Show me the steps to solve these problems:

A piston I 'can move along the $x$ axis of a cylinder. The piston rod $($i$.$e$.$ the segment $A{P}^{\text{'}}$ of length $b)$ connects the piston to the border of a disk of radius $R.$ The disk spins with constant angular velocity $.$ At the time $t=0$ you see $(t=0)=0.$

P$2-1$: Determine the position, velocity and acceleration of $A$ as a function of $t.$

P$2-2$: Determine the position of $P$ as a function of $$

P$2-3$: Without any calculation, at what values of $$ do you expect the velocity of $P$ to be vanishing? Also state whether at each of these values of $$ you expect the acceleration of P $($along hat$(x))$ to positive, null or negative. Explain why.

P$2-4$: Compute the velocity of I $\text{'}$ at a generic time $t,$ and verify that your result matches what you expected in P$\text{'}2-3.$

P$2-5$: Compute the velocity of A and P when $t=9s$ if $=(\frac{}{2})s^{-1},b=30cm$ and $R=1m.$