$(2)$ The lower spring in the previous problem is replaced by a viscous damper with damping constant $c$ and the stiffess of the upper spring is $k_1=k.$ The equation of motion is then:

$$ equivalens

speng

$($a$)$ Let $L=1m,M=3kg,k=14\mathrm{.}28\frac{N}{m},c=4N\frac{s}{m}$ and $g=1\mathrm{.}62\frac{m}{s^2}($surface gravity on the Moon$).$ Determine the natural frequency ${}_n,$ the frequency of damped oscillations ${}_d,$ and the damping ratio $.$ Leave your answers in terms of radicals such as $\sqrt[\phantom{2}]{2},\sqrt[\phantom{2}]{6},$ etc.

$(15$ marks$)$

$($b$)$ The bar is given an initial counterclockwise angular velocity ${}_0^{}=1ra\frac{d}{s}$ at the vertical position. Hence, determine the angular displacement $(f)$ completely.

$(20$ marks$)$

Hint: The equation of motion shown above has one of the following three solutions:

$(t)=C_1{e}^{n\text{'}}+C_2{e}^{?}$

$()\text{'}$