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 46

Chapter 17, Problem 46

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

Verified step by step guidance

The fundamental frequency of a string under tension is given by the formula:

f = \frac{1}{2} l \sqrt{\frac{T}{ρ}}

, where

f

is the fundamental frequency,

l

is the length of the string,

T

is the tension, and

ρ

is the linear mass density of the string.

Notice that the length

l

and the linear mass density

ρ

remain constant in this problem. Therefore, the fundamental frequency depends on the square root of the tension:

f \propto \sqrt{T}

If the tension is doubled, the new tension becomes

T' = 2 T

. Substituting this into the proportionality, the new frequency

f'

is related to the original frequency

f

by:

\frac{f}{'} = \sqrt{\frac{T}{'}}

Simplify the ratio of tensions:

\frac{T}{'} = 2

. Therefore,

\frac{f}{'} = \sqrt{2}

, which means the new frequency is

f' = f \sqrt{2}

Finally, substitute the given fundamental frequency of 220 Hz into the equation:

f' = 220 \sqrt{2}

. This gives the new fundamental frequency.

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Key Concepts

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

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system oscillates. In the context of a vibrating string, it is determined by the string's length, tension, and mass per unit length. This frequency is crucial for understanding how strings produce sound, as it corresponds to the pitch of the note played.

Tension in a String

Tension refers to the force applied along the length of a string, which affects its vibration characteristics. Increasing the tension in a string raises its fundamental frequency, as a tighter string vibrates faster. This relationship is essential for predicting how changes in tension will influence the sound produced by the string.

Relationship Between Tension and Frequency

The relationship between tension and frequency in a vibrating string is described by the formula f = (1/2L)√(T/μ), where f is the frequency, L is the length of the string, T is the tension, and μ is the mass per unit length. Doubling the tension results in an increase in frequency, specifically, the frequency increases by a factor of √2, illustrating how tension directly influences the pitch of the sound produced.

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