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 the time taken for its mechanical energy to drop to half its initial value.

Explanation

Mechanical energy of a damped oscillatior at time t
`therefore E(t)=E_(0)e^((-bt)/(m)" where "E_(0)= (1)/(2) kA^(2)`
If `E(t)= (E_0)/(2) implies t= t_(1/2)`
`therefore (E_0)/(2)=E_(0)e^(((-bt)_(1/2))/(m))`
`therefore 2=e^(((-bt)_(1/2))/(m))`
`ln_(2)= (bt)_(1/2)/(m)`
`ln_(2)=(ln2xxm)/(b)`
`= (0.693xx 200)/(40)`
`therefore t_(1/2) = 3.465s`.