Two masses connected by spring, find period of oscillation

[Aziza]

Homework Statement

Two masses are connected by spring and slide freely without friction along horizontal track. What is period of oscillation?

Homework Equations

The Attempt at a Solution

My solution:
let x1 be position of mass 1 (m1) and x2 be position of mass 2 (m2) and L be length of spring in equilibrium.
Then, the total stretch of the spring is x2-x1-L. Also, F1 = -F2. Thus:

m1(x1)'' = -k(x1 - x2 + L)
m2(x2)'' = -k(x2 - x1 - L)

Solving for x2 from first eqn and substituting back into second eqn yield:

\frac{d^2}{dt^2}[\frac{m1 m2}{k}(x1)''+(m1+m2)x1] = 0

I am unsure how to proceed from here, any hints? I would like to just multiply both sides by (dt^2)/d^2 but I am unsure if this is mathematically correct? It does simplify the problem though and gives me right answer...

 

[vela]

You should be well aware that $\frac{d^2}{dt^2}$ isn't a fraction.

 

[bigerst]

I don't think you really need differential equations for this one, since there is no external force there is a point on the spring that neither stretches nor compresses. on either side of this point is 2 springs with different spring constants, so effectively you have two different oscillators, with the same period.

 

[vela]

Do you know about reduced mass and how to convert a two-body problem into a one-body problem?