$(10$ pts$./$part$)$ A shock absorber in an automobile is essentially turns the

car into a damped oscillating system. $($a$)$ Examine the motion of a car

of mass $m=1500kg,$ whose suspension has an effective spring constant

of $k=6000\frac{N}{m}.$ If the shock absorber is filled with a viscous oil that

has a drag coefficient of $b=6000k\frac{g}{s},$ write and solve the equations of

motion for this system using the following initial conditions: the system

is initally displaced by $10$ cm at $t=0,$ and is released from rest. Plot the

motion of the system in a phase plot. $($b$)$ The oil in the shock absorber

leaks from the system, reducing the drag coefficient to $b=3000k\frac{g}{s}.$ As

before, write and solve the equations of motion for the system using the

same initial conditions, but the new drag coefficient. Then plot the phase

plot for the motion of this system.