An automobile, weighing $1000kg$ empty and $4000kg$ fully loaded, vibrates in a vertical

direction while traveling at $72k\frac{m}{h}r$ on a rough road having a sinusoidal waveform with an

amplitude $12cm$ and a wavelength $5m.$ Assuming that the automobile can be modeled as a

single$-$degree$-$of$-$freedom system with stiffness $400\frac{N}{m}m$ and damping coefficient is

$8000N.\frac{s}{m}.$ Determine the amplitude of vibration of the automobile when it is:

a$)$ Empty.

b$)$ Fully loaded.

Neglecting damping $($consider $=0),$ when it is empty Find:

c$)$ The range of speeds which produce amplitude more than $16cm.$

Neglecting damping $($consider $=0),$ when it is fully loaded Find:

d$)$ The suitable speed range to keep the amplitude less than $9\mathrm{.}6cm.$