For the damped oscillator, the mass m of the block is `200g, k = 90 Nm^(-1)` and the damping constant b is `40 gs^(-1)`. Calculate time taken for its amplitude of vibrations to drop to half of its initial value.

Explanation

`x(t) = Ae^(-(bt)/(2m)) cos (omega. t+ phi)`
`t= 0 implies cos (omega. t+ phi)= 1`
`therefore x(t) = Ae^(-(bt)/(2m))`
`therefore (A)/(2)= Ae^((-bt)_((1)/(2))/(2m))`
`therefore 2= e^((bt)_((1)/(2))/(2m))`
`therefore ln2= (bt)_(1/2)/(2m)`
`(0.693 xx 2m)/(b)= t_(1/2)`
`(0.693xx 2xx 200)/(40)= t_(1/2)`
`therefore t_(1/2) = 6.93s`.