Spring with two masses

Question

Two masses of mass m are connected by a spring with spring constant k (diagram below).
a) What is the resonant frequency of the system in terms of m and k?
b) If the masses are each 1AMU (atomic mass units) and the frequency is 125THz, what is the
spring constant? 

Answer 1

The formula for the angular frequency ( ω )  is   ω =  sqrt( k / μ )  where μ is the reduced mass.  

 In general, the reduced mass is product / sum, so for the equal mass case  μ = m/2.   

  The linear frequency, f,  is  f =  ω /(2 pi).   Putting this all together

   f =  sqrt( 2 k/ m) / (2 pi)

 

    125 THz =  1.25 x 10¹⁴  Hz.      1 AMU =  1.66 x 10^-27 kg.    So

 

      1.25 x 10¹⁴   =   sqrt( 2 k / 1.66 x 10^-27) / 6.28     Solving for k gives

   k  =  13 N/m.        The unit of k is Newtons per meter.