Homework H$\mathrm{.}5.$K

Given: A homogeneous disk $($with mass $M$ and outer radius of $R)$ is placed on a rough surface. When placed on this surface, the center of the disk A is moving to the right with a speed of $v_{A1}$ and has a counterclockwise rotation rate of ${}_1,$ as shown in Figure $1$ below. In Figure $2$ below is shown the instant at which the disk ceases to slip as it continues to move on the horizontal surface.

Find: For this problem:

$($a$)$ Determine the speed of $A,v_{A2},$ at the instant in Figure $2$ when the disk ceases to slip on the horizontal surface; and

$($b$)$ Determine the elapsed time during the motion as the disk moves from the position in Figure $1$ to the position in Figure $2.$

Use the following parameters in your analysis: $M=50kg,R=0\mathrm{.}5m,{}_k=0\mathrm{.}3,v_{A1}=5\frac{m}{s}$ and ${}_1=8ra\frac{d}{s}.$