$5\mathrm{.}7$ The concrete foundation $($unit weight $=23\mathrm{.}5k\frac{N}{m^3})$ of a machine has the following dimensions $($refer to Figure $5\mathrm{.}18)$: $L=3m,B=4m,$ height of the foundation $=1\mathrm{.}5m.$ The foundation is subjected to a sinusoidal horizontal force from the machine having an amplitude of $10kN$ at a height of $2m$ measured from the base of the foundation. The soil supporting the foundation is sandy clay. Given $G=30,000$kPa, $=0\mathrm{.}2,$ and $=1700k\frac{g}{m^3}.$ Determine

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

b$.$ the amplitude of rocking vibration at resonance.

$($Note: The amplitude of horizontal force is not frequency dependent. Neglect the moment of inertia of the machine.$)$