Q: The amplitude of a damped oscillator decreases to 0.9 times its original value in 5 s. In another 10 s it will decreases to α times its original magnitude, where α is

Sol: $\large A = A_0 e^{-\frac{b t}{2m}}$

after 5 sec ,

$\large 0.9 A_0 = A_0 e^{-\frac{5 b}{2m}}$

$\large 0.9 = e^{-\frac{5 b}{2m}}$

After 10 more sec (i.e., t = 15 sec) its amplitude becomes α A_{0}.

$\large \alpha A_0 = A_0 e^{-\frac{15 b}{2m}}$

$\large \alpha = e^{-\frac{15 b}{2m}}$

$\large \alpha = (e^{-\frac{5 b}{2m}})^3 = (0.9)^3$

= 0.729