Ch. 15 - Wave Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 15 - Wave Motion

Problem 57

Chapter 15, Problem 57

A particular violin string plays at a frequency of 294 Hz. If the tension is increased 22%, what will the new frequency be?

Verified step by step guidance

The frequency of a vibrating string is given by the formula:

f = \frac{1}{2} L \sqrt{\frac{T}{μ}}

, where

f

is the frequency,

L

is the length of the string,

T

is the tension, and

μ

is the linear mass density of the string.

Since the length

L

and the linear mass density

μ

remain constant, the frequency is proportional to the square root of the tension:

f ∝ \sqrt{T}

Let the initial tension be

T

and the initial frequency be

f

₁ = 294 Hz. The new tension is increased by 22%, so the new tension is

T

₂ = 1.22T.

The new frequency

f

₂ can be expressed as:

f

₂f₁ = T₂T₁. Substituting

T

₂ = 1.22T and

T

₁ = T, we get:

f

₂f₁ = 1.22.

Finally, solve for

f

₂ by multiplying both sides of the equation by

f

₁:

f

₂ = f₁1.22. Substitute

f

₁ = 294 Hz to find the new frequency.

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Key Concepts

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

Frequency

Frequency is the number of cycles of a periodic wave that occur in one second, measured in hertz (Hz). In the context of a vibrating string, frequency is directly related to the pitch of the sound produced. The frequency of a string can be influenced by factors such as tension, length, and mass per unit length.

Tension in a String

Tension refers to the force applied along the length of a string, which affects its vibration and, consequently, the frequency of the sound it produces. Increasing the tension of a string generally raises its frequency, as a tighter string vibrates faster. The relationship between tension and frequency is described by the formula: frequency is proportional to the square root of tension.

Wave Equation for Strings

The wave equation for strings relates the frequency of a vibrating string to its tension and linear mass density. Specifically, the frequency (f) can be calculated using the formula f = (1/2L) * √(T/μ), where L is the length of the string, T is the tension, and μ is the mass per unit length. This equation highlights how changes in tension directly affect the frequency of the sound produced by the string.

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Energy & Power of Waves on Strings

Related Practice

Textbook Question

The displacement of a standing wave on a string is given by D = 2.4sin(0.60x)cos(42t), where x and D are in centimeters and t is in seconds. Give the amplitude, frequency, and speed of each of the component waves.

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Textbook Question

When you slosh the water back and forth in a tub at just the right frequency, the water alternately rises and falls at each end, remaining relatively calm at the center. Suppose the frequency to produce such a standing wave in a 45-cm-wide tub is 0.85 Hz. What is the speed of the water wave?

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Textbook Question

One end of a horizontal string is attached to a small-amplitude mechanical 60.0-Hz oscillator. The string’s mass per unit length is 3.9 x 10⁻ ⁴ kg/m. The string passes over a pulley, a distance ℓ = 1.50 m away, and weights are hung from this end, Fig. 15–38. What mass m must be hung from this end of the string to produce five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true.

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Textbook Question

(II) For any type of wave that reaches a boundary beyond which its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle θ_iM corresponds to an angle of refraction equal to 90°. If θᵢ > θ_iM, all the wave is reflected at the boundary and none is refracted, because this would correspond to sin θᵣ > 1 (where is the angle θᵣ of refraction), which is impossible.

(a) Find a formula for θ_iM using the law of refraction, Eq. 15–19.

Textbook Question

The displacement of a standing wave on a string is given by D = 2.4 sin ( 0.60x ) cos (42t) , where x and D are in centimeters and t is in seconds. What is the distance (cm) between nodes?

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