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Ch 15: Mechanical Waves
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 15: Mechanical WavesProblem 39a
Chapter 15, Problem 39a

A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the antinodes of 0.300 cm. What is the speed of propagation of transverse waves in the wire?

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First, understand that the wire is vibrating in its fundamental mode. In this mode, the wire forms a single loop with nodes at each end and an antinode in the middle. The length of the wire is equal to half the wavelength of the wave.
Calculate the wavelength (λ) of the wave. Since the wire is 80.0 cm long and vibrating in its fundamental mode, the wavelength is twice the length of the wire. Therefore, λ = 2 * 80.0 cm = 160.0 cm.
Use the formula for wave speed (v), which is given by v = f * λ, where f is the frequency of the wave and λ is the wavelength. Substitute the given values: f = 60.0 Hz and λ = 160.0 cm (convert to meters: 1.60 m).
Perform the multiplication to find the speed of propagation: v = 60.0 Hz * 1.60 m.
The result from the multiplication will give you the speed of propagation of transverse waves in the wire in meters per second (m/s).

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Key Concepts

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

Wave Speed on a String

The speed of a wave on a string is determined by the tension in the string and its linear mass density. It is given by the formula v = sqrt(T/μ), where T is the tension and μ is the mass per unit length. This concept is crucial for calculating the speed of transverse waves in the wire.
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Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system vibrates. For a string fixed at both ends, the fundamental frequency is given by f = v/(2L), where v is the wave speed and L is the length of the string. This concept helps relate the frequency of vibration to the wave speed and string length.
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Linear Mass Density

Linear mass density (μ) is the mass per unit length of a string or wire, calculated as μ = m/L, where m is the mass and L is the length. It is a key factor in determining the wave speed on a string, as it affects how mass is distributed along the string's length.
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