Ch 16: Traveling Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

Ch 16: Traveling Waves

Problema 62

Capitolo 16, Problema 62

A string that is under 50.0 N of tension has linear density 5.0 g/m. A sinusoidal wave with amplitude 3.0 cm and wavelength 2.0 m travels along the string. What is the maximum speed of a particle on the string?

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Step 1: Convert the linear density from grams per meter to kilograms per meter. Since 1 g = 0.001 kg, the linear density becomes $ \mu = 5.0 \times 10^{-3} \, \text{kg/m} $.

Step 2: Calculate the wave speed $ v $ using the formula $ v = \sqrt{\frac{T}{\mu}} $, where $ T $ is the tension in the string (50.0 N) and $ \mu $ is the linear density. Substitute the values into the formula.

Step 3: Determine the angular frequency $ \omega $ of the wave using the relationship $ \omega = \frac{2\pi v}{\lambda} $, where $ \lambda $ is the wavelength (2.0 m) and $ v $ is the wave speed calculated in Step 2.

Step 4: The maximum speed of a particle on the string is given by $ v_{\text{max}} = \omega A $, where $ A $ is the amplitude of the wave (3.0 cm or 0.03 m) and $ \omega $ is the angular frequency calculated in Step 3.

Step 5: Substitute the values of $ \omega $ and $ A $ into the formula $ v_{\text{max}} = \omega A $ to express the maximum speed of a particle on the string.

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Tension in a String

Tension refers to the force exerted along the length of a string or rope, which affects how waves propagate through it. In this context, the tension of 50.0 N influences the wave speed and particle motion on the string. Higher tension generally results in faster wave propagation.

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Percorso guidato

04:39

Energy & Power of Waves on Strings

Linear Density

Linear density is defined as the mass per unit length of a string, typically expressed in grams per meter (g/m). It plays a crucial role in determining the wave speed on the string, as it affects how much mass is being moved by the tension. In this case, a linear density of 5.0 g/m will influence the maximum speed of particles on the string.

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Percorso guidato

8:13

Intro to Density

Wave Speed and Particle Motion

The speed of a wave on a string is determined by the tension and linear density, described by the formula v = √(T/μ), where T is tension and μ is linear density. The maximum speed of a particle on the string is related to the wave's amplitude and frequency. Understanding these relationships is essential for calculating the maximum speed of particles in the given wave scenario.

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Percorso guidato

07:19

Intro to Waves and Wave Speed

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