Fixed Rope Fourier: Complete Expression of Motion

[Unicorn.]

Hi,

Homework Statement

A rope of length L, linear density u is streched with tension T between a wall and a ring of negligible mass free to move vertically along a rod without friction. As the ring is maintained at y=0, we give to the rod a y(x,t=0) = sin(πx/L) form. We release the ring and the rope at t=0.

Give the complete expression of motion of the rope in function of t in term of its Fourier components.

Homework Equations

The Attempt at a Solution

I'm really lost, I don't know from where I have to start.
Since it must be symetric around the point x=0 so Bn=0 and anti-symetric around x=L so An=0
I have to do the work twice and add them ? And integrate from 0 to the period P=4L ?

Thanks

 

[anathema]

for your question! it's important to have a clear and structured approach to problem solving. Here are some steps you can follow to tackle this problem:

1. Start by defining your variables and constants. In this case, you have the length of the rope (L), linear density (u), and tension (T). You also have the initial position of the rod (y(x,t=0) = sin(πx/L)).

2. Next, write down the equation of motion for the rope. This can be expressed as a wave equation, since the rope is being stretched and released. It will have a time-dependent term (t) and a spatial term (x). You can also include the constants and variables you defined in step 1.

3. Now, you can use the method of separation of variables to solve this wave equation. This involves assuming a solution of the form y(x,t) = X(x)T(t), and then plugging it into the wave equation to get two separate equations for X and T. These equations will have a constant (λ) that you can solve for.

4. Once you have the solutions for X and T, you can use the initial condition (y(x,t=0) = sin(πx/L)) to solve for the Fourier coefficients, An and Bn. Remember that these coefficients represent the amplitude of each Fourier component in the solution.

5. Finally, you can write the complete expression for the motion of the rope in function of t by using the Fourier series formula. This involves summing up all the Fourier components, which will be given by the coefficients you found in step 4.

I hope this helps guide you in the right direction! Remember to always start by defining your variables and equations, and then use a systematic approach to solve the problem. Good luck!