A block of mass 0 . 1kg is connected to an elastic spring of spring...

Given,

$E = \dfrac{Eo}{2}$
$⇒ A = \dfrac{Ao}{2}$
Amplitudeofdampedoscillation = A = {A}_{o}e^{- \dfrac{bt}{2m}}$<br /><br />$⇒ \dfrac{{A}_{o}}{\sqrt{2}} = {A}_{o}e^{- \dfrac{bt}{2m}}$~<br /><br />$\sqrt{2} = e^{\dfrac{bt}{2m}}$<br /><br />$\ln \left(\sqrt{2}\right) = \dfrac{bt}{2m} = \dfrac{{10}^{- 2}}{2 \times 0.1}t =\dfrac{0.1t}{2}$<br /><br />$\dfrac{t}{2} =10\ln \left(\sqrt{2}\right) = 10 \times \dfrac{1}{2} \times 0.693$<br /><br />$t = 10 \times 0.693 = 6.93$<br /><br />$t \approx 7s$.