Forces of Constraints: Solved Example in Gregory's Book

[sayf alawneh]

hello :)
my question is about a solved example in gregory's book
first of all , assuming that that a cylinder with radius "a" is rolling without slipping towrad the y direction in an incline plane
with angle θ with the horizon and the cylinder him self is rotating with angle Φ about him self then why the equation of constraint became
y=aθ why its not y=aΦ ?? does that mean θ=Φ for the rolling condition ??

also
another example about the constraints is for the same cylinder but this time heis moving horizontally (i uploaded a screen shot for the solution)
how could he get those equations am not looking for answers here but i can't imagine the movement of this cylinder and the relation between θ and Φ
i don't have any attempts because am not even able to imagine the movement of the cylinder , sorry :(

 

[drvrm]

nother example about the constraints is for the same cylinder but this time heis moving horizontally (i uploaded a screen shot for the solution)

the screen shot talks about a disc rolling on a horizontal surface? how come you say it to be a cylinder.
Moreover in first part -you should post a diagram with these two angles defined ...your work out must be posted.