Two spring-mass systems, system 1 and system 2, are oscillating vertically. System 1 has a mass m attached to a spring of spring constant k . System 2 has a mass 2m attached to a spring of spring constant 8k . If both systems oscillate such that their maximum accelerations are equal, the ratio of their total mechanical energies E₁ : E₂ will be :

Step-by-step solution

The angular frequency of a spring-mass system is given by = k m . For system 1: ₁^2 = k m For system 2: ₂^2 = 8k 2m = 4k m = 4 ₁^2 The maximum acceleration in simple harmonic motion is a_ max = ^2 A . Given that a_ max1 = a_ max2 : ₁^2 A₁ = ₂^2 A₂ Substituting the values of ^2 : ( k m ) A₁ = ( 4k m ) A₂ This gives A₁ = 4A₂ . The total mechanical energy of a spring-mass system is E = 1 2 kA^2 . The ratio of their total mechanical energies is: E₁ E₂ = 1 2 k₁ A₁^2 1 2 k₂ A₂^2 E₁ E₂ = ( k 8k ) ( 4A₂ A₂ )^2 E₁ E₂ = 1 8