text: # Two masses connected by spring, find period of oscillation
  
  1. Feb 2, 2013
  
  ### Aziza
  
  1. The problem statement, all variables and given/known data
  
  Two masses are connected by spring and slide freely without friction along horizontal track. What is period of oscillation?
  
  2. Relevant equations
  
  3. 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...
  
  Last edited by a moderator: Feb 3, 2013
  2. Feb 3, 2013
  
  ### vela
  
  Staff Emeritus
  You should be well aware that $\frac{d^2}{dt^2}$ isn't a fraction.
  
  3. Feb 3, 2013
  
  ### bigerst
  
  I dont 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.
  
  4. Feb 3, 2013
  
  ### vela
  
  Staff Emeritus
  Do you know about reduced mass and how to convert a two-body problem into a one-body problem?