A cantilevered column, having $E=200$GPa and $I=300({10}^6)m{m}^4,$ is fixed at its base. The length

of the column is $L=5m.$ At the top of the column, a lumped mass is $m=5k\frac{N}{m}\frac{?}{s^2}.$ The gravity

acceleration is $g=9\mathrm{.}81\frac{m}{s^2}.$ Damping ratio is $=0\mathrm{.}05.$

$(1)$ Calculate the natural period for the lateral vibration of the mass. $(4$ marks$)$

$(2)$ Assume a horizontal ground motion acceleration history $u_g^{}(t)=5sin(20t)\frac{m}{s^2},$ which lasting

$6\mathrm{.}28$ seconds. Calculate the peak lateral displacement of the mass with respect to the column

base or the ground. $(6$ marks$)$