(Solved): Equations of Motion: Cylindrical Coordinates Part A - The angular velocity for a smooth shaft Lear ...

Equations of Motion: Cylindrical Coordinates Part A - The angular velocity for a smooth shaft Learning Goal: If there is no friction between shaft $AB$ and the collar, what angular velocity $\stackrel{?}{?}$ must the mechanism have to keep the collar at $r=1320\mathrm{mm}$ from the vertical axis? To set up and analyze equations of motion in a cylindrical coordinate Express your answer to three significant figures. The mechanism shown in the figure below rotates about the vertical axis. The collar has mass $m=2.55\mathrm{kg}$. The spring has an unstretched length of $1185\mathrm{mm}$ and the spring constant is $k=160\mathrm{N}/\mathrm{m}$. The distance $d=385\mathrm{mm}$, and the collar is required to stay a fixed distance $r=1320\mathrm{mm}$ from the vertical axis.( Figure 1 of $1>$ Part $\mathbf{B}$ - The minimum required angular velocity when there is friction Consider the same mechanism again, with $m=2.55\mathrm{kg},d=385\mathrm{mm},k=160\mathrm{N}/\mathrm{m}$, only now, instead of being smooth, the collar and shaft have a maximum coefficient of friction of ${?}_s=0.615$. What is the minimum angular velocity required to keep the collar at a constant distance $r=1320\mathrm{mm}$ from the axis of rotation? Express your answer to three significant figures.