Problem $1$: In the mechanical delay switch shown in Figure $1,$an electromagnet releases the $1$kg $$slider at position $1.$Under the action of gravity and the linear spring, the slider moves along the smooth bar from position $1$to position $2,$closing the switch. The constant of the spring is k$=40$Nm$,$and its unstretched length is l$=50$mm$.$The dimensions are R$=200$mm $$and h$=100$mm$.$

a$)$Obtain the expression of velocity of the slider as a function of at position $1$and $\backslash$theta $=\backslash$pi $$at position $2).$

b$)$Determine the magnitude of the slider's maximum velocity analytically and $\backslash$theta $$at that instant.