What is the speed of the string? (wave problem)

[bored2death97]

Homework Statement

A wave travels along a spring at speed v0. 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.

Answer in book says (v0/2)

Homework Equations

v=√(T/μ)
speed= square root of (tension divided by mu)

μ=m/L
mu= mass/length

The Attempt at a Solution

With the first equation I filled in the equation for mu. Assuming twice the radius would mean twice the mass, I multiplied mass by 2.

v=√[T/(2m/L)]

v=√(TL/2m)

Then if I were to take the value of 2 out of the square root, I would get v=0.707[√(TL/m)]

Rearrange and get v/0.707

So where did I go wrong??

 

[Dick]

Homework Statement

A wave travels along a spring at speed v0. 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.

Answer in book says (v0/2)

Homework Equations

v=√(T/μ)
speed= square root of (tension divided by mu)

μ=m/L
mu= mass/length

The Attempt at a Solution

With the first equation I filled in the equation for mu. Assuming twice the radius would mean twice the mass, I multiplied mass by 2.

v=√[T/(2m/L)]

v=√(TL/2m)

Then if I were to take the value of 2 out of the square root, I would get v=0.707[√(TL/m)]

Rearrange and get v/0.707

So where did I go wrong??

Twice the radius doesn't mean twice the mass. If you double the radius of a circle by what factor does the area increase?

 

[bored2death97]

Twice the radius doesn't mean twice the mass. If you double the radius of a circle by what factor does the area increase?

It increases by a factor of 4. Thank you.