Explanation:
A) To prove the motion of the center of mass of the cylinders is simple harmonic:
System diagram for given situation is shown in attached Fig. 1
We can prove the motion of the center of mass of the cylinders is simple harmonic if
 
where aₓ is acceleration when attached cylinders move in horizontal direction:
<h3>PROOF:</h3>
rotational inertia for cylinders  is given as:
                                    -----(1)
 -----(1)
Newton's second law for angular motion is:
                                              ∑τ = Iα ------(2)
For linear motion in horizontal direction it is:
                                              ∑Fₓ = Maₓ ------ (3)
By definition of torque:
                                                τ  = RF --------(4)        
Put (4) and (1) in (2)
                                         
 
                                         
 
from Fig 3 it can be seen that fs is force by which the cylinders roll without slipping as they oscillate
So above equation becomes 
                                     ------ (5)
------ (5)
As angular acceleration is related to linear by:
                                           
Eq (5) becomes 
                                      ---- (6)
---- (6)
aₓ shows displacement in horizontal direction
From (3)
                                               ∑Fₓ = Maₓ
 Fₓ is sum of fs and restoring force that spring exerts:
                                    ----(7)
 ----(7) 
Put (7) in (3)
                                    [/tex] -----(8)
[/tex] -----(8)
Using (6) in (8)
                                
                                       --- (9)
 --- (9)
For spring mass system
                                    ----- (10)
 ----- (10)
Equating (9) and (10)
                                    
 

then (9) becomes
                                 
(The minus sign says that x and  aₓ  have opposite directions as shown in fig 3)
This proves that the motion of the center of mass of the cylinders is simple harmonic.
<h3 /><h3>B) Time Period</h3>
Time period is related to angular frequency as:
                                    
                                   