System A consists of a mass m attached to a spring with a force constant k; system B has a mass 2m attached to a spring with a force constant k; system C has a mass 3m attached to a spring with a force constant 6k; and system D has a mass m attached to a spring with a force constant 4k. Rank these systems in order of decreasing period of oscillation.

    T₂ > T₁ > T₃ >T₄

Explanation:

In a simple harmonic motion the angular velocity is

         w =

angular velocity and period are related

         w = 2π / T

we substitute

         T =

let's find the period for each case

a) mass m

   spring constant k

          T₁ = 2π

b) mass 2m

   spring constant k

          T₂ = 2π

          T₂ = T₁ √2

          T₂ = T₁ 1.41

c) masses 3m

   spring constant 6k

          T₃ = 2π

          T₃ = 2π

          T₃ = T₁ 0.707

d) mass m

    spring constant 4k

          T₄ = 2π

          T₄ = 2π

          T₄ = T₁ 0.5

now let's order the periods in decreasing order

           T₂ > T₁ > T₃ >T₄