Hey there! Sign in to join this conversationNew here? Join for free
x Turn on thread page Beta
    • Thread Starter
    Offline

    17
    ReputationRep:
    Name:  30.png
Views: 63
Size:  206.1 KB

    this is my working:
    http://imgur.com/Z8aqSLv&yUVHKS0


    i have no real idea what i'm doing
    Offline

    13
    ReputationRep:
    (Original post by StarvingAutist)
    Name:  30.png
Views: 63
Size:  206.1 KB

    this is my working:
    http://imgur.com/Z8aqSLv&yUVHKS0


    i have no real idea what i'm doing
    I've not worked through this, but I notice that you have immediately changed variables to x-y and x+y. Have you tried following literally the instructions of the question and working directly with  x, y via x = x_0 \cos(\omega t + \alpha) and y= y_0 \cos(\omega t + \beta)?

    A quick google suggested to me that this is the standard way to go for the coupled pendulum!
    Offline

    11
    ReputationRep:
    (Original post by StarvingAutist)
    Name:  30.png
Views: 63
Size:  206.1 KB

    this is my working:
    http://imgur.com/Z8aqSLv&yUVHKS0


    i have no real idea what i'm doing
    This looks like a normal modes problem which is the kind of problem which arises when you couple oscillators together. It turns out that you can find a number of independent modes of oscillation of the system, and the general motion of the system is a linear combination of those modes.

    A "mode" is a motion where, say, all of the masses are moving in the same direction at the same time, then in terms of your problem, the expression x+y is useful, or when 1/2 the masses are moving in the -ve direction and 1/2 in the +ve direction, then the expression x-y is useful.

    These expressions are "useful" because they allow you to write DEs purely in terms of those expressions e.g. you get say:

    \ddot{x}+\ddot{y} = k(x+y)
    \ddot{x}-\ddot{y} = k(x-y)

    and you can introduce new variables p=x+y,q=x-y and get the decoupled DEs

    \ddot{p}=kp \\ \ddot{q}=kq

    When you do this, the p,q (or their x,y equivs) are called the normal coordinates of the system. They are the combinations of the original coords that allow you to write solveable DEs.

    To formalise this, you usually write things in matrix form so you turn your eqns into, say,

    \begin{pmatrix} \ddot{x} \\ \ddot{y} \end{pmatrix} =\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} (*)

    and then take a trial solution of:

    \begin{pmatrix} x \\ y \end{pmatrix} =\begin{pmatrix} A \\ B \end{pmatrix} e^{i \omega t}

    When you plug this into (*), you get an eigenvalue problem, whose eigenvectors are the A,B column vector, and whose eigenvalues give you the fundamental frequencies of vibration of the system (i.e. you will get a set of possible values for \omega - those are the fundamental frequencies of the normal modes).

    It turns out that the eigenvectors specify the combinations of x,y which give you your normal modes, so for this problem, you would expect to find an eigenvector:

    \begin{pmatrix} A \\ B \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}

    You'll note that the process of solving for the eigenvectors only gives the ratio of A to B, rather than their specific values.

    At the end, you form a general linear combination of the normal modes to give the general solution, and at that point you can combine the complex e^{-i\omega t}s into real sine and cosine solutions.
    Offline

    19
    ReputationRep:
    (Original post by atsruser)
    ....
    very impressive
    Offline

    11
    ReputationRep:
    (Original post by TeeEm)
    very impressive
    I aim to please.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: February 11, 2016
Poll
Do you agree with the proposed ban on plastic straws and cotton buds?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.