Equation of motion of coupled springs

[Lucy Yeats]

Homework Statement

A system is connected as follows, going vertically downwards: (ceiling)-(spring with constant k)-(mass 1)- spring with constant k)-(mass 2)
Let x be the displacement from the equilibrium position of mass 1, and let y be the displacement from the equilibrium position of mass 2. Take downwards displacement as positive.

I'm trying to show that the angular frequencies of the normal modes are ω^2=(3±5)k/2m, but I'm stuck.

m(d^2x/dt^2)=mg-kx
m(d^2y/dt^2)=mg-k(y-x)

When I try to put this into matrix form, I can't get rid of the mg terms.

Homework Equations

The Attempt at a Solution

 

[vela]

Homework Statement

A system is connected as follows, going vertically downwards: (ceiling)-(spring with constant k)-(mass 1)- spring with constant k)-(mass 2)
Let x be the displacement from the equilibrium position of mass 1, and let y be the displacement from the equilibrium position of mass 2. Take downwards displacement as positive.

I'm trying to show that the angular frequencies of the normal modes are ω^2=(3±5)k/2m, but I'm stuck.

m(d^2x/dt^2)=mg-kx
m(d^2y/dt^2)=mg-k(y-x)

When I try to put this into matrix form, I can't get rid of the mg terms.

Homework Equations

The Attempt at a Solution

Note the bolded phrases. The mg terms shouldn't be there in the first place.

By the way, I've moved this thread to the advanced physics forum since it looks like a problem from an upper-division classical mechanics course or math methods course.

 

[Lucy Yeats]

So I don't need the mg terms because they only affect the equilibrium position?
So if I cross out those terms will the equations be right?

 

[vela]

Right. When x=0, mass 1 is at its equilibrium position, so the net force on it is equal to 0. The same holds for mass 2 and (y-x)=0.