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

[aa1515]

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?!

 

[Redbelly98]

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?

 

[thomate1]

I would approach this problem by first considering the fundamental equation for wave velocity on a string, v=squareroot(T/mu). This equation relates the wave velocity (v) to the tension (T) and the mass per unit length (mu) of the string. In this case, we are given that the tension remains the same, so we can focus on how the mass per unit length is affected by changing the radius of the string.

Intuitively, we know that if the string has a larger radius, it will have a greater cross-sectional area and therefore a greater mass per unit length. This is because the mass of the string is distributed over a larger area. However, we also know that the density of the string remains the same, as it is made of the same material. So, the mass per unit length (mu) can be calculated by dividing the total mass by the total length of the string.

To find the new mass per unit length for the string with twice the radius, we can use the equation mu=m/L, where m is the total mass and L is the total length. Since the material and tension remain the same, the mass of the string will change proportionally to the change in radius. So, if the radius is doubled, the mass will also double. This means that the new mass per unit length will be twice the original value.

Substituting this new value for mu into the equation v=squareroot(T/mu), we find that the new wave velocity will be v=squareroot(T/2mu). This means that the wave will travel at a slower speed on the new string, as expected. We can also confirm this by plugging in numerical values for T and mu and solving for v.

In conclusion, when the string is replaced by one with twice the radius, the wave velocity will decrease by a factor of the square root of 2. This is because the mass per unit length of the string is doubled, causing the wave to travel at a slower speed.