Coupled oscillator mass on spring question?

[coffeem]

Homework Statement

An object of mass m and another of mass M = 2m are connected to 3 springs of spring constant horixontally. The displacement of the two masses are defined as x and y. When x = y = 0, the springs are unextended.


a) Write down the two coupled equations of motion.

b) Define the term normal mode.

c) Find the frequencies of the normal modes.

d) Write down the time-dependence of x and y for the case of normal modes of motion and for the case of general motion.

e) Using appropriate sketched, describe the motion of the two masses for each normal mode.

The Attempt at a Solution

a) Write down the two coupled equations of motion.

mx1(dot dot) = -kx1 - k(x1-x2)

2mx2(dot dot) = -kx2 - k(x2-x1)

b) Define the term normal mode.

Mode at which all of the components in a couples system oscillate with the same frequency.

c) Find the frequencies of the normal modes.

I don't know how to do this but believe it involves finding a determinent.

d) Write down the time-dependence of x and y for the case of normal modes of motion and for the case of general motion.

Again I do not know how to do this.

e) Using appropriate sketched, describe the motion of the two masses for each normal
mode.

? Not sure.


Any help given would be appretiated.

 

[dx]

You said that the normal modes are motions where all parts vibrate with the same frequency. Can you write an general expression that represents a solution of this form?

 

[coffeem]

No. Sorry I don't know how to do this can you explain this to me? thanks

 

[dx]

This is a very straight-forward question, and you either know how to do it or you don't. If you've never seen a question like this before, just read a textbook.