$1.$ A spring$-$mass system $($with a mass of $10$ kg $,$ a spring of $k=2000\frac{N}{m})$ has its mass

subjected to a harmonic force of amplitude of $180$ N and a frequency of $3\mathrm{.}5$ Hz $.$ If the

initial displacement and velocity are $15$ mm and $5\frac{m}{s},$ find the complete solution

representing the motion of the mass.

$2.$ A torsional system is composed of a disk with mass moment of inertia of $6kg-m^2,$ a shaft

having a torsional spring constant $k_t=14000N-\frac{m}{r}ad,$ and a torsional damper of constant

$c_t=210N-m-\frac{s}{r}ad.$ A harmonic torque of amplitude $450N-m$ produces a steady angular

oscillation of amplitude of $2$ :

i$.$ Find the frequency of the applied torque

ii$.$ Determine the maximum torque transmitted to the support.