Normal Modes and Frequencies of Coupled Oscillators?

[philnow]

Homework Statement

Two identical undamped oscillators are coupled in such a way that the coupling force exerted on oscillator A is \alpha\frac{d^2x_a}{dt^2} and the coupling force exerted on oscillator B is \alpha\frac{d^2x_b}{dt^2} where \alpha is a coupling constant with magnitude less than 1. Describe the normal modes of the coupled system and find their frequencies.

The Attempt at a Solution

I know this isn't much of an attempt, but I've searched online and in the text... what am I supposed to do with this coupling constant?

 

[vela]

Start by writing the equation of motion for both oscillators.

 

[mathman44]

That's where I'm stuck...

m\frac{d^2x_a}{dt^2}=\alpha\frac{d^2x_a}{dt^2}
m\frac{d^2x_b}{dt^2}=\alpha\frac{d^2x_b}{dt^2}

?

 

[mathman44]

I'd be glad to show more work if I knew what to do with this coupling constant!

 

[vela]

That's where I'm stuck...

m\frac{d^2x_a}{dt^2}=\alpha\frac{d^2x_a}{dt^2}
m\frac{d^2x_b}{dt^2}=\alpha\frac{d^2x_b}{dt^2}

?

Those equations say the only force on the masses is the coupling force. What about the restoring force?

 

[mathman44]

m\frac{d^2x_a}{dt^2}=\alpha\frac{d^2x_a}{dt^2} - k(x_a)
m\frac{d^2x_b}{dt^2}=\alpha\frac{d^2x_b}{dt^2} - k(x_b)