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.

For the damped oscillator, the mass m of the block is 200g, k = 90 Nm^

1.	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.
Answer» Solution :`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`.