A spring$-$mass$-$damper transducer is to be used to measure the absolute horizontal displacements of the top story of a building, which has a steady$-$state sinusoidal motion at a period of $2\mathrm{.}0$ seconds. The undamped natural period of the transducer is $1\mathrm{.}0$ sec$.,$ and the damping is $50\%$ of critical. There is a magnifying lever system in the instrument which gives a scale reading of $10$ times the relative displacement between the transducer mass and the support. If when installed in the building, the maximum scale reading is $0\mathrm{.}50$ cm$,$ determine the amplitude of the displacement of the top story in cm