It is not clear whether the total weight of all washers is 5m or 10m.   I assume the latter so each washer weighs m.

When N washers are transfered from one side to the other, the difference in weight between the two sides becomes 2N * mg.   This force creates the acceleration that all ten washers (mass = 10m) will experience.

To calculate the acceleration based on time and distance use d = 1/2(a)(t)^2.   a = 2d/t^2.

To calculate the acceleration based on the weight difference.     a = 2*N*mg/10m.

Set these two expressions for a to be equal.   2(.706)/(0.6^2) = 2*N*9.8/10

N = (.706/.36)* 10/9.8 = 2 Washers transfered.