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Chapter 14: Problem 5

If you doubled the tension in a string, what would happen to the speed of waves on the string?

Short Answer

Expert verified
When the tension in a string is doubled, the speed of waves on the string is increased by a factor of \( \sqrt{2} \), approximately 1.41, not doubled.

Step by step solution

01

State the speed-wave equation

In this step, express the speed of the wave, \( v \), using the equation \( v = \sqrt{\frac{T}{\mu}} \).
02

Replace T with 2T

Since the tension in the string, \( T \), is being doubled, replace \( T \) with \( 2T \) in the equation. This means that the formula becomes \( v = \sqrt{\frac {2T}{\mu}} \).
03

Simplify the formula

Simplify the formula by taking out the constant (\(2\)) out of the square root to get \( v = \sqrt{2} \times \sqrt{\frac{T}{\mu}} \).
04

Interpret the result

This result means that if the tension is doubled in the string, the wave speed would increase by a factor of \( \sqrt{2} \), approximately 1.41. So the wave speed is not doubled, but increased by this factor.

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