$5\mathrm{.}11$ A concrete foundation $($unit weight $=23\mathrm{.}5k\frac{N}{m^3})$ supporting a machine has the following dimensions: length $=5m,$ width $=4m,$ height $=2m.$ The machine impart a torque $T$ on the foundation such that $T=T_0{e}^{\text{i}\text{a}\text{r}}.$ Given $T_0=3000Nm.$ The mass moment of inertia of the machine about the vertical axis passing through the center of gravity of the foundation is $75{10}^{3}kg-m^2.$ The soil has the following properties: $=0\mathrm{.}25,$ unit weight $=18k\frac{N}{m^3},$ and $G=28,000$kPa. Determine

a$.$ the resonant frequency for the torsional mode of vibration, and

b$.$ angular deflection at resonance.