A certain resonance curve describes a mechanical oscillating system with logarithmic damping decrement `lambda=1.60`. For this curve find the ration of the maximum displacemetn amplitude to the displacement amplitude at a very low frequency.

Explanation

In general for displacement amplitude
`a=(F_(0))/( m)(1)/(sqrt(( omega_(0)^(2)-omega^(2))^(2)+4 beta^(2) omega^(2)))`
`=(F_(0))/(m)(1)/(sqrt((omega^(2)-omega_(0)^(2)+2 beta^(2))^(2)+ 4 beta^(2)( omega_(0)^(2)- beta^(2))))`
Thus `eta=(a_(res))/(a_(low))=( omega_(0)^(2))/(sqrt(4 beta^(2)( omega_(0)^(2)-beta^(2)))). = (omega_(0)^(2))/( 2 betasqrt(omega_(0)^(2)-beta^(2)))`
But `( beta)/(omega_(0))=( lambda//2 pi)/( sqrt(1+ ( lambda//2 pi )^(2))), ( lambda)/( 2 pi )= ( beta)/( sqrt(omega_(0)^(2)- beta^(2)))`
Hence `eta=( omega_(0)^(2))/( 2 beta^(2)). ( lambda)/( 2pi )=(1)/(2)(1+((lambda)/( 2pi))^(2))/( (lambda)/(2pi))=2.90`