How Does Doubling the Radius of a String Affect Its Wave Speed?

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Doubling the radius of a string affects its wave speed due to changes in linear density (μ). The wave speed formula, v = √(T/μ), indicates that as the radius increases, the mass per unit length also increases, which in turn affects the wave speed. Specifically, the volume of the string increases with the square of the radius, leading to a higher mass for the same length. Although tension remains constant, the increased linear density results in a slower wave speed. Understanding these relationships is crucial for solving the problem effectively.
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Homework Statement



A wave travels along a string at a speed of 261 m/s. What will be the speed if the string is replaced by one made of the same material and under the same tension but having twice the radius?

Homework Equations



v=squareroot(T/mu) (where T=tension)

mu=m/L (where m= mass and L= length)

The Attempt at a Solution



I'm not sure how I need to manipulate mu to accomadate for twice the radius. Volume would change, but wouldn't mass as well? I thought maybe I would need to use density, but we are not given the density of the string. Do I assume the mass doesn't change because the tension is the same? Fatter strings are supposed to go slower. So I'm really stuck on what to do..Help please?!
 
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Doubling the radius will change the linear density by a certain factor, which you'll need to figure out. I.e., if the radius is twice as big, then the volume (and hence mass) of a 1 m long string will be larger by what factor?
 

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