The amplitude of a damped oscillator becomes ( 1 3 )^ rd of original amplitude in 2 seconds. If its amplitude after 6 second become ( 1 n ) times the original amplitude, the value of n is ( n is non zero integer)

Step-by-step solution

The amplitude of a damped oscillator follows A(t) = A₀ e^ -bt , where A₀ is the initial amplitude and b is the damping constant. Given that A(2) = A₀ 3 , substitution yields A₀ 3 = A₀ e^ -2b , which simplifies to 1 3 = e^ -2b . After 6 seconds, A(6) = A₀ n = A₀ e^ -6b , giving 1 n = e^ -6b . Since e^ -6b = (e^ -2b )^3 = ( 1 3 )^3 = 1 27 , it follows that 1 n = 1 27 . Thus, n = 27 .