Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 17: Superposition

Problem 40

Chapter 17, Problem 40

A violinist places her finger so that the vibrating section of a 1.0 g/m string has a length of 30 cm, then she draws her bow across it. A listener nearby in a 20°C room hears a note with a wavelength of 40 cm. What is the tension in the string?

Verified step by step guidance

Step 1: Start by identifying the relationship between the speed of the wave on the string and the frequency of the sound wave in the air. The frequency of the wave on the string is the same as the frequency of the sound wave in the air. Use the formula for wave speed in air:

v = f λ

, where

v

is the speed of sound in air,

f

is the frequency, and

λ

is the wavelength.

Step 2: Calculate the speed of sound in air at 20°C using the formula

v = 331 + 0.6 T

, where

T

is the temperature in Celsius. Substitute

T = 20

to find

v

Step 3: Use the speed of sound in air and the given wavelength of the sound wave (

λ = 40 cm

) to calculate the frequency of the wave using the formula

f = \frac{v}{λ}

Step 4: Relate the frequency of the wave to the speed of the wave on the string. The wave speed on the string is given by

v = f λ

, where

λ

is the wavelength of the wave on the string. For the fundamental frequency, the wavelength on the string is twice the length of the vibrating section of the string (

λ = 2 L

). Substitute

L = 30 cm

to find the wave speed on the string.

Step 5: Use the wave speed on the string to calculate the tension in the string. The wave speed on a string is related to the tension and linear mass density by the formula

v = \sqrt{\frac{T}{μ}}

, where

T

is the tension and

μ

is the linear mass density. Rearrange the formula to solve for

T

, and substitute the values for

v

and

μ = 1.0 g / m

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Key Concepts

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

Wave Speed

Wave speed is the rate at which a wave propagates through a medium. For a vibrating string, the wave speed (v) can be determined using the formula v = fλ, where f is the frequency and λ is the wavelength. Understanding wave speed is crucial for analyzing how tension and mass per unit length affect the vibrations of the string.

Tension in a String

Tension is the force exerted along the length of a string, which affects its vibration frequency and wave speed. The relationship between tension (T), mass per unit length (μ), and wave speed (v) is given by the equation v = √(T/μ). This concept is essential for calculating the tension in the string based on the observed wave properties.

Fundamental Frequency and Harmonics

The fundamental frequency is the lowest frequency at which a string vibrates, corresponding to the longest wavelength. For a string fixed at both ends, the fundamental frequency can be calculated using the formula f = v/λ. Understanding harmonics and how they relate to the length of the vibrating section of the string is key to solving for the tension in this scenario.

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Related Practice

Textbook Question

In a laboratory experiment, one end of a horizontal string is tied to a support while the other end passes over a frictionless pulley and is tied to a 1.5 kg sphere. Students determine the frequencies of standing waves on the horizontal segment of the string, then they raise a beaker of water until the hanging 1.5 kg sphere is completely submerged. The frequency of the fifth harmonic with the sphere submerged exactly matches the frequency of the third harmonic before the sphere was submerged. What is the diameter of the sphere?

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Textbook Question

A 2.0-m-long string vibrates at its second-harmonic frequency with a maximum amplitude of 2.0 cm. One end of the string is at x = 0 cm. Find the oscillation amplitude at x = 10, 20, 30, 40, and 50 cm.

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Textbook Question

A string under tension has a fundamental frequency of 220 Hz. What is the fundamental frequency if the tension is doubled?

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Textbook Question

A flute player hears four beats per second when she compares her note to a 523 Hz tuning fork (the note C). She can match the frequency of the tuning fork by pulling out the 'tuning joint' to lengthen her flute slightly. What was her initial frequency?

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Textbook Question

INT One end of a 75-cm-long, 2.5 g guitar string is attached to a spring. The other end is pulled, which stretches the spring. The guitar string's second harmonic occurs at 550 Hz when the spring has been stretched by 5.0 cm. What is the value of the spring constant?

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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?

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