The amplitude of a damped oscillator is known to decrease to 0.9 times its original amplitude in 5 ~s . Approximately, by how many times its original amplitude will decrease after another 20 ~s ?

Step-by-step solution

Initial amplitude, A_i=A₀ Final amplitude A_f=0.9 A₀ Case 1 Time t₁=5 ~s Case 2 Time t₂=20 ~s Let, amplitude after 20 ~s be A^ . As we know that, aligned & A=A₀ e^ - t & 0.9 A₀=A₀ e^ -5 & e^ -5 =0.9 & A^ =A₀ e^ -20 & aligned Again, A^ =A₀ e^ -20 Now on powering Eq. (i) by 4 , we get aligned & (e^ -5 )^4=(0.9)^4 & e^ -20 =(0.9)^4 & From Eq. (ii), A^ =A₀ e^ -20 =A₀(0.9)^4 & =0.6 A₀ 0.59 A₀ & aligned