The time taken for the amplitude of vibrations of a damped oscillator to drop to 25 % of its initial value is t . The time taken for its mechanical energy to drop to 50 % of its initial mechanical energy is

Step-by-step solution

aligned & A=A₀ e^ - b t 2 m & A=25 % of A₀= A₀ 4 & A₀ 4 =A₀ e^ - b t 2 m & (2)⁻²=e^ - b t 2 m & b t 2 m =2 2 t= 42 m n 2 b ... (1) & E(t)=E(0) e^ - b t 2 m & E(t)=50 % of E₀ & E₀ 2 =E₀ e^ - b t^ 2 m aligned aligned & (2)⁻¹=e^ - b t^ 2 m & b t^ 2 m = 2 & t^ = 2 m 2 b aligned Divide equation (2) by (1), we have t^ t = 2 m 2 b b 4 m 2 t^ = t 2