 90 mm^2 (convert to m^2: \( 90 \times 10^{-6} \) m^2), and the length \( L \) is 20 cm (convert to meters: \( 0.2 \) m). First, calculate the volume \( V \), then the mass \( m \), and finally the linear mass density \( \mu \).

Step 4: Substitute \( \mu \) and \( T \) into the formula for the fundamental frequency. The tension \( T \) is given as 500 N. Use the formula \( f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \( L \) is 0.2 m, \( T \) is 500 N, and \( \mu \) is the linear mass density calculated in the previous step.

Step 5: Simplify the expression to find the fundamental frequency \( f_1 \). Perform the square root and division operations as needed. The result will give the fundamental frequency in hertz (Hz).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. In the context of a tendon, these waves can form due to the tension and elasticity of the material, creating nodes and antinodes along its length. Understanding standing waves is crucial for analyzing how tendons can vibrate and transmit frequencies.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system can vibrate, corresponding to the first harmonic. For a fixed string or cord, this frequency depends on the length, tension, and mass per unit length of the material. In the case of the Achilles tendon, calculating the fundamental frequency involves using the tension and physical dimensions of the tendon.

Tension and Elasticity

Tension refers to the force applied along the length of an object, which in this case is the force exerted on the Achilles tendon. Elasticity is the ability of a material to return to its original shape after deformation. The relationship between tension, elasticity, and the physical properties of the tendon is essential for determining how it vibrates and the frequencies it can support.

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