Additional problem $4$

A mass $m$ is attached to a wall with a spring of stiff$-$

ness $k_1.$ The mass is free to slide on a surface with a

dynamic coefficient of friction $($ignore static fric$-$

tion$).$ The mass is attached to a pulley with mass,

$M,$ and outer radius $r$ at a radius of $\frac{1}{2}r.$ This pulley

is attached to two springs with stiffness, $k_2$ and $k_3,$

that are in parallel.

$($a$)$ How many degrees of freedom does the system

have? Motivate your answer.

$($b$)$ Determine the equation of motion of the sys$-$

tem in terms of the angle of rotation of the

pulley, $.$

$($c$)$ Find the angular displacement of the rod,

$(t),$ for the initial conditions: $(0)=\frac{}{10}$ and

${}^{}=0.$ Assume that $m_1=1kg,m_2=2$

$kg,M=0\mathrm{.}5kg,m=2kg,k_1=200\frac{N}{m},$

$k_2=k_3=30\frac{N}{m},=0\mathrm{.}4$ en $r=0\mathrm{.}25m.$

$($d$)$ Draw a plot of the system response to deter$-$

mine how long the vibration will take to stop.

$($e$)$ Now assume that the mass is able to slide on

the surface without friction and use the en$-$

ergy method to determine the system's equa$-$

tion of motion and the natural frequency.