Energy in a long inextensible string

[throneoo]

Homework Statement

Find the maximum kinetic energy and total energy of the system if a string(with uniform linear density) of length L and mass M is oscillating as a standing wave with two fixed ends in its fundamental frequency f with amplitude 2A

Homework Equations

let x=0 and x=L be the coordinates of each end .
Assuming the solution to the wave equation is A(cos(kx-wt)-cos(kx+wt))
=2Asin(kx)sin(wt)
Thus the amplitude as a function of x is 2Asin(kx)
and kL=pi

w=2*pi*f

The Attempt at a Solution

Treating the string as a series of harmonic oscillator ,
the max.KE of each oscillator is then (1/2)[(w*2Asin(kx))^2](M/L)(dx) <--basically just half mv^2

then , integrating the expression w.r.t. x from x=0 to x=L gives

(M/L)(wA)^2* the integral of 2(sin(kx))^2

=(M/L)(wA)^2* the integral of 1-cos(2kx)
=(M/L)(wA)^2*L
=M(wA)^2

and by conservation of energy , max kinetic energy=the total energy of the system

I'm not really sure if it's a correct approach

 

[collinsmark]

View attachment 74863

Homework Statement

Find the maximum kinetic energy and total energy of the system if a string(with uniform linear density) of length L and mass M is oscillating as a standing wave with two fixed ends in its fundamental frequency f with amplitude 2A

Homework Equations

let x=0 and x=L be the coordinates of each end .
Assuming the solution to the wave equation is A(cos(kx-wt)-cos(kx+wt))
=2Asin(kx)sin(wt)
Thus the amplitude as a function of x is 2Asin(kx)
and kL=pi

w=2*pi*f

The Attempt at a Solution

Treating the string as a series of harmonic oscillator ,
the max.KE of each oscillator is then (1/2)[(w*2Asin(kx))^2](M/L)(dx) <--basically just half mv^2

then , integrating the expression w.r.t. x from x=0 to x=L gives

(M/L)(wA)^2* the integral of 2(sin(kx))^2

=(M/L)(wA)^2* the integral of 1-cos(2kx)
=(M/L)(wA)^2*L
=M(wA)^2

and by conservation of energy , max kinetic energy=the total energy of the system

I'm not really sure if it's a correct approach

Yes, that looks correct to me. :)

One thing though, the problem statement stated the frequency in terms of simple frequency, f. You should make the appropriate substitution your representation with ω = 2πf before submitting your final answer.

But yeah, that looks to be a valid approach.

 

[throneoo]

alright. thank you very much