How to Determine Pulse Time in a Non-Uniform String?

[ssnakee]

Homework Statement

a non uniform string of length L and mass M, has a variable linear density given by μ=kx where k is the distance measured from one side of the string and k is a costant.
a) find that M=(kL²)/2
b) show that the time t required to a pulse generated from one side of the string arrives to the other side is given by
t=√8ML/9F; where F is the tension of in the string

Homework Equations

μ=m/L...(1)
v=√F/μ→→→t=Δx/√F/μ...(2)

The Attempt at a Solution

i got the first result
μ=kx=m/l→→→m=kx*L and becouse x=L→→→m=xL² and becouse the mass is not linear but it stil is distributed in all the string m=(xL²)/2
the second part of the problem is where i have no idea of what to do where the 8/9 come from?
and in the eq 2 the linear density is under the force but in the eq that the problem tells the linear density is above the force

 

[Spinnor]

Solved in general for given mass density μ(x) below,

http://www.math.vt.edu/people/russell/m2k_fsp04_nonunf.pdf

From,

https://www.google.com/search?safe=....1.1.0...0...1c.1.64.serp..0.1.76.xz1FCT7WnDQ